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LONGMANS MODERN MATHEMATICAL SERIES
General Editors
P. ABBOTT, B.A.
F. S. MACAULAY, M.A,, D.Sc.
EXERCISES IN ALGEBRA
(a + b) 2 - a tj + 2ab + b' J .
a negative, b positive ; a numerically less than b.
(a + b)(a + b) = a a + ba
+ ab + b 2
- a' 2 + 2ab + b 2 .
w + ^^^^
ail Fa
ra+b) 2 = (a^-b)a -I-
-b)a =
l
ba
.. . , 1
(a^b) 2
[See Kx. XXXI, p. 183.]
Xotiflmang' flDo&ern fl&atbematical Series
EXERCISES IN ALGEBRA
(INCLUDING TRIGONOMETRY)
BY
T. PERCY NUNN, M.A., D.Sc.
VICE-PRINCIPAL OP THE L.C.C. LONDON DAY TRAINING COLLEGE (UNIVERSITY, CP.
LONDON); FORMERLY SENIOR MATHEMATICAL AND SCIENCE MASTER
WILLIAM ELLIS SCHOOL
PROFESSOR OF EDUCATION IN THE UNIVERSITY OF LONDOK
PART 1.
WITH DIAGRAMS
NEW IMPRESSION (1923)
KEISS UE
LONGMANS, GREE^ AND CO.
39 PATERNOSTER ROW, LONDON, E.G. 4
NEW YORK, TORONTO
BOMBAY, CALCUTTA/ AI?D MADRAS
Made in Qreat Britain
NOTE.
Tips exercises in this volume^ are intended to supply
the materials for a course in Algebra to be completed
at about the age of sixteen. In addition to the subjects
usually required in the Entrance Examinations of the
Universities it contains exercises upon logarithms and
the elementary parts of trigonometry, and a simple
introduction to the methods of the differential and
integral calculus.
A discussion of the principles upon which the author
has selected and presented the subject-matter of the
exercises will be found in the companion volume on
"The Teaching of Algebra". That volume contains
also suggestions as to the order in which the exercises
should be taken, and a full statement of the preparatory
teaching pre-supposed in each. It includes a similar
treatment of the ground covered in " Exercises in Al-
gebra," Part II, in which the course commenced in
the present book is completed.
The author will be greatly obliged if readers who
detect errors or obscurities in the examples or the
answers will kindly direct his attention to them.
LONDON DAY TRAINING COLLEGE
(UNIVERSITY OF LONDON),
July, 1913,
CONTENTS.
SECTION . I.
NON-DIRECTED NUMBERS.
KXERCISE PAGE
I. THE SHORTHAND " OF ALGEBRA .... 1
II. GRAPHIC REPRESENTATION 4
III. THE WRITING OF FORMULAS 13
IV. THE READING AND USE OF FORMULAE ... 23
Y. FACTORIZATION (I) .... 32
VI. FACTORIZATION (II) 89
VII. SQUARE ROOT 44
VIII. " SURDS" 46
IX. APPROXIMATION-FORMULA (I) 48
X. APPROXIMATION-FORMULAE (II) 53
XI. APPROXIMATION-FORMULA (III) 55
XII. FRACTIONS (I) 57
XIII. FRACTIONS (II) 63
XIV. CHANGING THE SUBJECT OF A FORMULA (I) . . 70
XV. CHANGING THE SUBJECT OF A FORMULA (II) . . 74
XVI. SUPPLEMENTARY EXAMPLES .... 81
A. Formulation (p. 81) ; B. Substitution (p. 84) ;
0. Some Arithmetical Puzzles (p. 87) ; P
Graphic Representation (p. 90) ; E. F-
ization, etc. (p. 93) ; F. Approximate
G. Changing the Subject, etc. (p
vii
viil CONTENTS
EKEROISB PAGK
XVII. DIRECT PROPORTION , 103
XVIII. THE USE OP THE TANGENT-TABLE .... 108
XIX. THE USE OF THE SINE- AND COSINE-TABLES . . 112
XX. SOME NAVIGATION PROBLEMS 118
XXI. RELATION OP SINE, COSINE, AND TANGENT . . 120
XXII. LINEAR RELATIONS 122
XXfll. INVERSE PROPORTION 126
XXIV. PROPORTION TO SQUARES AND CUBES . . . 129
XXV. JOINT VARIATION ... .... 132
XXVI. SUPPLEMENTARY EXAMPLES ..... 135
A. Teat Paper 1 (p. 135) ; B. Test Paper 2 (p. 136) ;
0. Test Paper 3 (p. 138) ; D. Statistics
(p. 140) ; B. Surds (p. 144) ; F. Test Paper 4
(p. 146) ; G. Test Paper 5 (p. 148) ; H. Tost
Paper 6 (p. 150).
SECTION II.
DIRECTED NUMBERS.
XXVII. THE Usu OF DIRECTED NUMBERS . . . .155
XXVIII. ALGEBRAIC ADDITION AND SUBTRACTION . . .161
XXIX. DIRECTED PRODUCTS 168
XXX. SUMMATION OF ARITHMETIC SERIES .... 175
XXXI. ALGEBRAIC MULTIPLICATION 183
XXXII. THE INDEX NOTATION 187
XXXIII. NEGATIVE INDICES , 191
yXXIV. FACTORIZATION 193
Tr . ALGEBRAIC DIVISION 196
^METRIC SERIES ......... 201
NUMBER-SCALE 208
CONTENTS Ix
RXKROISB TAGK
XXXVIII. FURTHER EXAMPLES ON DIRECTED NUMBERS . . 213
XXXIX. LINEAR FUNCTIONS 217
XL. DIRECTED TRIGONOMETRICAL Rvrios .... 221
XLI. SURVEYING PROBLEMS 221
XLII. HYPERBOLIC AND PARABOLIC FUNCTIONS . . .229
XLIII. QUADRATIC EQUATIONS 235
XLIV. FURTHER EQUATIONS 230
XLV. INVERSE PARABOLIC FUNCTIONS (I) . . . . 242
XL VI. INVERSE PARABOLIC FUNCTIONS (II) .... 247
XLVII. ABEA FUNCTIONS 250
XLVIII. DIFFERENTIAL FORMULA 255
XLIX. GRADIENTS 253
L. THE CALCULATION OF v AND THE SINE-TABLE . 264
SECTION III.
LOGARITHMS.
LI. GROWTH FACTORS .,..,.. 269
LII. GROWTH PROBLEMS , 272
LIIL THE GUNTER SCALE r
LIV. LOGARITHMS AND ANTILOGARITHMS
LV. THE BASE OF LOGARITHMS ......
LVI. COMMON LOGARITHMS
LVII. THE USE OF TABLES OF LOGARITHMS
LVIII. THE LOGARITHMIC AND ANTILOGAKITHMIC FUN'
LIX. NOMINAL AND EFFECTIVE GROWTH-FACTORS
CONTENTS
BUPPLEMENTAEY EXERCISES.
BXEUCISB PAOB
LX. THE USE OF LOGARITHMS IN TRIGONOMETRY . . 803
LXI. POLAR CO-ORDINATES 308
LXII. SOME IMPORTANT TRIGONOMETRICAL IDENTITIES . 310
LXIII. THE PARABOLIC FUNCTION 317
LXIV. IMPLICIT QUADRATIC FUNCTIONS (I) . . . . 320
LXV. IMPLICIT QUADRATIC FUNCTIONS (II) . . . . 325
LXVL MEAN POSITION 331
LXVII. ROOT-MEAN-SQUARE DEVIATION 887
LXVIII. THE BINOMIAL THEOREM ...... 340
LXIX. THE GENERALIZATION OF WALLIS'S LAW . . . 849-
SECTION I.
NON-DIRECTED NUMBERS.
EXERCISE I.
THE SHOBTHAND " OF ALGEBBA.
Express in the " shorthand " of Algebra the rules mentioned
in Nos. 1-12.
1. The rule for calculating the area of an oblong floor when
you know its length and its breadth.
2. The rule for calculating the breadth of an oblong room
when you know its length and the area of the floor.
3. The rule for calculating the cost of a number of things
when you know the price of each. Also the rule for calculat-
ing the price of a single article when you know that a certain
number of them cost so much. (Let n^ u the number
bought," C="the total cost/' j9="the price of each
thing".)
4. The rule for calculating the cost (C) of a certain nmnber
of things (N) when you know how much (c) another number
of things (ri) costs.
5. The rule for finding the number of pence in a given
number of shillings. (Let 2? "the number of pence,"
s = " the number of shillings ''.)
6. The rule for reducing pence to shillings.
7. The rules for reducing pounds to shillings and! pounds
to pence. (Let L ~ " the number of pounds ''.)
8. The rules for reducing shillings to pounds and pance to
pounds.
9. The rule for calculating the total area of three oblang
rooms each of the same length and the same breadth. The
same rule written to suit any number (n) of rooms.
10. The rule for calculating the area of a square room.
(Let s ~ "the length of the side ".) The rule for the area of
a number of such rooms all of the same size.
11. The rule for calculating the volume of an oblong room
(i) given the area of the floor (A) and the height (h) ; (ii)
given the length, breadth, and height.
12. The rule for calculating the depth of water in an oblong
1
2 ALGEBRA
cistern (i) when you know the volume of water (V) and the
area of the bottom of the cistern ; (ii) when you know the
volume of water, the length and the breadth of the cistern.
13. A number of persons subscribed equal amounts to send
the children in a certain school away for a holiday. Write
a formula for the amount subscribed by each person, given
the number of children and the expense of sending each child
away. Use the symbols of No. 4.
14. How must the last formula be changed if the expense
per head is reckoned in shillings and the subscription in
pounds ?
15. The rule for making tea is : " One spoonful of tea for
each person and one for the pot ". Keduce this rule to a
formula letting = "the number of spoonfuls of tea," p~.
"the number of persons".
16. Write formulae for the rules used (i) in reducing
shillings and pence to pence; (ii) in reducing pounds and
shillings to shillings ; (iii) in reducing pounds and pence to
pence.
17. Give in a formula the Post Office rule for the cost of
an inland telegram. (Let = "the cost of the telegram in
pence, " n=" the number of words over twelve ".)
18. A, library charges an entrance fee of half a crown and
a subscription of twopence for every book borrowed. Write
a formula Ijor the total amount paid (in shillings) for borrow-
ing a given/ number of books.
19. Th-e following rule for cooking a joint of beef is often
given : " Allow a quarter of an hour for every pound and twenty
minutes over ". Express this rule in a formula for the time in
hours rieeded to cook a joint of a given weight.
20. I am twenty-seven years younger than my father.
Write in a formula the rule for finding my age when his age
is 'Known. (Let A = u my father's age," a= " my age ".)
21. My age is three years less than half my father's age.
Write the rule for finding my age, given my father's.
22. A second-hand bookseller bought a number of books
for 1 10s. and sold them at 9d. each. Write a formula
for his profit (in shillings) after selling a certain number of
them (ri).
23. A greengrocer buys a number of oranges at a certain
price per dozen and sells them at so much each, Write a
formula for his gain (G) after selling a certain number. (Let
EXERCISE I 3
P = "the price of a dozen/' p = "the price of a single orange,"
N^"the number of dozens bought," ft ^" the number of
single oranges sold".) How would you write the formula
if there was a loss (L) instead of a gain?
24. Write down as a formula the rule for finding the
simple interest for a given number of years on a given sum
of money at a given rate of interest per cent per antmm.
25. Change the foregoing formula into one for finding the
amount of the sum of money instead of merely the interest
on it.
EXEKCISE II.
GRAPHIC REPRESENTATION.
A.
1. Fig. 1 is a copy (J of the actual size) of the traces left by
two snails after wandering on a sheet of paper. The numbers
indicate their positions after 1, 2, 3, ... 9 minutes. Make a
table of the distances travelled by each snail during each
minute. Draw a diagram, in accordance with instructions,
FIG. 1.
showing by vertical lines the distances covered in each minute
by the snail which left the trace marked AB. Draw another
diagram showing the movements of the other snail.
2. Draw, in the way explained to you, a diagram or graph
by which the speeds of the two snails during any given
minute can be compared. Which had the greatest average
speed during (i) the first, (ii) the third, (iii) the fourth minute ?
Why is it necessary to say " average speed " ?
EXERCISE II
3. A cyclist travels along the same road in the same
direction on two occasions. On each occasion he keeps
account of the distances he rides in the first, second, third,
. . . hours of his journey. They are given in the following
table. Draw, as in No. 2, a graph by which his performances
in corresponding hours on the two occasions may be
compared :
Hour . .123
Miles ridden :
1st journey 11 8 9,
2nd journey 10 9 9.
4567
13 CJ' 10J 12
8 10| 11J llf
Fia. 2.
4. Fig. 2 is a series of pictures (J of the natural size) of a
growing tulip. The bulb began to germinate on a Monday
morning and the pictures were taken on successive Monday
mornings afterwards with the exception of the fourth which
was drawn a fortnight after the third. Draw, as instructed,
6 ALGEBRA
a diagram composed of vertical lines representing the total
length of the plant (from A to B) at each measurement.
Draw in its proper place a dotted line representing the length
which the plant probably had on the Monday when the
measurement was omitted. (Note. The process of supplying
in this way a missing observation is called interpolation.)
5. Draw a graph (like that of No. 1) showing the increase
in length of the plant in each week. Is there in this graph
anything which serves as a test of the correctness of the
interpolation in No. 4?
6. If during one of the hours in No. 3 the cyclist had neg-
lected to note the distance he rode, would it be safe to supply
the omission by interpolation ?
7. On September 27, 1911, a rod 6 inches high was
erected vertically upon a drawing-board and placed in full
sunshine. The following table gives the lengths of the
shadow of the rod at different times during the day. Draw
a horizontal " time-line " (as in No. 5) graduated in hours.
At the proper points draw perpendicular lines to represent
the various lengths of the shadow. Draw carefully the curve
which you would use for finding by interpolation the length
of the shadow at times not mentioned in the table.
Time .... 10.0 10.40 11.15 11.30 11.45
Length of shadow in ins. 8'60 779 7'35 7*25 7 '22
Time .... 12.0 12.20 1.0 2.15
Length of shadow in ins. 7'22 7*30 775 9'81
8. At what time on September 27, 1911, was the shadow
of an upright rod shortest ? (Note that it is not exactly at
12 o'clock. " Noon " coincides with 12 o'clock only very
rarely.) What would the length of the shadow have been
at (i) 11 o'clock, (ii) 2.30 p.m. ?
9. Divide the base line in No. 7 into half -hour intervals,
measuring right and left from the time of noon. Starting
with noon find how much the shadow lengthens during each
half-hour towards the evening. Make a graph of the half-
hourly increase as in Nos. 1 and 5, drawing in the smooth
curve which the measurements suggest.
10. Do the same with the half-hourly decrease in the
shadows from the morning on till noon. What facts do the
two graphs bring out?
EXERCISE II 7
B.
Note. In the graphs of Nos. 4, 5, 7, and 9, the most impor-
tant thing is the smooth curve. This curve shows how the
quantity rises or falls in size, and enables us by " inter-
polation " to find its probable magnitude at a time when we
did not actually measure it. The position of the curve could
be fixed without actually drawing the vertical lines if the
points where the ends of those lines would come were marked.
Moreover, the absence of the vertical lines would make the
graph clearer. In the following graphs tEe vertical lines are
not to be drawn. Mark the points where their ends would
come if they were drawn, and run the graph smoothly through
or among those points. The best way to tell whether you are
drawing the graph properly is to hold the paper horizontally
close to the eye so that you can look along the curve.
11. A wooden beam is supported horizontally on the edges
of two files and is loaded with weights at the point midway
between the supports. The following table shows how much
a given weight presses the middle of the beam down. Exhibit
the results of the observations in a graph. (How much would
the beam sink if no weight at all was placed on it ? What
observation, therefore, will you enter in the graph in addition
to those given below ?)
Weight in Ib. 2 4 6 8 10 12 14 16
Sag in inches 0-3 0-6 0-9 1-2 1*5 1-8 21 2*4
12. In No. 11 find the sagging produced in the beam by a
load of (i) 5 Ib., (ii) 11 Ib. Also find what load would make
the beam sag (iii) 4 inch, (iv) 2 inches.
13. A marble was allowed to roll down a smooth groove
cut in a sloping plank. The following table shows the dis-
tances it rolled in 1, 2, 3, ... seconds. Exhibit these
distances in a graph. Use the graph to find how far the
marble would roll in (i) 1-5 seconds, (ii) 4-3 seconds. Also
use it to find how long the marble would take to roll (iii) 50
cms., (iv) 100 cms. 1 (Eead the questions at the end of No.
no
Time in seconds 123456
Distance in cms. 8 32 72 128 200 288
1 This experiment was first performed by the great Galileo about
1638.
8 ALGEBRA
14. Draw a graph showing how far the marble rolled in
the first, second, third . . . seconds. What information
does the graph give with regard to the increase of the average
speed of the marble during each second ? Find by interpola-
tion how far the marble rolled (i) between 1*7 seconds and 2*7
seconds after starting; (ii) between 3*6 seconds and 4 -6
seconds after starting. Answer the same questions by means
of the graph of No. 13. Do the answers agree ?
Note. In the foregoing graphs the vertical lines have
always represented actual lengths. There is no reason why
they should not represent other quantities, for example, money.
(If you please you may think of the length of a vertical line
as representing the height of a pile of coins.)
15. A furnishing firm advertises the following rates at
which furniture may be bought by monthly payments. Draw
a graph representing the given rates. What monthly pay-
ments should secure furniture to the value of (i) 30, (ii) 40,
(iii) 85?
Value of furniture 10 20 50 75 100
Monthly payment 6s. 10s. 26s. 35s. 42s.
16. The table gives in degrees the height of the sun above
the horizon (" altitude ") as measured in London at different
times on August 23, 1912. Exhibit the observations in a
graph. Use the graph to determine (i) the greatest height
reached by the sun during the day ; (ii) the time of noon ;
(iii) the time of sunrise and sunset.
Time Alt. Time Alt.
a.m. deg. p.m. deg.
6.40 15 12.40 494
7.0 18 1.23 47
7.55 264 2.15 42
8.43 33J 2.43 384
9.5 364 3.15 34 4
9.45 414 4.2 274
10.27 451 4.30 234
11.15 49 4.43 21
11.26 494 5.15 164
Note. It is often unnecessary to represent^ the whole of
each vertical line. When all these lines are greater than a
certain length the excess above this length is all that need be
represented. For example, in No. 17 the vertical scale may
EXERCISE II 9
begin with the number 20. The lowest 20 inches of each
vertical may be supposed to be below the base.
17. A rubber cord, 23*7 cms. long, is hung up vertically and
a number of weights are attached to the lower end in succes-
sion. The table gives the length of the cord for certain
weights. Exhibit the observations in a graph. What weight
would make the cord 25 cms. long ? How long would it be if
18 gms. were attached ?
Weight in gras. 5 10 15 20 25 30
Length in cms. 24 "3 24'9 25-5 261 267 27'3
18. A heavy button was hung at the end of a piece of silk
thread and allowed to swing like a pendulum. The table
gives the time taken for 100 swings with different lengths of
silk. Draw a graph of the results, and use it to find what
length of silk would give a pendulum beating seconds. What
would be the time of one swing when the silk is 2 feet 6 inches
long?
Length in feet .1 2 3
Time of 100 swings 55 sec. 1 min. 18 sec. 1 min. 35 sec.
Length in feet .4 5 6
Time of 100 swings 1 min. 50 sec. 2 min. 4 sec. 2 min. 15 sec.
19. The "lighting up time " for cyclists and motorists is
given in the following table for certain dates in 1912. Draw
a graph by which the lighting up time on intermediate days
could be determined. When must a cyclist light up on (i)
February 12, (ii) May 3? When can the cyclist first ride
without a light until 8 o'clock ?
Jan. 1 Jan. 31 Mar. 1 Mar. 31 Ap. 30 May 30 June 29
4.58 5.43 6.38 7.29 8.18 9.3 9.19
20. If you look at the back of a Post Office Savings Bank
Book you will find the following table. It states the single
premium or sum which you must pay in order to receive
100 when you are 60 years of age or to secure 100 for
your relatives if you die before that age. The premium de-
pends upon the age at which you insure. Draw a graph
and use it to find the premium you would have to pay (i) at
18, (ii) at 32.
10 ALGEBRA
Age next birthday.
Premium.
s. d.
15
41 4 6
20
45 5
25
49 5 6
30
53 16 6
35
5!) 1 6
40
65 2
45
72 1
50
80 3 6
0.
21. During a very wet season certain fields in the Thames
valley were flooded. During the first day of the flood the
water covered 2 acres, on the second day it spread over 3
acres more. During the following five days 4*3 acres, 4 '8 acres,
3*6 acres, 2*3 acres, 0*7 acres were added in succession to the
area covered. Draw a column-graph showing the way in
which the area covered by the flood grew from day to day.
22. Convert the column-graph into a graph showing the
way in which the area of the flood probably grew from hour
to hour. Why must one say " probably "? What area did
the water probably cover (i) 1J days, (ii) 4i days after the
beginning of the flood ?
23. In a terminal examination in algebra out of a class of
28 boys the marks given ran as follows. Between 10 and
20 per cent., 2 boys; 20-30 per cent., 2 boys; 30-40 per cent.,
3 boys; 40-50 per cent., 4 boys; 50-60 per cent., 5 boys;
60-70 per cent., 5 boys; 70-80 per cent., 4 boys; 80-90 per
cent., 2 boys; 90-100 per cent., 1 boy. Draw a column-
graph exhibiting these results. What is the use of drawing
such a graph ? What is the total area of the rectangles ?
24. Obtain (i) the ages and (ii) the heights of all the boys
or girls in your form. Divide the greatest differences of age
and height each into the same number of steps, and find (as
in the preceding example) how many boys or girls are included
in each step. Draw two column-graphs, one to show the
distribution of ages and the other the distribution of heights
in your form. Compare the two graphs.
25. The directors of an exhibition published every four
weeks the number of persons who paid for admission during
the preceding four weeks. The numbers are given (roughly)
in the following table. Exhibit them in a column-graph.
EXERCISE II 11
Convert; this into a graph which will show the probable
number of paying visitors in any given time. About how
many people probably paid for admission during (i) the 6th,
(ii) the 14th, (in) the last week of the exhibition ? In which
week did the number first exceed 10,000, and in which week
did it again fall below this number ?
Month 1234567
Number admitted (in thousands) 22 36 42 30 22 15 8
D.
26. If the ends of a flexible chain are fastened to two pegs
on the same horizontal level the chain hangs in a character-
istic curve called the catenary. Hang up a chain (e.g. a
watch-chain or neck-chain or a dog-chain with small links)
against an upright blackboard or drawing-board. Fix the
position of various points of the chain by measuring their
distances above definite points on a base-line. Use these
measurements to obtain a reduced reproduction of the
catenary.
27. A steamer is travelling in an easterly direction across
a bay. At a certain moment it is due north of a battery on
shore and its distance (as determined by a range-finder) is
four miles. The following table gives the distance of the
steamer at subsequent times and its bearing, that is, the
angle which the line to it from the battery makes with the
north. Draw a diagram of its path across the bay.
Distance (in miles) 37 3*65 3-95 475 5'35 6'4 7'4 7*8
Bearing E. of N. 14J 35 50J 66J 74 82 90J 94J
28. Two lighthouses, A and B, are situated due north and
south of one another and a mile apart. The following table
gives the bearings of a ship taken simultaneously at A and B
every ten minutes. Give a diagram of the path of the ship.
Bearings from A 13 23J 34J 45 J 54 63 73J 83J 90
W. of N.
Bearings from B 11 19J 29 38J 46 54 62J 72J 80
W. of N.
29. A bicyclist is riding at constant speed along a road
across a common. He whistles to his dog who is on the
common at some distance from him. The dog constantly
directs his motion towards his master and runs half as fast
12 ALGEBRA
again as the latter rides. Find, according to the method
explained to you, the path of the dog.
30. A piece of mud on the rim of a moving cart-wheel
traces out a curve called the cycloid. Trace a cycloid on
paper b;y rolling a circle (e.g. a penny) along the edge of your
ruler and marking a number of the positions occupied by a
certain point on its circumference.
EXERCISE III.
THE WRITING OF FORMULA.
In order that your answers maybe corrected easily, you should
use the symbols placed in brackets after the statement of each
problem. The first letter is always intended to be the symbol
of the subject of the formula. The other letters are intended
to be the symbols of the other phrases in the order in which
they occur in the statement. The same symbols should be
used in each of the different parts of the same question.
In many cases an arithmetical example of the kind of
problem you are to deal with is first given. By considering
how you would work this example, you can always find out
the general rule for solving problems of that particular kind.
In other cases, where you cannot see what is the rule for solving
the problem, you should make up an arithmetical example for
yourself and consider how you would solve it. You will in
that way be able to find out the rule.
A.
1. Write down formulae to find :
(i) The weight of a bag containing 37 marbles, given the weight
of the bag and the weight of a single marble ( IV, b, m) ;
(ii) The weight of a bag containing any given number of
marbles (ri) ;
(iii) The weight of a bottle of ink, given the weight of the
bottle, the weight of a cubic inch of the ink, and the
quantity in the bottle (W, b, i, V) ;
(iv) The length of shelf that would be occupied on a bookshelf
by 8 volumes of a certain thickness, followed by 5 volumes
of another given thickness (I, t l9 2 ) ;
(v) The length of shelf that will be occupied by a certain
number of volumes f inch thick, followed by a given
number of volumes | inch thick (J, n i} n%) ;
(vi) The length of shelf that would be 'occupied by two sets
of volumes, given the number and the thickness of the
volumes in each sefe ;
13
14 ALGEBRA
(vii) The weight of a handful of new pence, shillings, and half-
crowns, given the number and the weight of each kind of
coin(TF, n b n^ n,, p, s, h)\
(viii) The value of the handful of coins expressed in pence (P).
2. (i) A boy goes into an office where his salary begins at 50
and increases by 5 every year. What will his salary be
after a given number of years ? (S, t).
(ii) Write down in symbols the general rule for finding the
salary when the commencing salary, the annual increase,
and the time of service are given (&, $ , i, t).
3. (i) A motor-car begins to descend a hill at the rate of 14
miles an hour and increases its speed by 3 miles an hour
every minute. How fast will it be moving after a given
number of minutes ? (s, t).
(ii) Write the rule so that it will apply to any initial speed
and rate of increase (s, % i, t).
4. (i) A man decides to spend 35 upon books for his library
and buys them at the rate of 2 a month. What sum will
be left at the end of a given number of months ? ($, t).
(ii) Write the rule for solving all such problems, given the
original sum of money and the monthly rate of expendi-
ture (8, So, r).
6. (i) The brakes of a railway train are put on when it is going
at 45 mls./hr. (miles per hour), and its speed now
diminishes at the rate of 10 mls./hr. every minute until
it stops. Write a formula for the speed of the train after
a given time expressed in minutes (P, t).
(ii) Write this rule so that it will apply to any original speed
and rate of decrease (F, F , r, t).
(iii) Write a formula for the speed of the train if, after its
speed has been decreased at a certain rate for a given
time, the brakes are taken off and the speed increases at
a given rate for a given number of minutes (F, F , r 1? ti,
r* fe).
& (i) A boy sells a model flying machine and a number of white
mice and spends most, but not all, of the proceeds in buy-
ing rabbits. Write a formula for the money left over
(the " residue "), given the sum he receives for the flying
machine, the prices of a mouse and a rabbit, and the
numbers sold and bought (R, /, m, r, n ly n^).
(ii) Give a formula for the money he still requires if the cost
of the rabbits is greater than the proceeds of his sale
<*)
7. Write formulas to find :
(i) The area of the wall of a room containing a window, given
the height and breadth of the wall and the height and
breadth of the window (A, H, B, h, 6);
(ii) The same when there are three windows of the same size ;
EXERCISE III 15
(iii) The same when there is a given number of windows of tho
same size (n).
8. A bath is fed by two taps each supplying 1/7 gallons per
minute, and is emptied by a waste pipe carrying away 2*1
gallons per minute. Write down formulae for finding :
(i) The number of gallons in the bath if to begin with it has
40 gallons in it and you turn on one tap for a given
number of minutes (F, t) ;
(ii) The same, if you turn on both taps ; -
(iii) The same, if you let the water run out with both taps off ;
(iv) Write down in symbols the rule for solving all problems
like (i), (ii), and (iii), given the number of gallons in the
bath to begin with, the amount supplied per minute by
each tap, and the amount carried away per minute by the
waste pipe (V, V Q , s, w).
9. Find formulae by which to express any given term of the
following series of numbers (T n , n):
(i) 3, 5, 7, 9, ...
(ii) 8, 13, 18, 23, ...
(iii) 21, 24, 28, ...
(iv) 1-4, 21, 2-8, 3-5, ...
(v) f, 1A, 1 T *, - . .
(vi) 94, 88, 82, ...
(vii) 18-2, 16-4, 14 '6, . . .
10. Calculate the 10th term of each of the foregoing series
of numbers.
11. Supposing that the bath in No. 8 contained 120 gallons
when full, how would you find :
(i) The time in which it could be filled by one of the taps ;
(ii) The same, both taps being turned on ;
(iii) ,The same, both taps and the waste being turned on ?
Write formulae expressing the rules for solving problems
like (i), (ii), and (iii), given the number of gallons in the bath
when full, the number of gallons delivered per minute by each
tap, and the number carried away per minute by the waste
pipe (t, F, 5, w).
12. How would you answer questions (i), (ii), (iii) if the
bath contained 40 gallons at the beginning?
Write three formulae expressing the rules you would follow,
using the same symbols as in the previous question.
16 ALGEBRA
13. Write down formulae for finding :
(i) The area of a rectangle, given the length and breadth ;
(ii) The length of a rectangle, given the area and the breadth ;
(iii) The breadth of a rectangle, given the area and the length ;
(iv) The weight of a figure (e.g. the map of Great Britain) cut
out from cardboard, given the area of the figure and the
weight of a square centimetre (or square inch) of the
cardboard (W, A, w) ;
(v) The area of a figure cut from cardboard, given its weight
and the weight of unit area of the cardboard ;
(vi) The weight per square centimetre of a sheet of cardboard,
given the length, breadth, and weight of a rectangle cut
from it ;
(vii) The area of a triangle, given the base and the altitude (A
b, h) ;
(viii) The weight of a triangle when cut from cardboard, given
the weight of a square centimetre (or square inch) of the
cardboard ;
(ix) The weight of a triangle cut from a brass plate weighing
3*4 grams/cm. 2 (grams per square centimetre), given the
base and the altitude ;
(x) The altitude of a triangle cut from the same brass plate,
given the weight and the base.
14. (i) Give the rule for finding the amount of air available for
each person in a class-room, given the number of cubic
feet of air and the number of persons (a, F, n) ;
(ii) The same, the number of persons being calculated by
counting the number of desks and adding one for the
master or mistress ;
(iii) The same, the number of persons being calculated by
counting the number of "dual" desks (i.e. desks that
hold two) and adding one for the master or mistress ;
(iv) The same, when the desks are single but three of them are
unoccupied. (The teacher is present) ;
(v) The same when the desks are dual and three are unoccupied
(teacher present).
15. (i) To find the weight of a single lead shot I propose to weigh
160 together in a beaker. What formula should I use to
find the weight of a shot, given the weight of the beaker
with the shot in it and of the beaker when empty?
(s, w l9 w*).
(ii) Write a formula for this problem for any given number of
shot (n).
16. There are a certain number of villas in a road, all of the
same size. Two roads lead out of this road on one side and
three on the other. These roads all haye the same width.
EXERCISE III 17
Given this width and the length of the road find a formula for
the frontage of each villa (/, n, w, I).
17. In many towns all the houses on one side of a street
have odd numbers while those on the other side have even
numbers.
(i) Write down the rule for finding the number of houses on
the " odd number " side when you know the number of
the last house on that side (n, N).
(ii) Give the corresponding rule for the " even number " side,
(iii) Give the rule for finding the number of houses between
those bearing two given odd numbers (not counting the
houses that have those numbers) (n, JV l5 N%).
(iv) Find whether the same rule holds good for the <( even
number " side.
18. (i) New-laid eggs are 2s. a dozen. Write down a formula for
the cost in shillings of a given number of eggs (c, n).
(ii) Write a formula that will serve whatever be the price per
dozen (c, n, p).
19. A business firm sets aside at Christmas a sum of
money to be distributed equally to the three senior clerks in
each of their departments. Write down f ormulsB for :
(i) The amount of the bonus given to each department
(6, A, n) ;
(ii) The amount received by each clerk (c).
20. Upon a piece of land of a given area (in square yards)
a given number of blocks of workmen's dwellings have been
erected. Each block contains 22 tenements. Write down
formulae for :
(i) The number of square yards of the site per block (6, A 9 n) ;
(ii) The number of square yards of the site per tenement (t).
B.
21. The value of a piece of land in a London suburb has
risen in ten years from 120 to 430. How would you
calculate the rise in value of another similarly situated piece
which was worth 240 ten years ago ?
(i) Write down in symbols a rule for finding the increase of
value in all such cases, given the former and the present
value of one piece of land and the former value of the
other (i, v ly i> 2 , F,).
(ii) Write down a formula for the present value of the second
piece of land (F 2 ).
2
18 ALGEBRA
22. In another part of London the rent of a house has gone
down from 85 to 70 during the last fifteen years. If the
value of other houses has decreased in the same proportion,
how would you find the decrease in the rent of a house now
rented at 42?
(i) Write a formula for the decrease of the rent in such cases,
given the former and the present rent of one house and
the present rent of the other (d, r i9 r 2 , R 2 )<
(ii) Write also a formula for the former rent of the second
house
23. A number of friends go on a holiday together, sharing
equally the living expenses and the travelling expenses.
Given the amount of each kind of expense and the number of
the friends, there are two ways in which they could " settle
up " at the end of the holiday.
(a) Each person could hand over to the treasurer first his
share of the living expenses then his share of the travelling
expenses.
(b) The treasurer could add together the expenses of both
kinds, and each person could pay his own share of the whole.
Write down two formulae to express these two ways of
proceeding (I, t, n, S).
24. A man leaves his house property and his bank shares,
each to be divided equally among his children. Given the
value of the house property and the bank shares and the number
of children, write, as in the last example, two formulae for the
value of the legacy received by each child (Z, h, b, n).
25. A man left property of a certain value to be sold and
divided equally among his nephews and nieces. A certain
amount had to be paid to the Inland Revenue Office as
Estate and Succession Duties. Given the value of the property,
the amount of the duties, and the number of persons, write
down two formulae for the share ultimately received by each
(S, p, d, n).
26. Two families go on a holiday together with the under-
standing that each person shall pay the same proportion of
the expenses. If there were 5 in the one family and 3 in the
other, what share of the total expenses would be borne by
each family?
Write a formula for calculating the share of each family,
given the number in each and the total expenses (S 19 /S 2 , n v
w 2 , E).
EXERCISE III 19
27. In the following examples the population of the town
is supposed to be changed only by deaths and new births.
Emigration and immigration are supposed to balance one
another.
The number of births in a town is 86 per month and the
number of deaths 64 per month. What is the easiest way of
finding the increase of the population in (say) 7 months?
(i) Write a formula for the increase of the population of the
town in a certain time, given the monthly number of
births and deaths (i, t, b, d).
(ii) The population of a town is increasing. Find a formula
for its value after a certain number of months, given
the original population and the monthly number of births
and deaths (P, P , t, b, d).
(iii) Write a formula to suit the case of a town whose popula-
tion is decreasing.
28. Two motor-cars start from the same place and travel
in opposite directions at speeds of 18 and 16 miles per hour
respectively. How would you most easily calculate their
distance apart after 4 hours?
(i) Write a formula for the distance apart after a given time
of two cars which start from the same point and travel
opposite ways at a given speed (d, t, s ly s 2 ) ;
(ii) The same, the cars being already a certain distance apart
to begin with (d ) ;
(iii) The same, the cars starting from the same point but in the
same direction ;
(iv) The same, the cars going in the same direction and the
faster being a given distance ahead of the slower at the
beginning (d Q ) ;
(v) The same, the slower car being a given distance ahead of
the faster at the beginning.
29. Between two mile-stones on a hill-side the road gradually
rises. Given the numbers on the mile- stones and the total
rise in feet write formulae to give :
(i) The rise in feet per mile (r, m t , n^ t R) ;
(in The rise in feet per foot ;
(iii) The rise in feet per 100 feet.
30. Between two given dates a certain total number of
inches of rain has been recorded at Kew Gardens. Given
the dates and the total rainfall write down formulae for :
2*
20 ALGEBRA
(i) The average rainfall per year (r, d lt d. 2t R) ;
(ii) The average rainfall per month.
31. A man owns a certain number of houses receiving from
each the same rent and having to pay out of this rent the
same amount of ground rent.
(i) What is his net income from the houses ? (I, n, r, g).
(ii) How much (in shillings) does he pay in income tax at Is. 2d.
in the pound ? (T).
(iii) He leaves the property to his children. What income
does each receive ? (Lot c represent no. of children.)
(iv) How many shillings income tax does each pay at Is. 2d. in
the pound ?
32. In the great Paris flood of 1910 the depth of water was
observed to increase at one place from 1*8 metres to 3*2 metres
in 12 hours. How would you find the amount of increase to
be expected in (say) 16 hours ?
(i) Write a formula to find in all such cases the increase
during a given time, given the first and second observed
depths of the water and the interval of time during which
the increase took place (i, t, d lf d 2 , T).
(ii) Write a formula for the total depth of the water after a
given time (d).
(iii) Write a formula for the depth after a given time supposing
that the depth of the water is decreasing instead of in-
creasing (i.e. d l > d> 2 ). Use the symbols of (i) and (ii).
(iv) Write a formula for the time the flood would take to dis-
appear if it continued to subside at the same rate (T).
33. How many planks 10 inches wide would there be
side by side across the floor of a room 16 feet wide ? The
number could be determined either by reducing the 16 feet to
inches or expressing the 10 inches in feet. What would the
working be if you chose the latter way ?
(i) Find the number of 10 inch planks when the width of the
floor is given in feet (n, w).
(ii) Find the number of 8 inch planks when the width of the
floor is given in feet.
34. (i) How many books J inch thick can be set side by side on a
shelf of a given length in inches ? (n } I).
(ii) The same, the length being given in feet ?
(iii) The same, the books being 1J inch thick and the length
given in inches ?
(iv) The same, the length being given in feet ?
EXERCISE III 21
35. How many planks 8 inches wide and 13 feet long
would cover a floor whose area is 391 square feet? Write a
formula for :
(i) The number of planks 8 inches wide and of given length
(in feet) that would cover a floor of a given area (in square
feet) (n, I, A)
(ii) The same, the planks being 9 inches wide ;
(Hi) The same, the planks being 10 inches wide.
36. The subscribers to a library pay a subscription of 2s. 6d.
a year and 2d. for every volume borrowed.
(i) Write a formula for the year's cost in pence to a person
who borrows a certain number of books (<7, n).
(ii) Write a formula for the year's cost in shillings,
(iii) Write a formula for the number of books borrowed, the
cost being given in pence ;
(iv) The same, the cost being given in shillings.
37. Men were admitted to a football match for 3d. and
boys for 2d. Thore were 2036 men present and the receipts
amounted to .32 9s. How would you find how many boys
were present ?
Write a formula for finding :
(i) The number of boys when the number of men and the total
receipts (in pounds) are given (6, m, T) ;
(ii) The number of men when the number of boys and the total
receipts in pounds are given ;
(iii) The total number of persons present in case (i) (n) ;
(iv) The total number present in case (ii).
38. At a cricket match 6d. is charged for admission to the
ground and a further charge of Is. for a seat in tbc grand-
stand. Given the number of persons in the grandstand and
the total receipts (in pounds), find a formula :
(i) For the number admitted but not to the grandstand (n ly
a, T) ;
(ii) For the total number admitted (n).
(iii) Write formulae to calculate the same two numbers in any
case, given (in shillings) the charge of admission to the
ground and the extra charge for the grandstand (a, s).
39. (i) A bookshelf containing a certain number of volumes of the
Temple Shakespeare is hanging from a nail on a wall.
Write a formula for the pull on the nail, given the weight
of the shelf and cord and the average weight of a volume
(P, n, W, w).
22 ALGEBRA
(ii) Write a formula for the average weight of a volume, given
the pull on the nail, the weight of the shelf and cord, and
the number of volumes.
40. (i) A man has to walk to a place 24 miles away. How far
will he be from his destination after a given number of
hours if his rate of walking is 3J miles an hour ? (d, t).
(iH How many more hours will his journey take him ? (T).
(iii) Write these rules so that they will apply to any given dis-
tance and rate of walking (d Q , r).
EXEKCISE IV.
THE BEADING AND USE OF FORMULAE.
A.
1. Each of the following formulae tells you how to obtain
the various terms of some series of numbers. Calculate the
first four terms of each series and the 10th term.
(i) T n = 2n - i.
(ii) T n - 4n - 3.
(iii) T n = 100 - 6n.
(iv) T n = 4(3n - 2).
(v) T n = i(8n + i).
(vi) T n - (n- i)(n- i).
(vii) T n - (2n - i) (n + 3).
(viii) T n = n 2 - 2n + 7.
(ix) T n = 2 n( 3 n 2 - i).
- (n + 3).
n" (an ,) (n + 3)' n
2. A young man has offers of three appointments. The
salary in pounds (S) which he would earn after a given number
of years' service (n) is given for each post by one of the follow-
ing formulae. In each case the salary ceases to rise after
20 years. Describe in words the various conditions offered,
and calculate which will eventually give him the highest
salary,
(i) S = 120 + |n.
(ii) S = 90 + 411.
(iii) S - 85 + ^n.
23
24 ALGEBRA
3. The "rise" in inches (R) in the steps of a staircase is
connected with the " tread" (T) by the following formula
(called "the French rule"), (i) Explain the rule in words;
(ii) calculate the rise for a tread of (a) 9 inches, (b) lOf
inches ; (iii) find whether this rule has been followed in
making the school staircase.
R 12 > T > 9
V
4. The load that may be safely attached to an iron chain is
given by the following formula :
L = 711d 2
L = safe load in tons ; d = diameter of chain-iron in inches.
Find the greatest load that can be lifted by a chain in
which the diameter of the iron is (i) half an inch, (ii) 1-2
inches.
5. The rule used by military engineers in blowing a breach
in a wall by means of blasting powder is
p = (ML 3
p = charge in Ib. ; L= half-thickness of wall in feet.
Explain the rule and calculate the charge i necessary to
make a breach in a wall (i) 3 feet thick, (ii) 14 feet thick.
Note. The wall is bored half through.
6. The distance to which you can see from the top of a sea-
cliff is given by the formula
d = 1-22 Jh
d = distance in miles ; h = height of cliff in feet.
Find how far you can see from a cliff (i) 25 feet high,
(ii) 256 feet high, (iii) 400 feet high.
7. In the following formula d = the depth of the crown of
a brickwork arch, r == the radius of the arch. Both are
measured in feet. For a single arch n = 0*4, for a series of
arches n = 0*45. Calculate the depth of the crown (i) of a
single arch of radius 16 feet, (ii) of a series of arches of radius
9 feet.
d = n r.
EXERCISE IV 25
Note. In this example n is called a " coefficient " or a
"constant". What are the coefficients or constants in Nos.
4, 5, 6?
8. The velocity of the stream at the bottom of a river (v) is
connected with the velocity at the surface (V) by the formula
Calculate the velocity at the bottom when the velocity
at the surface is (i) 4 ft./min. ; (ii) 9 ft./min. ; (iii) 121 ft./min.
9. Suppose you were asked to find by thef formula of No. 6
the distance visible from a cliff 40 or 56 or 75 feet high. You
could not do it easily because it is not easy to see the square
roots of 40, 56, 75, etc. They must, no doubt, lie between the
square roots of 36 and 49, 49 and 64, 64 and 81, respectively.
It is possible, therefore, that they may be obtained by draw-
ing a graph. Draw a graph as you are instructed and test ifc
by choosing any number you please and seeing whether the
graph gives the square root correctly.
How could you use the graph to find the square roots
of numbers 100 times those which you have plotted ?
Note. This graph will be required in solving many of
the following problems. It must therefore be drawn very
accurately.
10. Find the distance visible from a cliff (i) 40, (ii) 56, (iii)
75, (iv) 200, (v) 770 feet high.
11. From a cliff I can just see the lights of a seaport 15
miles across the sea. Supposing that the lights are 20 feet
above the sea, what is the height of my eye ? (Use the graph
of No. 9.)
12. Find the velocity at the bottom of a river when the
velocity at the surface is (i) 30, (ii) 88 feet per minute.
13. Draw a graph to show how rapidly the charge of
powder required to blast a wall increases as the thickness of
the wall increases.
14. Use this graph to find the thickness of wall that re-
quires a charge of (i) 20 lb., (ii) 35 Ib.
How can you use this graph for finding cube roots?
15. Surveyors use the following rule to calculate roughly
the heights of buildings, etc., when they have no instruments.
Take on level ground a line AB of convenient length pointing
towards the building. Set up at A a staff higher than
yourself. Walk back from it until the line of sight through
26 ALGEBRA
the top of the staff meets the top of the building. Move the
staff to B and repeat. Let H = ht. of building, h = ht. of
staff, B = ht. of eye above ground, D = distance AB, d l and
d 2 = distances from A and B at the first and second observa-
tions respectively. Then :
" E) h
To find the height of a church tower I set up a staff 12
feet high at two points distant 100 feet from one another.
The line of sight cuts the top of staff and tower when I stand
11 feet from the staff at one end of the base, and when I
stand 8 feet from it at the other end. My eye is 5 feet
from the ground. Calculate the height of the tower.
Calculate the height of a tree given D = 50 feet, h = 10
feet, E = 4: feet 9 inches, d l = 9 feet, d 2 = 6 feet 6 inches.
16. Find the height of the school or a neighbouring building,
etc., by the method of No. 15. Compare your results with
those obtained by others in the class, and, if possible, with
the known height of the object.
B.
17. A beam is fixed in masonry at one end and sticks out
horizontally into a room. It is loaded at the free end. The
following formula gives the weight it will just bear without
breaking :
W = breaking weight in cwt. ; b = breadth of beam, and Z =
length of beam, d = depth or thickness of beam all in inches.
A; is a coefficient. For wrought iron k = 68, for cast iron k
46, for English oak k = 15.
(i) Find the greatest weight that can be hung on the end of a
wrought-iron bar 10 feet long and 2 inches square ;
(ii) The same, the bar being of cast iron.
(iii) What weight will break an oak beam 12 feet long, 3 inches
wide, and 4 inches deep ?
18. The quantity of water raised per minute by a pumping
engine (e.g. from the hold of a ship or from a well) is given
by the formula :
g -. 0-034 nld 2
EXERCISE IV 27
g = no. of gallons per minute ; n = no. of strokes per minute ;
I = length of stroke in feet ; d = diameter of pump in inches.
(i) Find how much water can be raised hy a pumping engine
making 20 strokes a minute, 2 feet 6 inches in length, the
diameter of the pump barrel being 22 inches ;
(ii) The same, when n 15, I = 3 feet, d 18 inches.
19. In using a rope to lift heavy weights, etc., it is im-
portant to know (a) the working load (i.e. the weight that the
rope can be used to lift constantly) and (b) the breaking load.
These are given by the formulae :
(a) L - kj& ; (b) B - fc 2 C 2
L = working load in tons ; B = breaking load in tons ; C = cir-
cumference of rope in inches; k and & 2 are "constants"
depending on the material.
jf &
Common hemp . . . 0'032 0'18
Best hemp . . . O'lOO 0'60
Iron-wire rope . . . 0'290 1*80
Steel- wire rope . . . 0'450 2-80
Calculate the working load (i) for a common hemp rope
4 inches in circumference; (ii) for a steel- wire rope of the
same circumference. Calculate (iii) the breaking load for an
iron- wire rope 2 inches in circumference, (iv) Could a steel
rope 1 inch in circumference bear safely the working load of
an iron- wire rope of 3 inches in circumference ?
20. Draw, on a single sheet of squared paper and with the
same axes, graphs showing the working load of iron-wire and
steel-wire ropes for different circumferences up to 4 inches.
Be careful to distinguish the curves by labels.
21. Answer the following questions from the graphs :
(i) What are the working load's of an iron-wire rope and of a
steel- wire rope of 2J inches circumference ?
(ii) What should be the circumference of a steel-wire rope for
a crane which has to lift 4J tons ? If the builder had to
use an iron-wire rope what should be its circumference ?
Measure, if you can, the circumference of some thick hemp
or a wire rope and calculate its working and breaking
loads.
22. When a ship springs a leak the quantity of water (in
tons per hour) is given by the formula :
W - A./203"
28 ALGEBRA
A = area of hole in square inches ; d = depth of hole in
feet.
(i) A rivet with an area of J square inch has fallen out of a
ship's bottom 20 feet below the water-line. How many
tons of water must be pumped out of the ship per hour so
that it may not accumulate ?
(ii) There is a hole 4 square inches in section in the side of a
ship 4 feet 3 inches below the water-line. How many
tons an hour will leak in ? (Answer to the nearest ton.)
23. If you look at a crane you will see a large hook at the
end of the rope. The weakest part of the hook is the " shank,"
i.e. the upright part that passes through the iron attachment
to the rope. By the following formula the engineer calculates
the thickness of shank necessary if a given load is to be lifted.
D = diameter of shank in inches ; W = weight to be lifted
in tons.
D = x/(>45W + 0-2
What is the size of shank needed for dealing with the follow-
ing loads? (i) 20 tons, (ii) 10 tons, (iii) 16 tons. (Use the
graph of No. 9.)
24. Piles are driven into the ground by a "pile-driver," a
machine that lifts a heavy weight called a "ram " and drops
it on to the pile. If a house or other load is to be laid upon
the piles we must know how much each pile can support.
The formula is :
+ P)
L ^ greatest load in tons pile will bear ; W = weight of ram
in cwt. ; h = height in feet from which ram falls ; d = distance
in inches that the pile was driven in by the last blow ; P =
weight of pile in cwt.
A ram weighing 6 cwt. falling a height of 4 feet drove a
pile weighing 15 cwt. 1 inches into the ground. What is
the greatest load the pile will safely bear?
(Note. It is usual actually to load the pile with not more
than about ^ of this load. Compare the working and greatest
load of ropes in No. 19.)
25. Fig. 3 represents one span of a telephone wire sup-
ported by standards at A and B. An engineer uses the
symbols I for the bare length of the span between A and B,
EXERCISE IV 29
* 3
i i
L for the actual length of wire in the span, d for the dip or sag
of the wire in the middle, all these
measurements being made in feet.
He also uses the symbol 5 for the
pull with which the wire is stretched
over the standards and w for the
weight of every foot of the wire, FIG. 8.
both being measured in pounds. Explain the meaning and
use of the following formulas taken from his pocket book :
1 2 W
(\ j * "
1} d== 8s-
<") S " S
(iii) L = 1 + ?
(iv) d *'' vx ~ " !)
26. An iron telephone wire weighs 0*072 lb./ft, and is
stretched with a pull equal to the weight of 270 Ib. between
standards 100 feet apart. Calculate the dip of the wire in
inches.
27. The dip of the same wire stretched between standdsra
120 feet apart is observed to be 6 inches. Calculate the
stretching force.
28. What is the actual length of telephone wire in the loop
between two standards 150 feet apart when the dip is 18
inches ?
29. The size of the barrel of the winding engine used to
raise and lower the " skips " in a coal-pit is given by the
formula :
D = diameter of winding barrel in feet ; p = depth of pit in
feet ; n = no. of revolutions of engine per min. ; t = thickness of
rope in inches.
Calculate the size of the winding barrel for a pit -shaft 800
feet deep, the rope being 3 inches thick, and the number of
revolutions of the engine 30 per minute.
0.
Note. The most important parts of an ordinary steam
engine are (a) the cylinder C (fig. 4) into which the steam
30
ALGEBRA
is admitted from the boiler by a valve ; (b) the piston P which
is driven up the cylinder by the steam ; (c) the piston-rod K x ;
(d) the crank-shaft S which turns the wheel round ; (e) the
connecting-rod B 2 ; (/) the crank-pin p which fastens the
crank-shaft and the connecting-rod together.
When the steam enters the cylinder it presses against the
piston with a certain force. Imagine the cylinder held upside
down and weights to be packed on it till they press the piston
down exactly as hard as the steam did. Then the weight
lying on each square inch of the piston is called the steam-
pressure. As the piston moves out the steam-pressure will
not remain steady, but we can imagine a steady steam-pressure
which would produce on the whole the same effect. This is
called the " mean " steam -pressure.
James Watt found that the work performed by his engines
S /
Pro. 4.
could, as a rule, be accounted for by supposing a steady
steam -pressure of 7 lb. per square inch. He always as-
sumed this to be the pressure, therefore, in calculating the
" horse-power " of his engine. In these days much greater
steam-pressures are used, but makers in describing their
engines sometimes still follow Watt's plan. The horse-power
calculated according to Watt's rule is called the " nominal
horse-power " of the engine (N.H.P.). The actual horse-
power is called the "indicated horse-power " (I.H.P.).
The following symbols are used in Nos. 30-33 :
P = mean steam-pressure in Ibs. per square inch.
D = diameter in inches of cylinder.
I = the length of stroke of the piston in inches, i.e. the dis-
tance it is driven forward at each revolution of the
wheel.
n = number of revolutions of the wheel per minute.
H = the indicated horse-power.
EXERCISE IV 31
N = the nominal horse-power.
30. The tractive force or pull of a locomotive engine is
given by the formula :
T = tractive force in Ibs. ; w = diameter of the engine's driving
wheel in inches.
(i) Find the tractive force of an engine with a driving wheel
5 feet high, the cylinders being 2 feet in diameter,
the length of stroke 3 feet 6 inches and the mean steam-
pressure 50 Ib. ;
(ii) The same, given that P = 55, D = 27, I = 45, w = 66.
31. A certain amount of the pressure of the steam is
necessarily wasted in overcoming the friction of the piston
against the cylinder. The amount of this pressure is given
by the formula :
p.
Draw a graph showing the loss of steam-pressure to be ex-
pected in cylinders from 1 foot to 4 feet in diameter. (Take
y 1 ^ inch to represent an inch in the diameter of the cylinder.)
What is the loss with cylinders of the following diameters :
(i) 16 inches, (ii) 23 inches, (iii) 40 inches? What is the
smallest cylinder for which the loss of pressure will be not
more than (iv) 4 Ib., (v) 3 Ib. ?
32. The old Admiralty formula for calculating the N.H.P.
of a paddle engine was
=
3000
(i) Calculate the N.H.P. of an engine when the diameter of
cylinder is 54 inches, stroke 36 inches, and revolutions per
minute 30 ;
(ii) The same, substituting D = 48 inches, I =* 40 inches, n = 35.
33. The indicated horse-power may be calculated by the
following formula. Use it to find the I.H.P. of a marine
engine hi which (i) the diameter of the cylinder is 5 feet
10 inches, the length of stroke 4 feet, the mean steam -pressure
30 Ib., and the number of revolutions 15; (ii) D = 5 feet,
I = 5 feet 3 inches, n - 20, P = 20 Ib.
21,000
EXEKCISB V.
FACTORIZATION (I).
1. Calculate the total floor space of two rooms arranged as
in fig. 5 (p. 34) when the dimensions are as follows :
AB BO AD
(i) 24ft. *6ft. 13ft.
(ii) 119 ft. 81 ft. 64 ft.
(iii) 27'3 ft. 327ft. 16 '5 ft.
(iv) 21 ft. 4 in. 18 ft. 8 in. 14 ft. 6 in.
2. Calculate the area of a room shaped like fig. 6 when the
dimensions are as follows :
AB AF ED DC
(i) 40ft. 20ft. 35ft. 20ft.
(ii) 25 ft. 17 ft. 15 ft. 17 ft.
(iii) 18-6 ft. 13-2 ft. 11 '4 ft. .13-2 ft.
(iv) 21 ft. 17 ft. 21 ft. 13 ft.
(v) 26ft. 22ft. 26ft. 18ft.
(vi) 36 ft. 6 in. 21 ft. 7 in. 36 ft. 6 in. 28 ft. 5 in.
3. Fig. 7 represents the plan of a picture gallery. Find its
area when the dimensions are as follows :
AB AH DE CD
(i) 47ft. 16ft. 53ft. 15ft
(ii) 52ft. 23ft. 52ft. 27ft.
(iii) 44 ft. 16 ft. 28 ft. 32 ft. 1
(iv) 57ft. 14ft. 26Jft- 28ft.
(v) 120 ft. 37 ft. 60 ft. 26 ft.
(vi) 88ft. 17ft. 44ft. 16ft.
4. Write down a formula for the total floor space in the
rooms represented in fig. 8. Express the formula in the form
most suitable for calculation. (Use a, b, c as symbols for the
lengths of the rooms, d for their common width.)
1 Imagine the rectangle CE to be divided into two rectangles 16
feet wide.
82
EXERCISE V 33
5. Obtain formulae for calculating easily the area of a
T-shaped room like fig. 7 :
(i) When AB (a) and DE (b) are unequal but AH and CD are
equal (c) ;
(ii) When AB and DE are equal (a) but AH (6) and CD (c) are
unequal ;
(iii) When AB (a) and DE (b) are unequal and CD is double
AH (c) ;
(iv) When AB (a) and DE (b) are unequal and AH is three
times CD (c) ;
(v) When AB is double DE (a) and AH (6) and CD (c) are
unequal ;
(vi) When DE is four times AB (a) and AH (6) and CD (c) are
unequal. ^
6. The wall on one side of a passage is 36 feet long and
14 feet high. It is faced with bricks up to a height of 4 feet,
but above this height is covered with paint. Calculate :
(i) The area of the painted surface in square feet ;
(ii) The area of the same in square yards ;
(iii) The cost of painting it at 2d. a square yard.
7. Write down formulae (i) for the area (measured in square
feet) of the painted surface of the last question ; (ii) for
the cost of painting it. Let a = the length in feet, b = the
height in feet of the passage, c = the height in feet of the
brickwork, p = the cost in pence .of the paint per square yard,
C = the total cost in shillings. The formulae are to be expressed
in the form most suitable for calculation.
Note. Instead of writing " Let a be the symbol for the
length in feet of the passage," "let p be the symbol for the
cost in pence of the paint per square yard/ 1 it is often con-
venient (because shorter) to write " Let the passage be a feet
long," " let the paint cost^p pence per square yard". Upon this
plan the foregoing question could be expressed as follows :
"Find the total cost in shillings (C) of painting a passage
a feet long and b feet high, but faced with brickwork up to a
height of o feet, the cost of the painting being p pence per
square yard ".
8. Across the upper part of the wall in No. 6 there runs
a board d feet wide to carry hat-pegs. Write down formulae
for calculating in the easiest way (i) the area of the painted sur-
face ; (ii) the cost of painting it. (Use the same symbols as
in No. 7.)
3
34
ALGEBRA
D E F
FIG. 5.
C, ,D
A B
A E
FIG. 9.
FIG. 6.
FIG. 8.
1
26
i
FIG. 10.
D 1 'C
F E
T
{
T
b
1
Fia. 11.
<c>
Sc
t
b
i
Fia. 12.
EXERCISE V
35
FIG. 13.
FIG. 14.
FIG. 15.
-c+b
FIG. 16.
3c
--i
Fia. 17.
FIG. 18.
36 ALGEBRA
9. A strip of carpet c feet wide runs along the middle of a
passage a feet long and b feet wide. Write down formulae for
calculating in the easiest way (i) the area of the uncovered
part of the floor ; (ii) the total cost in pence (C) of polishing
it at a cost of p pence per square foot ; (iii) the total cost in
shillings (C) of polishing it at a cost of p pence per square
yard.
10. Across a room 97 feet long and 35 feet wide (AF in
fig. 5) a partition is thrown so as to cut off a room 47 feet long
(AB). Find the area of the room (BF) on the other side of
the partition.
11. Across a room a feet long and c feet wide a partition is
thrown so as to cut off a room b feet long. Find l the area of
the room on the other side of the partition in a form suitable
for easy calculation.
12. Find the cost in shillings of covering with linoleum at
Z shillings per square yard the room whose area was calculated
in the last question.
13. Fig. 9 is the plan of a hall AC containing a platform
EG. Seats are to be placed to the right and left of the plat-
form as well as in front of it. Calculate the area available for
seats when the dimensions are as follows :
AB
AD
EF
EH
(i) 109 ft.
(ii) 107ft.
40ft.
32ft.
18ft.
14ft.
20ft.
16ft.
(iii) 89 ft.
57ft.
27 ft.
19 ft.
(iv) 117ft.
46ft.
23ft.
34ft.
(v) 99 f fc.
55 ft.
15 ft.
33 ft
14. Write down formulae for the available floor space in
the hall represented by fig. 9 in the following circumstances.
(The formula is in each case to be in the form most suitable
for substitution) :
(i) AB (a) and EF (6) are unequal while AD is double EH (c) ;
(ii) AB is double EH (a) while AD (b) and EF (c) are unequal ,
(iii) AB (a) and EH (b) are unequal while AD is three times EF
(iv) AB (a) and EF (c) are unequal while EH (6) is two-thirds
of AD;
(v) AB (a) and EF (c) are unequal while EH is three -quarters
of AD (b).
1 That is, write down a formula for finding the area of, etc. The
expression " find the area, etc.," is used for brevity.
EXERCISE V 37
15. Write down formulae for the areas exhibited in figs.
10-18. In each case throw the formula into the shape in
which it is most convenient for use in calculating the area.
Note. If you cannot easily find the formula it is often a good
plan to copy the figure on a piece of paper or thin cardboard,
and to seek a way of cutting it up and re-arranging its parts
so as to form a simple area.
16. Calculate by short methods the value of each of the
following numerical expressions :
(i) 17-4 x 8*6 - 7*4 x 8'6.
,(ii) 17*4 x 8*6 - 14*8 x 4-3.
(in) 24 x 13 + 9 x 13 - 3 x 13.
(iv) 24 x 13 -f 9 x 26 - 4 x 39.
(v) Z2 x 14 + 4 x 28.
(vi) 21 x 16 - 35 x 8.
(vii) 38 x 55 - 57 x 33.
(viii) 14 x 15 x ii - 22 x 7 x 5.
,. v 13 x 18 + 26 x 12
7x23-14x5*
9 x 34 - 12 x 17
V ' 39 x 7 + 21 x 21*
17. Find identities by the aid of which formulae containing
the following expressions can be made more suitable for cal-
culation :
(i) a 2 b + a 2 c. (ii) pqr - q 2 . (iii) pq 3 - qr.
(iv) a! 2 m + blm 2 . (v) a 3 - pa 2 . (vi) pa 3 + q 2 a a .
(vii) abc + bed - cde. (viii) 2ab + 3ac. (ix) fp a q - 4pq 2 .
(x) a 2 b' 2 - 2ab. (xi) ap 2 q - pq. (xh) ar 3 - 2br 2 + r.
18. Keduce the following formulae to the forms most con-
venient for substitution :
(i) A = d - sld.
(ii) M = ia*p + *apq.
(iii) V
(iv) P = a2bc - ab 2 c + abc 1 .
(v) Q = 2a 3 b + 3ab s - ab.
(vi) T = ap 2 q - 2bpq 2 -f 3q 3 .
(vii) W = 27rr 2 w - 4raw.
(viii) B 12 m 2 n + g mn 2 - 311111.
(ix) C - i(6a 8 - I5a 2 + 3a).
(x) V ~ 3?ra a c + 47rabc.
Write down formulae in the most convenient form for the calcula-
tion of the following quantities. (The linear measurements
38 ALGEBRA
indicated in the figures may all be supposed to be made in
centimetres) :
19. The weight (W) of a brass plate cut to the shape of fig.
14 (the weight of a square centimetre being w grams).
20. The same for a plate cut to the shape of fig. 18.
EXEEGISE VI.
FACTORIZATION (II).
1. Fig. 19 is the plan of a courtyard. AC is a square and
FD if completed by producing the lines AF and CD would be
also a square. Find the area of . o
the courtyard when the dimen-
sions are as follows :
(i)
(ii)
(iii)
(iv)
AB
129 ft
78ft.
146 ft.
57 '4 ft.
FE
29ffc.
18ft
54ft.
32-6 ft
2. Fig. 20 (p. 41) represents a
square metal plate with a square D C
hole in the middle. Find the Fio. 19.
area of the plate when the dimensions are as follows :
AB CD
14*5 cms. 5 '5 cms.
6| in. 1J in.
7f in. 2| in.
16*3 in. 4'3 in.
(Imagine the hole to be moved from the middle to the
corner of the plate.)
3. Fig. 20 may also be supposed to represent the cross-
section of a hollow metal bar. Find its volume when AB =
6-8 cms., CD = 3*2 cms., and the bar is 20 cms. long.
4. Write down formulae for calculating quickly
(i) The volume (V) ;
(ii) The weight (W) of the hollow bar of No. 3, using
the symbols a and b for the length of AB and CD, I for the length
of the bar, and w for the weight of 1 c. cm. of the metal.
5. Find the most useful formula for the weight w of a cubic
centimetre of the material of the bar, given its total weight
and its dimensions.
39
40 ALGEBRA
Before trying the next example answer the following
questions :
(i) How can 4 equal squares whose sides measure 3 inches be
arranged so as to make a single square ?
(ii) What will be the length of the side of this square ?
(iii) Answer similar questions with regard to sots of 9, 16, and
25 equal squares ;
(iv) By what symbols will you represent the side of a square
made up of 4 equal squares the length of whose sides is
represented by 6 ?
(v) Answer the same question with regard to squares made up
of 9, 16, and 25 of the smaller squares.
6. Fig. 21 represents a square metal plate measuring a cms.
each way, in which four square notches have been cut, each
measuring b cms. each way. Find in a form suitable for
calculation the area of metal left.
7. Do the same for the plate represented by fig. 22.
8. Do the same for the plate represented by fig. 23.
9. Give convenient formulas for calculating :
(i) The weight (W) of the plate represented by fig. 22, given
that the material weighs w grams/cm. 2 ;
(ii) The weight (W) of a bar I cms. long whose cross-section is
represented by fig. 21, given that the material weighs w
grams/cm. 3 ;
(iii) The weight (w) of a square centimetre of the plate from
which fig. 23 is supposed to be cut, given that the whole
plate weighs W grams.
10. Give the best formula for calculating the cost in shil-
lings (c) per cubic foot of a rod whose cross-section is repre-
sented by fig. 23, given the dimensions in feet and the total cost
(G) in pounds.
11. Write down the identities which you would use in
simplifying for calculation formulae that contain the following
expressions. Show that the identities are always true no
matter what quantities or numbers the symbols reter to :
(i) p a - q 2 . (ii) 4* 2 ~ t> 2 - (iii) m 2 - 9 n 2 .
(vi) u 2 -
(iv) 3 6a 2 - 25b*. (v) p 2 a 2 - b 2 . (vi) u 2 - v y t 2 .
(vii) pV - rt 2 . (viii) a 2 b 2 - i6c 2 . (ix) a 2 - 16.
(x) 81 - b 2 . (xi) p 3 q 2 - 25. (xii) i - mV.
Note. Expressions such as ac + be and a 2 - b* can, as you
have seen, be replaced for the purposes of easy calculation
by expressions such as (a + b)c and (a + b) (a - b). These
EXERCISE VI
B
C D
FIG. 20.
< a
ID
FIG. 21.
FIG. 23.
FIG. 24.
FIG. 25.
42 ALGEBBA
expressions describe factors which when multiplied together
will give the result of the calculation. For this reason the task
of discovering them is often called factorizing 1 the original
expressions.
12. Factorize the following expressions. (You need not
prove that the identities are always true) :
(i) pa 2 -pb a . (ii) 4 ap 2 - 9 aq 3 . (iii) *T? - TrrV
(iv) a 3 - ab a . (v) p 2 t - t 3 . (vi) 4 P 2 t - 25!'.
)
.
(vii) i2a 2 - 27b 2 . (viii) a 2 b-s6b 3 . (ix) 8a 2 - 50.
(x) 4p 3 - 9 p. (xi) a 2 b 2 - c 2 . (xii) 9 a s b 2 - a.
13. Factorize the following expressions :
(i) (a 2 - b 2 ) + 3 (a + b). (ii) (a 2 - b 2 ) + 7(a - b).
(iii) (a 2 - b 2 ) - 4 (a + b). (iv) (a 2 - b 2 ) - p(a - b).
(v) (a 2 - b 2 ) + (a + b). (vi) (a 2 - b 2 ) + (a - b).
(vii) 2(a 2 - b 2 ) + 6(a - b). (viii) 2 (a 2 - b 2 ) - 3(3 + b).
(ix) a(p 2 - q 2 ) + b(p - q). (x) a(p 2 - q*) - b(p + q).
14. Factorize the following expressions :
(i) (a + b) 2 - c 2 . (ii) (p - q)* _ r *.
(ju) (* + b) 2 - 4. (iv) (p - q) 2 - 9.
(v) (p + q) 2 - or 2 . (vi) ( a - b) 2 - i6
i6c*.
(vn) 4(u + v) 2 - w a . (viii) 9 (u - v) 2 -
(ix) (a + b) 2 - a 2 . (y) (a + b) 2 - b 2 .
(xi) (a - b) 2 - b 2 . (xii) (a - b) 2 - a*.
(xiii) 4 (a + b) 2 - b a . (xiv) 25(a - b) 2 - a*.
(xv) (a + b) 2 - 4 b 2 . (xvi) (a - b) 2 - 9 b 2 .
(xvii) i6(p + q) 2 - 36p 2 . (xviii) 2s(p - q) a - 9q .
Note. Before doing the next set of examples answer the
following questions :
(i) If from a group of 12 things I take all of them except 5 how
many are left ?
(ii) If from a group of 29 things I take all except 13 (i.e. 29 - 13)
how many are left ?
(iii) What is left when I take 32 - 17 from 32 ?
(iv) Complete the identity a - (a - b) = ;
(y) What is the value of 48 + (48 - 7) ?
(vi) Complete the identity a + (a - b) = .
15. Factorize the following expressions :
ft a 2 - (a. - b)'. fin p 2 - (p - q)'. (iii) r 2 - (r - 3)".
(iv) p 2 - (p - 6'7} a . (v) ap 2 - a(p - q)'. (vi) Trr 2 - 7r(r - w) a .
(vii) a 2 - (a - 3b)*. (viii) s 2 - (s - rt) 2 . (ix) a a - (a - pb) a .
W P 3 - (P.- aq) a .
Note. In the following examples the formulaB are always
to be thrown into the form most suitable for substitution.
EXERCISE VI 43
16. Find the area of a figure shaped like fig. 19, given
that FE =a a cms. and DC = b cms.
17. Find the area of the same figure, given that DC = a cms.
and DB = b cms.
18. Find the area of fig. 22, given that AB = a inches and
BC = b inches.
19. Find the area of the same figure, representing the length
of AB by p and the length of BC by q.
20. Find the area of fig. 22 ; representing the length of AD
by p and the length of BC by q.
21. Find the area of the same figure, if a = the length of BC
and b = the length of AB.
22. Find the area of figure 19, given that AB a cms. and
DC = b cms.
23. Calculate the area of the ring represented in fig. 24,
given that r x = 13*9 cms. and r 2 = 11-1 cms. (Take ?r =
22/7.)
24. Write down formulae for calculating (t) the area of a
ring-shaped surface whose external and internal radii are r^
cms. and r 2 cms. respectively ; (ii) the weight (W) of such a ring
cut out of a brass plate weighing w grams/cm. 2 ; (iii) the volume
of I cms. length of a pipe whose external and internal radii
are r l and r 2 cms. ; (iv) the weight (W) of such a pipe, given
that the material weighs w grams/cm. 3 ; (v) the weight (w) of
a cubic centimetre of the material of such a pipe given the
weight (W) of a I cms. length of it.
25. Fig. 25 represents a circular disc pierced by four equal
circular holes. Write down a formula for the surface left,
representing the radius of the disc by a and the radius of a
hole by b.
EXEBCISB VII.
SQUARE BOOT.
1. Find correctly to two decimal places (if necessary) the
length of the side of a square whose area is (i) 23-04 square
yards, (ii) 62-41 square inches, (iii) 94*09 square cms., (iv) 72
square feet, (v) 3 -26 square miles, (vi) 28*7 square feet, (vii)
81-263 square yards, (viii) 1-024 square miles.
2. Find the square root of each of the following numbers
(the roots are to be given correctly to three significant
figures): (i) 348; (ii) 1562-8; (iii) 41616; (iv) -043; (v)
000294.
3. The total area of the British Isles is 121,377 square milea
If the land could all be arranged in the form of a square what
would be the length of its side ?
4. The area of England and Wales is 58,324 square miles.
Calculate the length of the side of a square having this area.
5. Using the results of the last two examples construct on
squared paper a diagram (arranged like fig. 20, p. 41)
showing the proportion of the area of England and Wales to
the area of the whole of the British Isles.
6. The total population of the British Isles in 1901 was
41,609,091, that of England and Wales also was 32,527,843.
Construct a diagram, like the one of No. 5, showing what
part of the total population lies within the boundaries of Eng-
land and Wales.
7. Use the formula d 1-22^/S (see Ex. IV, No. 6) to
calculate the greatest distance you could see across the water
from a cliff 527 feet high, allowing 5 feet more for the height
of the eye above the ground. Compare this result with that
obtained graphically.
8. Make a similar calculation for the distance visible from
a mountain 5320 feet high.
9. The Peak of Teneriff is 12,180 feet high. What is the
greatest distance from which it could be seen by an observer
at the mast-head of a ship 54 feet above the sea ?
44
EXERCISE VII 46
10. The velocity (V) of the water at the surface of a river is
18*5 feet per minute. Calculate the velocity at the bottom (v)
by the formula :
v = (v + i) - 9 y?
11. The top of the inner dome of St. Paul's Cathedral is so
high above the floor of the nave that a small but heavy weight
could swing beneath it at the end of a wire 285 feet long.
Find how long each swing would take using the formula :
t = 1-11 7!
in which t = time of a single complete swing in seconds, I =
length of- wire in feet.
12. How would you calculate the radius of a circle if you
knew its area? Write down a formula.
13. Use the formula of the last question to calculate the
radii of the circles whose areas are (i) 126-2 square inches, (ii)
1-82 square feet.
14. Calculate the radii of the two circles whose areas
would be equal to those of the whole British Isles and
England and Wales respectively (see Nos. 3 and 4). Use these
circles to construct a diagram on the same principle as that of
No. 5.
15. Give a formula for calculating the radius of a cylinder
when you know its volume, and its height.
16. The volume of a cylinder 4*8 cms. high is 2'67 c.cms.
What is its radius ?
17. To find the thickness of some copper wire I immerse a
piece of it 10 metres long in water in a measuring cylinder.
The water rises 4%5 c.cms. Calculate the thickness of the
wire.
EXBECISE VIII.
SURDS.
Note. It may be assumed in the following examples that
V2 = 1-41, v/ 3 = 1'73, V5 = 2-24, Jl - 2-65; V 11 =
3-32, V 13 =* 3 ' 6 - What do these statements mean?
1. Find to two decimal places the value of the following :
V8, V 12 > V 6 ' V 39 ' V"> V 567 ' V 2 ' 75 > V' 52
V6 - V3, V12 + J8, 6^/26 - 4^22.
2. nationalize the denominators of the following fractions
and then find their values to three significant figures :
. 1 8 1 /3 /8 0-8
3. Throw the following expressions into the form most
suitable for calculation but do not actually find their values :
3V5
' 26^/2' V 2
4. Calculate the value of the following expressions when
= 75, b - V2, c = V 3: ~
(i) 2a 2 - b 2 , (ii) J(a 2 - be), (iii) a 2 b + b 2 c + c 2 a,
(vii) a ~ a > (viii) ^-, (ix) ~,
(x) (a + b) c.
Note. The answers to the following problems are to be left
in a surd form.
5. The sides of a square are of length a. What is the
length of the diagonal ?
6. Bisect an equilateral triangle by a perpendicular drawn
from one of the vertices to the opposite side. Call the length
of half the base a. What is the length of the perpendicular ?
46
EXERCISE VIII 47
7. One of the adjacent sides of a rectangle is three times
the other. Call the length of the smaller a. Give an expres-
sion for the length of the diagonal.
8. A ladder of length la is placed against a house with its
foot 3& away from the wall. How high is the top of the
ladder above the ground ?
9. (i) A spider comes out of a hole in the corner of a room, runs
6 feet along the foot of the wall, then climbs vertically up
the wall which is 10 feet high, and having reached the
ceiling crawls 7 feet out along it at right angles to the
wall. How far is it from the hole ?
(ii) Give a formula for the distance in all similar cases, calling
the three movements a, 6, and c.
(\\\) If he spins a thread and descends from the ceiling a dis-
tance h, what is now his distance from the hole ?
10. An aeroplane is above my head at a height of a feet.
It flies due north and upwards until it is over a point p feet
away and at a height of b feet. It then flies eastwards at the
same level a distance of q feet. How far is it now from me ?
11. A carriage wheel of radius r picks up a piece of paper
from the road and carries it round one-twelfth of a revolution.
How high is it now above the ground?
How high will it be when it has gone A of the way round ?
EXERCISE IX.
APPROXIMATION-FORMULAE (I).
A.
1. Draw a square and call the length of its side a. Convert
it into a square of side a + b by the addition of two identical
rectangles and a square. Write inside each figure the ex-
pression for its area.
2. Use the figure *of No. 1 to complete the identity
(a + b)^ =
3. (i) Calculate the area of each of the rectangles and of the
square required to convert a square measuring 10 inches
each way into one measuring 12 inches each way. Show
that the resulting square has the correct area.
(ii) Repeat the calculations, substituting the measurements 3'6
cms. and 4 cms.
(iii) Repeat, substituting the measurements 14 cms. and 14'7
cms.
4. In 3 (i) what fraction of the side of the completed
square (12 inches) is the added portion (2 inches) ? What
fraction of the area of the complete square (144 square inches)
is the area of the added square (4 square inches) ?
Answer the same questions with regard to 3 (ii) and 3 (iii).
What do you notice about the two sets of fractions? Will
this property always hold good ?
5. I want to cover with linoleum a room 12 feet square.
I have a square of linoleum measuring 11 feet each way and
also a strip 22 feet long and 1 foot wide. What fraction of
the whole floor space must I leave uncovered ?
6. A square shed occupies the corner of a square play-
ground. The length of its wall is one-twentieth of the length
of the playground. What fraction of the area of the play-
ground does it cover ?
7. A square of length a is converted into one of length
a -f b by adding a "gnomon " of width b. Compared with
a , b is so small that the square b 2 may be neglected. What
48
EXERCISE IX 49
formula for the increased area of the square may in this case
be used instead of A = (a + 6) 2 ? Throw it into the form
most convenient for calculation.
Note. A calculation in which you take account of a rela-
tively small number b but neglect its square is said to be
carried to a first approximation. The word "approxi-
mately " when used in this and the next exercise means
"to a first approximation ".
In writing a statement of an approximation, whether in
symbols or in numbers, you should use, instead of the sign
" =," the sign "==". This symbol means "is approximately
equal to ".
8. (i) A square lawn 40 feet long has a path 3 feet wide along two
sides. Use the formula of No. 7 to calculate approxi-
mately the area of the garden.
Repeat the calculations, replacing the former dimensions
(ii) by 53 feet and 3J feet ;
(iii) by 43 feet 6 inches and 3 feet 3 inches respectively.
9. (i) A square lawn 50 feet long has a path 2 feet wide all round
it. Calculate the area of the garden to a first approxi-
mation. About what fraction of the whole area is
neglected ?
(ii) Repeat the calculation, replacing the former dimensions by
77 feet and 3 J feet respectively.
10. A metal tray is to be made a inches square with
vertical sides b inches deep abutting directly on to one another
without overlapping. The metal plate weighs w ounces per
square inch. Write a formula (convenient for calculation)
giving the total weight (W) of the tray.
Write also a formula for the percentage waste of metal
when the tray is cut out of a square sheet measuring a -f 26
inches each way.
11. A square mirror measuring 17 inches each way is
mounted in a frame f inch wide. Find approximately the
area of wall covered by the mirror and frame.
12. (i) Write down a, formula for the exact increase in area in the
square of No. 7.
(ii) Write a formula which will give the increase to a first
approximation.
13. (i) A square sheet of paper measuring 24 cms. in the side
stretches 3 mms. each way when wetted. Find approxi-
mately the increase of its area,
(ii) About what fraction of the whole area is being neglected ?
50 ALGEBRA
(iii) What fraction of the increase of area is being
neglected ?
(iv) Which of those fractions would you give if you were asked
how exact your answer is ?
14. Draw (or, if it is more convenient, draw to half-scale)
a rectangle 3 cms. high and equal to the total increase of area
of the wetted paper in No. 13. Mark off a square equal to
the part neglected in taking the first approximation. (The
diagram will help you to realize its relative unimportance.)
15. The number 10 2 can be regarded as (9 + I) 2 , (7 + 3) 2 ,
(5 4- 5) 2 , (4 + 6) 2 , etc. Show that when the identity of No. 2
is applied to these expressions the result is 100 in each case.
Choose another example of your own and find whether the
identity works when applied to it.
16. Take any one of the cases in No. 15 and show by an-
alysis why the identity holds good. Describe the steps of
your analysis in symbols. What use do you now consider
that you are entitled to make of the identity ?
17. A piece of thin brass tube has an internal radius r and
a thickness t. Write down a formula for calculating to a
first approximation the area (A) of the ring of metal.
18. A metal ball has a thickness t and an internal radius r.
Calculate approximately the area (A) of its external surface.
How much greater is it than the internal surface ? [The area
of a sphere of radius r is given by the formula A = 4?rr 2 .]
19. A boiler consists of a cylinder of length I and radius r,
capped at each end by a hemisphere. The thickness of the
metal is t. Show that the area of the outside surface is given
approximately by the formula
A - 27r{(2r + 1) (r + t) +
20. Complete the following identities :
(i) (a + 2b) 2 - ; (ii) ( 3 p + q) -
(iii) (p + jq) 2 = ; (iv) (a + v/sb) 2
(v) (i
i + I) - ; (vi) (i + o-oo 3 t) 2 - ;
(vii) (i + ct) 2 = ; (viii) (a + pb) 2 = ;
(ix) {l(a + b)} 2 - ; (x) {r(i + ipt)} 2 - .
21. Suppose a straight line 1 unit long to be drawn any-
where on a sheet of metal, and let the metal then be heated
till it is one degree warmer. The line will increase in length
by a definite but very small amount c (called the " coefficient
EXERCISE IX 51
of linear expansion "). If it is warmed 2, unit length in-
creases 2c, and so on. Write formulae for calculating :
(i) The length (I) of an original unit when the metal is warmed
t degrees,
(ii) The length (If) of a line originally 4 units long when the
metal is warmed t degrees,
(iii) The same, the original length having been I*
22. A unit square is marked out on a sheet of metal which
is then warmed t degrees. Find, to a first approximation :
(i) The area (A,) of the original unit square ;
(ii) The area (A,) of a square which originally measured I
units each way ;
(iii) The area of a piece of metal originally containing A square
units.
23. Find to a first approximation the increase of area (I t ) of
a circular disc of metal of radius r when it is warmed t, the
coefficient of linear expansion being c.
Use your formula to find the increase in area of a metal
disc of radius 21-7 cms., when warmed 50, the coefficient of
expansion being 0*00006.
B.
24. Suppose you want to know approximately only the
length of the side of a square containing 20 square cms. It is
obviously between 4 cms. and 5 cms. If you take a square of
16 square cms. away from the original the residue will have
an area of 20 16 = 4 sq. cms. Arrange it as a long strip.
In calculating its height to a first approximation what will you
ignore ? What then is the approximate height of the strip ?
The approximate length of "the side of the square ? Find out
how far the square of your result differs from 20.
25. Find approximately (i.e. to one decimal place or to two
if the second is a 5) the square roots of the following
numbers : 10, 18, 54, 110, 175, 27, 410, 17-6, 83-5.
26. Calculate approximately the radii of the circles whose
areas are (i) 13 square cms. ; (ii) 40 square inches ; (iii) 125
square feet.
27. Write down identities by means of which the values of
the following expressions may be calculated approximately.
(The second quantity mentioned is in each expression con-
siderably smaller than the first) :
4*
52 ALGEBRA
(i) Vl
(iii) V
(V) 2N
i' + p-
(a + b 2 ) -
*
5
>
(ii) V(a 2 + 2p)
(iv) \/r' J + i =
/p' 2 + q' =
(vii)
(vi) Vi6 + a =
x/8i
+ nt = .
28. Deal similarly with the following expressions :
(i) \/i3 + 2a = ; (ii) ^24 + p = ;
(iii) Va + b = (AI) ; \/p + 2q =
(v) V(a + 2 v /b)^= ; (vi)
(vii) V(9a -r 6\/b) = ; (viii)
29. (i) You start from a point H foot above the sea and climb a
further height of h feet. Write a formula for D, the
radius (in miles) of the circle of sea visible from the
higher point (See Ex. IV, No. 6 )
(ii) Suppose that h is small compared with H. Write a formula
suitable for calculating D when \/H is known.
30. A boy standing on a cliff 400 feet above the sea climbs
a tree 20 feet high so as to see as far as possible, (i) What
is the greatest distance visible from the higher point? (ii)
What additional distance was he able to see by climbing the
tree? Write a formula for d the additional distance rendered
visible by climbing h feet up from a point H feet above the
sea h being small compared with H,
EXERCISE X.
APPBOXIMATION-FOBMUL^E (II).
1. Draw a square measuring a each way. Mark off in one
corner a smaller square measuring a - b each way. Indicate
in the gnomon two rectangles each of area ab. Suppose that
you begin to reduce the larger square to the smaller square by
taking away one of the rectangles ab ; what must you do before
you can take away the other?
Complete the identity (a - &) 2 ~
2. The length of the side of a square is reduced from a to
a - 6. Give a formula for calculating approximately the area
of the reduced square when b is small compared with a.
3. The outside measurement of a square picture frame is
26*4 inches. The frame is 0*7 inches deep. Use the formula
of No. 2 to find the approximate area of the picture.
4. A square sheet of paper measures 31*2 cms. each way
when wet. In drying it shrinks 6 mms. each way. Find to
a first approximation its area when dry.
Note. Imagine the following. There are a number of piles
of atlases on a shelf in this room, each pile containing nine.
You are asked to carry 20 atlases into a neighbouring room
and place them in piles on a table. By mistake you take
away 4 complete piles, i.e. 9x4 atlases. Noticing your
mistake you carry back 4 from each pile, i.e. 4 x 4, and put
them on the shelf again. Thus you have removed altogether
just as many as if you had originally taken 5x4, i.e. (9 4)
x 4 atlases. It is clear, then, that
the total no. on shelf \ __ / the same no.
- (9 - 4) x 4 J~\-9x4 + 4x4. .
5. Show that the identity of No. 1 holds good of all numbers
by first considering (9 - 4) (9 - 4) and then analysing the
steps by symbols.
63
54 ALGEBRA
6.
Complete
the identities :
(i) (a -
2b) a = ;
(ii)
(2
- b) a =
(iii) (p - q') a = ;
(v) (I - 2 P *) 2 = ;
(vii) A(x - ct) 2 = ;
(ix) {a(i - bt a )} a - ;
(iv)
(vi)
(viii)
(x)
(i
^i
(r 2
- ct) 2 =
i - ct)}
- ^ab) 2 =
7. The outer radius of a thin metal organ pipe is r inches,
the thickness t inches ; write a formula for calculating ap-
proximately the volume (V) of the metal employed, considering
the pipe as a cylinder of length I feet. Write also a formula
for the cost (0) in shillings of the material employed, if 1 Ib.
contains v c. inches and costs c pence.
8. A sheet of metal of area A is cooled t, the coefficient
of expansion being c. Give an approximation-formula (i) for
its area after cooling ; (ii) for d, the decrease in its area.
(Compare Ex. IX, No. 22.)
9. Write a formula for the area of the total surface of a closed
metal cone, the radius being r and the slant height I. Find
approximately how much it decreases when the cone is cooled
t t the coefficient of expansion being c.
10. Show by the figure of No. 1 that *Ja 2 - p = a - ^
Aa
approximately. (Compare Ex. IX, No. 24.)
11. Write down approximate equivalents to the following
expressions :
(i) V 9 a 2 - b ; (ii) 4*3 x/(p 2 - J) ;
(iii) V(i6 - 2'it 3 ) ; (iv)_ Va' 2 - b 2 ;
(v) x/(a - 4 \/b).
12. The bob of a pendulum of length a is pulled out a small
distance b from the vertical. Write a formula for calculating
approximately how much the bob rises.
Apply your formula to the case of a pendulum 2 feet
long pulled 4 inches out of the vertical.
EXEECISB XL
APPROXIMATION-FORMULA (III).
A.
1. What formula would you use to calculate (a + b) 3 to a
first approximation, b being small compared with a ?
2. A cubical biscuit tin measures 10 inches each way inside.
The metal is -% inch thick Calculate the volume of the
metal exclusive of the solder and the side pieces of the lid.
3. An inch cube is heated t. Find approximately its new
volume, the coefficient of linear expansion being c.
4. Write a formula for calculating approximately the in-
crease in volume, I, of a sphere of radius r when it is heated
4
r, the coefficient of linear expansion being c. [V = o 77 *?* 3 .]
5. Show by the use of symbols that the identity (a + b) 3 =
a 3 + 3a z b + 3aZ> 2 + b' 6 can always be used.
6. Complete the following identities sufficiently for approxi-
mate calculation the second number described being small
compared with the first :
(i) (a + 2b)* = ; (ii) (2a + b) 8 = ;
(iii) (i + |) 3 - ; (iv) (a + pt)"' = .
7. (i) Complete the identity (a 2 - 2ab + b 2 )a =
(ii) Complete the identity (a 3 - 2ab + b 2 )b =
(iii) Complete the identity (a 3 - 2ab + b 2 ) (a - b)
(Bead the note before Ex. X, No. 5.)
8. How could the cube model have been modified so as to
show the existence of the identity
(a -6)3 = a 3 - 3a*b
9. Complete the following identities sufficiently for approxi-
mate calculations the second number being small compared
with the first :
(iii) (i - 2- 3 k) 3 = j (iv) (a - *f) - ,
55
56 ALGEBRA
10. Show by the model how to find approximately the cube
root of a number which does not differ much from a number
the cube root of which is already known.
11. Complete for approximate computation the identities
(i) J/iFTd - ; (ii) 4/aT^d = .
12. Express each of the following numbers in one of the
forms of No. 11 and find its cube root to a first approximation :
10, 30, 60, 1030, 322, 0-528, 0-023, 1-674.
B.
13. An inch-cube of metal is warmed t ; find its new
volume, the (very small) coefficient of linear expansion being c.
Also find its volume after being cooled t.
14. A sphere of the metal of No. 13 has a radius r. Find
the increase in its volume when it is warmed t.
15. A cube of metal measuring 1 inch each way is heated
until its edges are 1-002 inch long. What is now (i) the area
of each of its faces ; (ii) its volume ?
16. A cube of metal measuring 1 inch each way is cooled
until its volume is reduced to 0*991 c. inch. What is now (i)
the length of an edge ; (ii) the area of a face ?
17. The volume of a piece of metal at different tempera-
tures is given by the formula V = V (1 -f 0-000182), V being
its volume at 0. Write down formulae connecting (i) the
length, L, of a rod of the metal with L its length at 0, and
(ii) the area, A, of a plate of the metal with A its area at 0.
18. Given that 3/3" = T44, s/5_= l-TVfind _the value of
3/24, 3/40, */81, /320, 3/15, /25, 3/120, */45.
19. Express the following in the form most suitable for
computation :
yi6, ^108, ^2160, 3/9000, 3/1-6, 3/0-648, 3/0'0875.
20. The following expressions describe certain combinations
of numbers symbolized by the letters a, 6, etc. Write down
expressions which indicate the easiest way of finding approxi-
mately the cube root of each kind of combination : a*b, ab z ,
dW, a*b, ab*, a*b*, a 2 6 2 c 3 , a 3 6 3 c 2 , a 5 b*c, dWc\ 27p*r, lOSrrm 3 ,
8(p - q), 16(a 2 - 6 2 ), 64p 3 (1 - ) f (a 4 6 + a 3 ), (p 3 - p*q*) 9
(a 4 6 3 - a 3 6 4 ).
EXEECISE XII.
FRACTIONS (I).
1. Convert the following formulae (in which the subject is
1/K) into formulae in which the subject is E :
/\ JL _ l _ x
W R ~ p q'
..... I 2,3
( 1IJ ) R = p + q*
y _ 3 5
V lv ) R p q'
i p
( v ) R J ~ q'
_L P
(Vi) R = q - *
<vii)R- = ? + '-
^n; R q
i q
(viii) R = r - -.
C ) - - P + J
1 ap I'
W R-^-?
jc_ 2q
(xi) R ~~ "p 3P-
/ -N x J
(xn) R = - - pq.
;. The formulas of Nos. 2 to 5 are to be thrown into
the shape most convenient for substitution.
(If the numerator or denominator of a fraction is in the form
ab + ac, or a 2 Z? 2 will it be more convenient to leave it in
that form or to factorize it ?)
67
68 ALGEBRA
(iii) P - p - 23.
(vi> - - -,
<*> i - 1 - "
/ ... x 2b
(vin) A = ab H -- .
v " "
ii) p = JL _ 3.
y pq p
- P p q pq
(i) V - -i-.
- + -
U V
(ii) v = 2 /(! - I),
(iii) v - ^
EXERCISE XII 69
6. (i) V = uv(j + I).
(ii) V = u(i - I).
(v) V = ^ ^.
U V
(vi) V - (u 4- i)/(i + [).
(vii) V = (u - v)/(i - ?).
i i i^
t .... .. u v uv
(vm) V = .
V ' U + V - I
u +
(!) V = (l - J)/(U + V).
(xi) V = (u - v)/(| - i).
Find simpler expressions by which the following may be
replaced when they occur in formulae :
/\ J x
< n) w - a-
\ -
W be ab*
.. v i 2 i
60 ALGEBRA
7 - <*> 3? ~ ^ -
<) I +
/...v m n
(in) .
an am
r \ JL _ 2
p a pqab
W 3 T - &
8. (i) Two rectangular rooms are thrown into one room. The area
of one of the rooms is A lf of the other A 2 . The breadth
of the first room is bi, of the second 6. 2 . Find the length
of the combined rooms ;
(ii) The same, given thai the area of the first is n times the
irea of the second room (A).
9. A man has a number of envelopes to address. The
work would take him 5 hours. He engages the help of a boy
who could address the whole of the envelopes in 7 hours.
How would you find how long they would take together ?
(i) Write a formula for calculating most easily the joint time
(T), given the time taken by the man (m) and the boy (b)
respectively when working alone.
(ii) Write a formula for the joint time if there are 2 men and
3 boys at work on the addressing.
(iii) Find the joint time taken by a man and a boy when the
boy, working alone, would take n times as long as the man
(T, n, m).
(iv) Find the joint time taken by a man and a boy given that
the man, working alone, could do the work in I/nth of the
time taken by the boy (T, n, 6).
(v) Find formulae for the joint time in each of the last two
cases supposing that there are p men and q boys engaged
on the work.
10. (i) When p equal marbles are dropped into a cylindrical jar of
water the level of the water rises a height a. When q
equal marbles of another size are dropped into an identi-
cal jar the change of level is b. How many times (n)
is one of the former marbles as large as one of the latter ?
(ii) The same, supposing that the area of the water surface in
the second jar is 3 times as great as it is in the first ;
(iii) The same, the radius of the first jar being 3 times as great
as that of the second ;
EXERCISE XII 61
(iv) The same, s times the sectional area of the first jar being
the same as t times the sectional area of the second ;
(v) The same, s times the diameter of the first being equal to t
times the diameter of the second jar.
11. A bath has two taps. One is found to deliver a gallons
of water in_p minutes, the other b gallons in q minutes. Write
formulae for calculating :
(i) The quantity (V) which is delivered into the bath in t
minutes when both taps are running ;
(ii) The quantity delivered in a time which is n times q
minutes ;
(iii) The time (T) which the first tap alone would take to run in
Q gallons ;
(iv) The time which both taps together would take to run in Q
gallons ;
(v) The time necessary to increase the quantity in the bath from
Qo to Q gallons by the second tap only ;
(vi) The same when both taps are running.
12. A boat is rowed upon a lake for a distance of 1/p of a
mile from east to west and tben for 1/q of a mile to the north.
How far is it from its starting-point ?
13. A boy rows a boat for 1/p of a mile towards tbe east.
Then turning round, he rows for 1/q of a mile in a direction
between west and north until he is exactly due north of his
starting-point. How far away is the starting-point ?
14. Two cyclists ride together round a circular track, one
along the outside edge where the radius is E, the other along
the inside edge where the radius is r. A single revolution
of the pedals carries the former's bicycle forward P feet, and
the latter's p feet. Write down expressions for calculating :
(i) The difference (d) between the number of pedal-revolutions
made by the cyclists in going n times round the track ;
(ii) How many times (t) the pedal-revolutions of the former will
be as numerous as the pedal-revolutions of the latter.
15. Two boys, X and Y, find that they can cover 100 yards
in p paces and q paces respectively. X takes m paces in walk-
ing from the centre to the outer circumference of the track of
No. 14 ; Y takes n paces in walking from the centre to the
inner circumference. Write formulae for :
(i) The width (w) of the track in feet ;
(ii) The difference (d) in yards between the outer and inner
circumferences ;
62 ALGEBRA
(iii) The difference (d) in miles between 200 revolutions along
the outer and 200 revolutions along the inner circumfer-
ence ;
(iv) The area (A) of the track in square feet ;
(v) The cost (C) in pounds of preparing the track at c pence
per square yard.
10. (i) V c.cms. of water are poured from a narrow cylindrical jar
into a wider one, the areas of cross-section being re-
spectively A! and A 2 . Find how many centimetres (d)
the level of the water sinks ;
(ii) The same, given that the radii of the jars are r t and r a re
spectively.
17. (i) In the preceding question if, in pouring the water from on-
jar to the other, 1/p of it was spilt, what fraction would
remain ?
(ii) To what height would it fill a jar of sectional area A ?
(iii) To what height would it fill a jar of radius r 1
(iv) What would be the answers to (ii) and (iii) if q/p of the
water were spilt in transference ?
18. V c.cms. of water are poured from a jar of sectional area
A into a jug, and then poured back again into che jar. At
each pouring q/pihs of the water are spilt. Show how to
calculate :
(i) How much water (v) is returned to the jar ;
(ii) How much lower (d) the level is than at first.
19. Instead of being poured back into the original jar the
water is passed into a second cylinder of smaller sectional
area, a, g/^ths being lost at each passing as before. Find
how much higher (d) the water is here than it was in the
original cylinder.
20. (i) Taking the data of No. 19, find how many times (n) the
water in the second cylinder is as high as it was in the
first cylinder ;
(ii) Answer the same question, given that the radii of the
cylinders are R and r.
EXEKCISB XIII,
FRACTIONS (II).
A.
1. Complete the following identities with a view to easier
evaluation :
(i) 16 - (a + 7) - ; (ii) x6 - (a - 7) - ;
(iii) 3a - (2a + 5) = ; (iv) 5p - (sp - 8) = ;
(v) 5P + (3P - 8) = ; (vi) 24 + (18 - r) = ;
(vii) 24 - (18 - r) = ; (viii) 24r + (18 - r) = ;
(ix) 24r - (18 - 6r) = ; (x) ( 3 p + 4 b) - (2p - 4b) - ;
(xi) a 2 - (p - 3 a 2 ) - J (xii) (xom' + p) - ( m + 5 p') - ;
(xiii) 13 - (18 - 3r) = ; (xiv) ap - (sp - 20) = .
2. Write in full the explanations of your answers to No. 1
(ii), (vi), and (xiii) which you would give to a person who did
not know how the results were obtained.
3. What restrictions are there upon the values of numbers
that can be represented by the symbols in No. 1 (i), (ii), (vi),
(xi), (xiii)?
4. Simplify the following expressions :
(i) 14 - 2(7 - 3p). (ii) 19 - 3(5 - t).
(iii) 14! - 4(3! + 2). (iv) 3(a - 2b) + 2(a + 3b).
(v) 3(a - 2b) - 2(a + 3b). (vi) 5(2? - 3q) - 3(4? - 7q).
(vii) 6(3m - 2n) - 7(4111 - 5n). i(viii) a(3p -t- 4) - 2ap.
(ix) a( 3 p + 4) - Sap. (x) a(4 + b) - b(a - b).
(xi) ^
(xii) 2(7 V2 - 4\/3) - 4(3^2 - 2N/3).
5. What restrictions are there upon the values of the
symbols in No. 4 (v), (vi), (vii) ?
6. Write full explanations of the answers which you give to
No. 4 (vii) and (x).
7. (i) Owing to a decrease in the pressure a tap which usually
delivers q gallons of water per minute into a tank actually
supplies 8 gallons a minute less. Give the best formula
for calculating how much longer (t) it will take to run in
100 gallons.
68
64: ALGEBRA
(ii) Adapt the formula to suit any given decrease of 'supply per
minute (d) and any quantity of water to be supplied (Q).
8. When the tire is fully inflated the rear wheel of a bicycle
has a radius of 14 inches. Find a formula for the extra
number of revolutions per mile (ri) when, owing to a leakage
of air, the radius is reduced by a given amount (d).
9. In consequence of an increase in the duty on tobacco
the retail price of cigarettes is to be raised by a penny the
ounce packet, regardless of the quality of the cigarettes.
A certain smoker allows himself 3 per annum for cigar-
ettes. How many ounce packets fewer must he buy in a
year if he still purchases the same brand ? Let p^" the old
price of a packet in pence ".
10. The smoker of No. 9 determines to increase by 7s. 6d.
his annual allowance for cigarettes, but the larger sum does not
yet permit him to buy as much tobacco as he formerly pur-
chased for 3. Find how many packets short he is in the
course of a year.
11. Throw the following formulae into the shape most suit-
able for computation :
12. Simplify the following fractional expressions with a view
to computation :
i
(i) - - L (ii)
a a + 2b 3P - q 3P
(iii) 2 * (iv) ,~S_ - 3.
v ' 5a 5a + b v ; 3 - p q
EXERCISE XIII 65
13. A dishonest tradesman measures off carpet for sale with
a yard-measure which is really s inches short. Of how many
pounds has he defrauded his customers by the time he has
sold at a price of p shillings a yard a roll of carpet which
really contains I yards ?
14. A metal yard-measure is of the correct length at a
temperature of 60 F. Obtain expressions for the error in-
volved in using it to measure a length which is really I yards
when the temperature is (i) t above 60 ; (ii) t below 60.
(The coefficient of linear expansion of the metal is c.)
B.
15. Show that ^-i = 1 + ^ ^
and that
1 - a I - a
Hence show that = 1 + a +
1 - a 1 - a
16. Obtain a similar expression for -= "
17. If a 2 is so much smaller than a that it may be
neglected, what expressions may be substituted for
and = respectively ?
18. Complete the following identities for the purpose of
approximate calculation (the second member of each denomin-
ator is very small compared with the first) :
/\ r /"\ d
rr-p = ; ("> FTTt " ;
(iii) d ^ .. j _x =
v ' i -f <rooo6t ' x _ 5
a
I + -
a
a + b
a
66 ALGEBRA
19. Write formulae for the approximate solution of (i) No. 8,
(ii) No. 14.
20. On the morning of a certain day Consols can be bought
for 80 per 100 stock ; at the end of the day the price has
risen to 80. Find approximately the difference in the
amount of stock purchasable for 3200.
21. On account of a collision between two of their trains
the stock of a railway company falls from 102 to lOlf per
100. What difference, approximately, does this fall make
to the amount of stock that can be bought for 5100 ?
22. At a temperature of 60 the length of a rail is I feet,
but in laying the railway line space must be left between the
rails to allow for the expansion due to a possible rise of tem-
perature of (say) 40. Find 1 what difference this allowance
makes to the number of rails used in laying down a line
L miles long, the coefficient of linear expansion of the rails
being c.
23. A steamer is timed to cover a certain journey of m miles
at an average speed of s mls./hr. This speed is subject,
according to the direction of the wind, to a relatively small
change of d mls./hr. Find a formula for the difference in
hours between the longest and the shortest journey.
24. In each of the following expressions the second member
of a denominator is very small in comparison with the first.
Write down a series of expressions which are equivalent to
the given ones for the purpose of approximate calculation :
a + bt c + dt"
I
IOO - ct)
EXERCISE XIII 67
.3
(xvii)
r 3 (i + ct) vxvm; ( a _ b) *-
/ \ * I *
(xix)
xx a ~ ^(a 3 + IT)'
25. A number of brass cylinders are to be made of exactly
the same volume, V, and of approximately the same radius, r.
It is found, when they are finished, that the greatest and least
radii differ from r only by a small quantity a. Find h, the
difference in height between the longest and the shortest of
the cylinders.
26. The same, when the volume of the cylinder may in
extreme cases be either greater or less than V by the small
amount v.
27. The diameter of the cylinder of a marine engine is D
inches. The formula of Ex. IV, No. 31, shows how to calcu-
late the loss of steam pressure (P) due to friction of the piston
against the cylinder. Obtain a formula for the difference in
the loss of pressure (p) when the diameter of the cylinder is
increased by an amount h inches, small compared with the
whole diameter.
Find p when D = 81 inches and h 1 inch.
28. Obtain a similar formula for p when the diameter of
the cylinder is decreased by h.
29. Write a formula for N, the number of times a pendulum
of length I swings in T seconds. (See Ex. VII, No. 11.)
The pendulum is made shorter by a small amount h.
Write a formula by which to calculate approximately the
additional number (n) of swings in T seconds.
30. Obtain a formula for the case in which the pendulum's
length is increased instead of being decreased.
31. Obtain a formula for calculating = to a second
approximation.
i
32. Obtain a similar formula for
1 + a
33. Obtain similar formula for
(0 r-^K* (">
b viiv rr^t*
(The second member of each denominator is small compared
with the first.)
68
ALGEBRA
0.
34. Express the following in forms more convenient for
substitution :
(i)
(ii)
(iii)
(iv)
w
(Ti)
(vii)
(Tin)
P =
P =
P =
i
i
i
P
i
2
P
i
3*
p
i
2
P
i
3
P
i
a
P
i
b*
P
i
P
i
b
r
P
i
a"
P
i
p
i
a '
P
+
a
i
P
Q =
Q =
I
2p
i
a
2p
i
+ a
n
i
a
m
i
- a
n
I
a
m
a
i
Q
i
qn -f
m
b
pm
n
+ a
Q
q - m
P
-
n
Q q - a p - a'
2m - 311 301 - 2n*
x ' 2m - 3n 3m - 2n
/ \ I pa qb
(xlv) M - r^ia - r^b-
35. Reduce each of the following to a single equivalent
fractional expression :
(i) a~Tb + (a + b) 2 '
(ii) '
a + b (a + b) a *
a + b
- b-
EXERCISE XIII
69
a + b "*" a - b "*" a 2 - b 1 '
a + b a - b a 2 - b 2 '
(xviii)
36- Simplify the following algebraic fractions :
- b a (a + b) 2 a + b'
a a - b a "*" (a + b) a a + b'
a 2 - b 2 (a - b) a a - b*
a + b (a + b) 3 -
/vii^> l + a + b 2ab
(V11) r^b + (Tn^ + (a-T 5?.
/ ""v i _ a ~ b _ 2ab
(V111) a + b " (a + b) 2 ~ (TTTb) 1 '
EXEECISE XIV.
CHANGING THE SUBJECT OF A FORMULA (I).
A.
Note. Whenever your answer to a problem is a number
you should test its correctness.
1. I am thinking of a number. I multiply it by 3 '6 and
add 14-9. The result is 23'18. What is the number?
2. I am thinking of a number. I multiply it by 3f and
subtract 1 from the product. The remainder is J. Find the
number.
3. I am thinking of a number. I add to it 2*3 and multiply
the total by 8*9. The result is 133-5. What is the number?
4. I am thinking of a number. I subtract from it f and
multiply the remainder by -|. The result is -J-. What is the
number?
5. I am thinking of a number. I subtract 4 -32 from it
and divide the residue by 3*24. The quotient is 4. Find the
number.
6. I am thinking of a number. I divide it by 4*4 and add
7 '35 to the quotient. The result is 13'6. Calculate the
number.
7. Write down in symbols a statement of the various steps
followed in working out each of Nos. 1 to 6. Put n = " the
number thought of " and use a, b> c as symbols for the other
numbers, whether integers or fractions.
Note. In the following six examples (Nos. 8 to 13) you are
to calculate the number described either (1) mentally or (2)
by setting down the statement in symbols and applying to it
the rules by which a number may be moved from one side ol
the sign = to the other.
If you find the number mentally you must afterwards set
down the steps by which you found it, and see whether they
followed the rule.
8. I add 6 to twice a certain number and multiply the sum
by 7. The product is 84.
70
EXEKCISE XIV 71
9. Taking a certain number I diminish it by 5 and multiply
the residue by 9, obtaining 99 as the product.
10. I multiply a certain number by 5, subtract 4 and divide
the residue by 7. The quotient is 8.
11. A certain number is multiplied by 7, the product added
to 1 12, and the sum divided by 4. The result is 17.
12. I think of a number. I add seven to it and divide it
into 60. It goes 5 times.
13. I think of a number, and divide 72 by another number
which is 5 less than double the former number. The quotient
is 8.
14. Describe in symbols the steps followed in solving any
three of the last six examples. Put n for the number to be
calculated and a, 6, c, d for the other numbers mentioned.
Note. The following eight examples (Nos. 15 to 22) are
statements of " Think of a Number" problems. Express the
problems in words and solve them.
15. 3-4(7'ln - 12-8) = 53-04.
14-91
' (3-7w - 4-9) - >
18. 6(7 - 5) - 50 = 46.
20. i(4n - 9) + 18 = 25.
qq
22. - 8-3 = 6-5.
23. Describe in symbols the steps by which you could solve
any problems like (i) No. 17, (ii) No. 19, (iii) No. 21.
Write down in symbols the steps by which you would in
each case calculate the number (n) described in the following
eight examples (24 to 31).
24.
25.
72 ALGEBRA
26. ~(bn - c) - p = q.
27.
28.
29.
30.
31.
Note. The formulas of the following four examples are
all taken from Ex. III. After changing the subject of one
of these formulae you should in each case express in words
the meaning of the new formula derived from it, and make up
your mind whether it is true.
32. Write down the formula of Ex. Ill 1 (ii). Obtain from
it formulae (i) for the number of marbles in the bag ; (ii) for
the weight of a single marble.
33. Write down the formula of No. 1 (vi). Obtain from it
formulae (i) with t 1 as the subject ; (ii) with t 2 as the subject.
34. Write down the formula of No. 2 (ii). Change its
subject to t.
35. Write down the formula of No. 4 (ii). Obtain from it
formulas (i) with S , (ii) with n, (iii) with t as subject.
Note. The formulae of Nos. 36 to 41 are taken from Ex.
IV. Be prepared to explain the meaning of the formulae
you derive from them.
36. From the formula of No. 17 obtain formulae (i) for Z, (ii)
fork
Find the greatest length of wrought iron bar 4 inches broad
and 3 inches deep that can safely support a weight of half a ton
hung at the end of it.
37. The indicated horse-power of a marine engine is 150.
The diameter of the cylinder is 5 feet, the length of piston
stroke 4 feet 2 inches, the number of revolutions 30 per min.
EXERCISE XIV 73
Turn the formula of No. 33 into one from which the mean
steam pressure can be calculated.
38. Derive from the formula of No. 4 a formula which will
enable you to calculate the diameter of the chain-iron appropri-
ate to a given load. Use it to calculate the diameter necessary
for lifting 10 tons.
39. Change the subject of the formula of No. 5 from p to L.
What does the new formula tell you ?
40. Change the subjects of No. 25 (i) and (ii) to L Do the
results agree ? What length of span is possible in the case of
a wire weighing 0'12 lb./ ft., if it may not be stretched with a
force greater than 256 Ibs. wt. nor allowed to dip more than
8 inches.
41. Change the subject (i) of No. 6 to h ; (ii) of No. 22 to
d; (iii) of No. 31 to D.
Note. The formulae of Nos. 42 to 47 are taken from Ex.
III.
42. Write the formula of No. 3 (ii). Change its subject
to i.
43. From the formula of No. 15 (ii) obtain formulae with
(i) w v (ii) n as subject.
44. Change the subject of the formula of No. 28 (iv) from
d to (i) d , (ii) t, (iii) s r
45. Find a formula for the total rainfall (R) during the years
between two given dates (d 1 and d 2 ) given the monthly average
M-
46. Derive a formula for R from No. 30. Does it agree
with the one you have just found ?
47. Take the formula of No. 32 (i) and change its subject to
(i) T, (ii) t, (iii) d g .
Note. The formulae of Nos. 48 to 50 are taken from Ex. IV.
48. Change the formula of No. 24 into one by which you
may calculate the height from which the rain should fall,
given its weight, etc.
49. From the formula of No. 23 derive a rule for finding
the weight that can safely be lifted by a crane hook, given the
thickness of the shank.
50. In No. 25 show (i) that formula (iv) could be derived
from (iii) ; (ii) that formula (iii) could be derived from (iv).
EXEKCISE XV.
CHANGING THE SUBJECT OF A FORMULA (II).
A.
1. A certain number is multiplied by 2*3 and the product
is taken from 12 -1. The residue is 4*28. Find the number.
2. A humorist when asked his age replied, " Multiply my
age by 6, subtract the product from 840, and divide the residue
into 1899. Add 7 to the quotient and the answer will be 10."
What was his age?
3. Asked the same question next day by another person, he
replied, " Take 10 times my age away from 840 and divide
the difference into 1530. Subtract the quotient from 10 and
you will have 7 left." Does this answer agree with the
former ?
4. Divide a certain number by 26, subtract the quotient
from 8-2, multiply the difference by 12 and you will obtain
20'4. What is the number ?
5. What number would you obtain as the answer to No. 4
if, by mistake, you thought that you were to subtract 8 -2 from
the quotient by 26 ?
6. I take 4 times a certain number from 28*6 and subtract
the result from 11*3. The residue is 3 '5. Find the number.
7. If in doing No. 6 you had read " subtract three times
the result" for "subtract the result," what answer would
you have obtained ?
8. Describe in symbols the steps taken in solving any two
you please of Nos. 1 to 7.
Show how you would in each case find the number n de-
scribed in Nos. 9 to 13.
9.
10.
EXERCISE XV 75
11. a - p(b - en) = q.
12. a 2 + b(p - q*n) = c 2 .
13. (a 2 - b*n) = r 2 .
Note. The formulae in Nos. 14 to 17 are taken from Ex.
III. See that you understand the meaning of the formulae
which you derive from them.
14. Write the formula of No. 27 (iii) and change the sub-
ject to b.
15. From the formula of No. 28 (v) obtain a formula for
the speed of the slower car (s. 2 ).
16. From the formula of No. 32 (iii) derive a formula for
dy the second observed depth of the water.
17. Derive a formula for d< 2 from that of No. 32 (iv).
The formulas in Nos. 18 to 21 are taken from Ex. IV. Bo
ready to explain the meaning of the formulae you derive from
them.
18. In No. 3 obtain a formula for T.
19. Obtain from No. 15 a formula by which to solve the
following problem : A person whose eye is 5 feet 6 inches
above the ground sets up a pole 12 feet high and on stand-
ing back 14 feet can just see over the end of the pole the
top of a church spire 180 feet high. He now moves the
pole 80 feet nearer to the church. Where must he stand
so as to see the spire as before ?
20. Change the subject of the formula of No. 29 (i) to t ;
(ii) to n.
21. Change the subject of No. 25 (iv) to L. Does the new
formula agree with 25 (iii) ?
22. The following formula is taken from the Pocket Book of
an electrical engineer. Derive from it a formula for l r
23. The same Pocket Book yields the following formula.
Change the subject to d.
B.
24. A certain number when multiplied by 6 gives the same
result as when it is multiplied by 4 and 14*3 is added to it.
What is the number ?
76 ALGEBRA
25. A certain number when multiplied by 3*4: is equal to
its product by 1*8 together with 6-4. Find the number.
26. What are the numbers described in the following
statements ?
(i) sn + 4 = n 4- 10.
(ii) 411 - 7 = ii - 2n.
(iii) 28 - 311 = 43 - 6n.
(iv) 7*3 + 5n = I2n - 7-4.
27. Describe in symbols the steps taken in solving problems
like (ii) and (iv) of No. 26.
28. The humorist of Nos. 2 and 3, when shown that his
answers did not agree, said, " I will tell you my real age. If
you multiply it by 2 or if you divide it by 3 and subtract the
quotient from 350 you get the same number ". What was
his age according to this statement ?
29. My age is three times my son's. If you multiply his
age by 5 and subtract 30 you will again have my age. How
old are we ?
30. A man who was asked by a foolish young sportsman
to sell his horse said, " I will let you have it either for four times
as much as I gave for it less 130 or for 220 less three
times what I gave for it. Choose which price you will pay.' 7
As a matter of fact the seller knew that the two prices were
the same. Find how much he gave for the horse and at what
price he was offering it for sale.
31. I am thinking of a number. Three times the residue
when it is subtracted from 81 is the same as four times the
residue when 60 is subtracted from it. What is the number?
32. Find the numbers referred to in the following state-
ments :
(i) 8n = 3(n + 20).
(ii) 7(n + 2) = 5(n + 8).
(iii) 7(n + 4) = n(n - 2).
(iv) 4(n + 3-4) = 3(14-1 - n).
(v) 8(20 - n) = 9(n - 3).
(vi) 4(311 - 4) = 7(n + 2).
(vii) 5(211 + 5) - 7(411 - 17).
(viii) 5(4 - an) - 6(5 - 40-
33. Describe in symbols the steps by which problems like
(i), (iii), (v), and (vii) of No. 32 can be solved.
34. X had a friend C who went to live in a road in which
X's friend A occupied No. 20 and his friend B No. 41. C
gave X a rule for remembering his number. Unfortunately
EXERCISE XV 77
X forgot whether the rule was " Three times the difference
between my number and A's is the same as four times the
difference between my number and B's," or whether the
words " three " and "four" should be interchanged. What
may C's number have been ?
35. A man learnt that a casket of jewels had been hidden
in a certain hedge and that twice its distance from one of two
trees growing out of the hedge was three times its distance
from the other. He found that the two were 60 yards apart.
Calculate the various possible positions of the jewel case and
show them in a diagram.
36. I am thinking of a number. One-fifth of the residue
when it is subtracted from 100 is the same as one-sixth of
the residue when 12 is subtracted from it. Find the number.
37. Find the numbers referred to in the following state-
ments :
d\ n " 2 - 5Q - n
(l) ~5~~~ 7 *
(ii) "-S-^- 1 .
(Hi\ *4 " 3n = 4" - 5
v ' 13 10 "
fiv) Si - an . 3^_5
38. Describe in symbols the steps to be taken in solving
problems like (i) and (iv) of No. 37.
39. A bought a house for 360 and a year later sold it to
B. When B left it he sold it to C for 480. B afterwards
boasted that his profit was half as large again as A's. At
what price did he sell the house?
40. What values of n will make the following statements
true ?
(i) ?(n + 7) - |(2n - 3).
(ii) 3(28 - S n) - f(4n - i).
4 5
(iii) 4(5n _ 3 ) . |( 8 i - n).
5^7 -i-
M 4 "7 _ 5.
w yn + 4 la'
78 ALGEBRA
(vi)
(vii)
(viii)
(ix)
to
23 - 3n
I
23 + 311
3
22
^ 7
n + 3
4
9
5 n 3
30 - n
2*4
3n - 9*5
2n - 47
39
4n ~~ 3
2n + 3
41. A cyclist A rides a distance of 28 miles in the time that
another cyclist B takes to ride 18 miles. My own usual rate
of riding is 3 miles an hour less than A's and 2 miles an hour
greater than B's. How fast do I ride ?
42. Find the values of n which satisfy the following state-
ments :
(i) 8(71 - n) = 5(n - 14) + i.
(ii) 4(2 - n) + 3(2 + n) = 13.
(iii) 7(2n - 3-2) - 4(147 - 411) - 2-5.
(iv) i in + 6*5 = 42*9 - 3(n - 12 '8).
(v) 4(28-5 - 311) - 3(2n - n -3) + 4-5 - 2(487 - 5n) - 3-4.
(vi) 5(2n - 7) = 28 - 2(2n - 7.
(viii) 6(3n - 4) + 5(7 - 211) - 12 6(7 - 2n) - 5(311 - 4) + 7.
43. I am thinking of a number. The difference between
one-third and one-sixth of the number is 5. What is the
number ?
44. Find a number the fourth part of which exceeds the
fifth part by 3.
45. Find the values of n which satisfy the following state-
ments :
H-*
(ii) ?n - 7 - ^n.
v ' 3 '4
, x n -f 19 n
(v) __j> = _ + 3 .
(vi) |(n - an) - i( 3 n - 2) - I
EXERCISE XV 79
- i(n- n) = ?(n + 4)
46. The following are general descriptions of statements of
the kinds given in the preceding examples. Exhibit the steps
by which the value of n could be determined in the case of
each form of statement :
p.
q
b
v pn - q p - qn
(iii) a(pn + q) - b(rn - s) = c.
(iv) - = - + c (a + b).
n n
(v) a(pn - q) = b(pn - q) + c.
, . x n - a n-b n - c
( Vl) __ + __ = __.
C.
47. From No. 46 (ii) derive formulae for calculating^ and q.
48. From No. 46 (iv) derive a formula for calculating a/b.
49. From No. 46 (v) derive a formula for calculating a - b.
Note. The formulae of Nos. 50-57 are taken from a book
on electricity.
50. Change the subject of the following formula to h :
c =
2 1 - h *
51. Change the subject of the following formulae from i
to V 2 :
52. Obtain formulas having h and a respectively as subject
from the formula
53. Obtain a formula for n fron; the statement :
80 ALGEBRA
54. Change the subject of the following formula to V x :
V - V /I IN
Q__ G _ 111 i *
K-/1 . I -j~~ f ~^~ 1
1 4:7T Wj
55. Find a formula for q, given that
56. Convert the following formula into one with r 2 as
subject :
r i ~ r 2
57. Obtain an expression for Y x from the formula
!_ _ CZ V
-p>
58. From the formula e = A ^ derive one with r as
subject.
59. From the mechanical formula Fs = W . ~- derive
one with -w as subject.
60. Change the subject of the following formula to r :
//I ~~ fJf)r o <i\
ev =* ^ l - A ^ 2 (a 2 - r 2 ).
EXEECISE XVI.
SUPPLEMENTARY EXAMPLES.
A. FORMULATION.
1. A certain sum is required to restore a church, and sub-
scriptions are coming in a constant monthly rate. Write
formulae to find :
(i) The amount still to be collected after a given number of
months (S, S , m, t) ;
(ii) The number of months after this before the fund will be
closed (T).
2. Given the area of a square plot of grass write a formula
for the length of the side of the plot (s, A).
An oblong plot of grass in the middle of a garden is to be
altered into a square by cutting off from its length and adding
to its breadth, its area being unchanged. Write formulae
for:
(i) The length of the side of the square, given the length and
breadth of the oblong (s, I, b) ;
(ii) The amount to be cut off from the length (p) ;
(iii) The amount to be added to the breadth (g).
3. Write down formulae for finding :
(i) The circumference of a circle, given its diameter ((7, d) ;
(ii) The circumference of a circle, given its radius (<7, r) ;
(iii) The radius of a circle, given its circumference ;
(iv) The radius of a circle, given the number of times that it
revolves in rolling through a certain distance (r, n, I) ;
(v) The number of times that a circle of given radius will
revolve in rolling a given distance ;
(vi) The area of a circle, given its radius (A 9 r) ;
(vii) The radius of the circle, given its area ;
(viii) The volume of a cylinder, given its radius and height (K, r,
fr) ;
(ix) The height of a cylinder, given its volume and radius ;
(x) The radius of a cylinder, given its volume and height.
4. (i) Give a formula for finding how much the surface of water
in a rectangular cistern will sink when a given number
81 6
82 ALGEBRA
of cubic feet of water are drawn off, the length and
breadth of the cistern being given in feet (/i, V, , 6) ;
(ii) The same, the cistern being cylindrical and of a given
radius (r).
(iii) A gallon of water occupies 0'16 cubic feet. Find how much
the water will sink in a rectangular cistern when the
quantity drawn off is measured in gallons ;
(iv) The same, the cistern being cylindrical.
6. (i) Two motor-cars start together on the same road but one
travels faster than the other. How far will they be apart
after a certain number of hours (d, s lt s 2 > ?
(ii) Write a formula that shall show how far the slower has
gone when the faster has gone a certain distance (d z , di).
(iii) Write a formula to show how far the slower car is behind
the faster when the latter has gone a certain distance
(A, d,).
(iv) Write another formula to show how far the faster is ahead
of the slower when the latter has gone a certain distance
(A, *)
6. Write down formulae corresponding to (i), (iii), and (iv)
of the preceding question supposing that the faster motor-car
starts a certain distance ahead of the slower (d Q ).
7. The " Lusitania" leaves New York to cross the Atlantic a
certain number of sea-miles behind an ordinary liner. Given
the number of knots (i.e. sea-miles per hour) at which each of
the steamers is travelling, write formulae to find :
(i) How many miles nearer the steamers will be after a given
number of hours than they were at starting (m, d , s x ,
**, ;
(ii) What will then be their distance apart (d) ;
(iii) How many hours it will take the " Lusitania " to catch the
other liner up (T) ;
What distance they will then be from New York (L) ;
How many hours it will take to reduce their distance apart
by a given number of miles (M) ;
(vi) How many hours it will take to reduce their distance apart
to a given number of miles (D) ;
(vii) The distance of the " Lusitania " from New York ; given the
distance of the other steamer from New York (L lt L 2 ) ;
(viii) The distance of the slower steamer from New York when
the " Lusitania " is a given number of miles from New York
(I, L,) ;
(ix) The distance of the slower steamer from New York when
the two steamers are a given distance apart (L 2) d) ;
(x) The distance of the " Lusitania " from New York when the
two steamers are a given distance apart (!>i, d).
(iv}
(v;
EXERCISE XVI S3
8. Write down a formula for :
(i) The length of wall-paper of given width required to cover
a wall of given length and height. All the dimensions
being in feet (L, w, I, h) ;
(ii) The same, the width of the paper being given in inches ;
(iii) The same, the length of the paper being required in yards,
but the dimensions of the room being still given in feet,
and the width of the paper in inches ;
(iv) The same, the width of the paper being the standard width
of 21 inches ;
(v) The number of " pieces " of standard width required for the
wall, each piece having the standard length of 12 yards ;
(vi) The cost of the paper in shillings, given the price of the
paper per piece in shillings (C, p) ;
(vii) The cost of the paper in shillings, the price of a piece being
given in pence.
9. (i) Into a jug of uniform cross-section containing a certain
amount of water, I drop a number of marbles of equal
size. Write a formula for the height through which the
surface of the water will rise, supposing that all the marbles
are covered (h, A, n, v).
(Does the shape of the cross-section make any difference ?)
(ii) Given the original depth of the water, find the depth after
a given number of marbles have been dropped in (d n , d ).
(iii) Write a formula for finding the number of marbles required
to raise the surface of the water from one given level to
another.
(iv) The Crow in ^Esop's Fable drank water from a deep vessel
by dropping in pebbles until the water was sufficiently
high. Supposing the vessel to be a uniform jug of given
cross-section, and the pebbles to be of equal size, and given
the length of the crow's bill, and the original distance of
the surface of the water from the top of the jug, find a
formula for the number of pebbles that he must drop in
BO as to drink comfortably. (You should allow one more
pebble than the least number necessary) (n, A, v, 6, d).
10. Write down a formula for :
(i) The weight of a cylinder, given the radius of the base,
the height, and the weight of a cubic inch of the material
(W, f, h, c) ;
(ii) The weight of a cylindrical bottle containing ink, given
the weight of the empty bottle, its internal radius, the
depth of the ink and the weight of a cubic inch of it
(W, w, r, d, c) ;
(iii) The weight of a cubic inch of the ink, given the weight of
the bottle, with the ink in it, the weight of the empty
bottle, the internal radius of the bottle and the depth of
the ink ;
6*
84 ALGEBRA
(iv) The cost of the ink per cubic inch, given the price of the
bottle when full, the internal radius, and the depth of the
ink (0, P, r, d).
B. SUBSTITUTION.
11. Where a railway line goes round a curve the outer rail
is always raised above the inner rail a height given by
, _ wv 2
~ iWr
h = elevation of outer rail in inches ; w = width of gauge
in feet ; v = greatest speed of train in miles per hour ; r =
radius of curve in feet.
(i) Find how much the outer rail must bo raised round a curve
of 2260 feet radius on an ordinary English " narrow
gauge " railway (w = 4 feet 8 inches), the greatest speed
allowed being 60 miles per hour.
(ii) Solve tho same problem for a railway with the Irish gauge
of 5 feet 3 inches, tho radius being 2800 ft.
(iii) Light railways in India have a metre gauge (practically 3'3
feet). Find the elevation of the outer rail to allow a
speed of 24 miles per hour round a curve of 264 yards
radius.
12. A moving train experiences a resistance due to the air,
friction of the rails, etc. Harding's formula gives
E = W (6 + 0-33v) + 0'0025At; 2
R = resistance in Ibs. on the level in calm weather ; W = wt.
of train in tons ; A = area of frontage of train in square feet.
(i) Calculate the resistance to a train weighing 30 tons, pre-
senting to the air a frontage of 60 square feet, when it is
going at 50 miles an hour ;
(ii) The same, when the train weighs 24 tons and is going at 40
miles per hour.
13. The horse-power required to drive 'a ship at a given
speed can be found (approximately) by means of the
formula :
H = 0-0088s 3 (0-05A + 0-005S)
8 = speed in knots ; A = immersed cross-section in square feet ;
S = area of wetted surface in square feet.
(i) What horse -power is needed to drive a steamer at a speed
of 10 knots, when A = 200 square feet and S = 2000 square
feet?
(ii) A steamer in which A = 240 square feet and S = 3000 square
feet is travelling at 12 knots. What horse-power is being
used?
EXERCISE XVI 85
14. The diameter in inches of a crank- shaft l (when made
of wrought iron) is given by the formula :
/83H
Find the thickness of crank-shaft that should be used in a 16
horse-power gas engine making 120 revolutions per minute.
15. If you look in at the door of an electric lighting works or
a factory you can generally see near the roof a long shaft with
large pulley wheels on it conveying the power from the engine
to the dynamos or other machines by means of leather belts.
A sufficient thickness for this shaft is given by :
_, 3/G5H
a
Find the diameter of a shaft which is made to revolve 180
times a minute and is transmitting 120 horse-power from an
engine to a number of machines.
16. What information is given by the following formula ?
B = the pressure of steam in the boiler; D~ diameter of
cylinder; d^ diameter of crank-shaft; I = length of piston
stroke, all three in inches.
What should be the boiler pressure if the cylinder's dia-
meter is 50 inches, the length of the piston-stroke 30 inches,
and the shaft 8 inches in diameter ?
17. There should also be a certain connexion between the
diameter of the crank-shaft and the dimensions of the cylinder.
The connexion is :
What is the diameter of the cylinder suitable to a crank-
shaft 12 inches in diameter, the length of the stroke being
3 feet ?
18. The rate at which the draught rushes up a factory
chimney is given by the formula :
(l -4
s = 2-42
J For Nos. 14-17 refer to fig. 4 and the accompanying note
(p. 30).
86 ALGEBRA
s = speed in ft. /sec. ; h = height of chimney above fire-bars ; T =
internal temperature (degrees Fahrenheit) ; t = outside tem-
perature.
(i) Calculate the draught in a chimney 169 feofc high, the in-
side and outside temperatures being 360 and 60 respec-
tively ;
(ii) The same, the chimney being 289 feet high, the internal
and external temperatures being respectively 580 and
50.
19. It has been found by observations in Madras that the
greatest number of cubic feet of water discharged per second
in a time of flood is given by the formula :
w = k /A*
A = no. of square miles drained by the river ; k = a constant,
which has the value 450 for places within 15 miles of the
sea, 562-5 between 15 and 100 miles from the sea, 675 for
limited areas near hills.
(i) Find the greatest flood discharge to be expected in a Madras
river at a point 10 miles from the sea, the drainage area
being 8000 square miles ;
(ii) The same, substituting 60 miles and 3700 square miles ;
(iii) The same, for a river in hill country draining 1200 square
miles.
20. The following formulas state the rules for determining
the income-tax (in shillings) payable on earned incomes. (In-
comes entirely or partly unearned are subject to different rules.)
(1) 160<I> 400 : T = 3(1 - 100 - P - 160).
(2) 400<I> 500 : T = (I - 100 - P - 150).
(3) 500<I> 600 : T = 2(1 - P - 120).
(4) 600<I> 700 : T = 3(1 - P - 70).
(5) 700<I>2000 : T = 'i(l - P).
(6) 2000<I>3000 : T - I - P.
(7) 3000<I>5000 : T = 5(1 - P).
(8) I >5000 : T = J(I - P) + j(I - 3000).
(I = earned income; C = no. of children under 16; P =
amount of insurance premiums or superannuation fund pay-
ment.)
Explain the rules and apply them to calculate the tax pay-
able :
(i) By A, who earns 360 per annum, has 4 children, and pays
12 in insurance premiums on the lives of his wife and
himself ;
(ii) By B, who earns 475, has 5 children, and pays 16 to a
superannuation fund and 10 in insurance premiums ;
EXERCISE XVI 87
(iii) By C, who earns 660, has 3 children, and pays 70 in
insurance premiums ;
(iv) By D, who earns 1800 ;
(v) By E, who earns 2400, and pays 225 in insurance pre-
miums ;
(vi) By F, who earns 7600, and pays 500 in insurance pre-
miums.
G. SOME ARITHMETICAL PUZZLES.
21. I take two numbers, each less than 100. I multiply the
first by 100 and add the second. The result is in various
cases (i) 1426, (ii) 2853, (iii) 947, (iv) 4308, (v) 1008. What
were the numbers in each case ?
Note. The principle of No. 21 is used in the following
puzzles, Nos. 22-25.
22. Taking the number of the month in which I was born
I multiply by 20, add 12, multiply the sum by 5, and then add
the day of the month. The result is 578. When is my
birthday ? What is the rule for solving tho puzzle ?
23. I am thinking of a date in English history. I take 4
from the number which marks the century and multiply the
residue by 10. I then add 4 to this product and again
multiply by 10. Finally, I add the number of years in the
century. The result is 1289. What was the date ? What is
the rule for solving the puzzle ?
24. I am thinking of a number. To its double I add 4. I
multiply the sum by 5 and add 3. I now multiply by 10
and add another 3. The result is 26733. What was the
number ? The rule ?
25. I open a book at a certain page and choose one of the
first 9 words in one of the first 9 lines. To 10 times the
number of the page I add 25 together with the number of the
line. To 10 times this sum I add the number of the word.
The result is 12697. Tell me the numbers of the page, line,
and word. What is the rule ?
26. Invent puzzles similar to Nos. 22-25 but involving
different calculations.
27. A number N is composed of two digits a and b. Show
that
N 9a + (a + b)
Hence prove (i) that if the sum of the digits is divisible by 9
the number itself is divisible by 9 and conversely ; (ii) that
88 ALGEBRA
if the sum of the digits is divisible by 3 the number is divis-
ible by 3 and conversely.
28. A number is composed of three digits a, 6, c, and is
divisible (i) by 9, (ii) by 3. Show that the sum of the digits
is divisible (i) by 9, (ii) by 3.
29. Do the rules of Nos. 27 and 28 apply to a number of
four or more digits ?
30. Find which of the following numbers is divisible by 9,
and which of the others are divisible by 3 : 426, 891, 3024J
12765, 42936, 824315, 927654727, 824510682.
31. In order to find whether 726 is divisible by 11 I add
the 7 to the 26. Since 33 is divisible by 11 I conclude that
726 is also divisible by that number. Show that this rule
holds good for all numbers containing three digits.
32. To find whether 5379 is divisible by 11 I add 79 to 53
and obtain 132 which is divisible by 11. I conclude that 5379
is also a multiple of 11. Show that this rule is true for all
numbers composed of four digits.
33. What are the corresponding tests for numbers com-
posed (i) of five digits, (ii) of six digits ? Apply them to the
numbers 21736, 932085.
34. Show that any number of four figures, such as 8228, in
which the second half is the first half reversed, must be de-
visible by 11.
35. Show that the same rule holds for numbers containing
six figures such as 827728.
Note. A number whose digits are a and b can be written
lla + (b - a) or lla - (a - b) according as b or a is
greater.
36. Show that if a number of three digits is divisible by 11
the difference between the middle digit and the sum of the
other two must be zero or 11.
37. Show that if a number of four digits is divisible by 11
the sum of the first and third digits is either the same as the
sum of the second and fourth or differs from it by 11. Can
the rule be extended ?
38. Test the divisibility by 11 of the following numbers :
102036, 151602, 9243871, 8654768.
39. (i) Factorize (at 4- b) - (bt + a), a being greater
than b.
(ii) Factorize (at 2 4- bt + c) - (ct 2 + bt + a), a being greater
than o.
EXERCISE XVI 89
40. Use the results of No. 39 to prove the following : Take
any number composed either of two or of three digits. Obtain
another number by reversing the order of the digits. The
difference between the numbers will be a multiple of nine.
41. Show that when there are three digits the middle digit
of the difference is always 9. What must be the sum of the
other two digits? (See No. 28.)
42. I take a number of three digits and form another number
by reversing the digits. On subtracting I find that the first
figure of the difference is 4. What are the other two figures ?
43. In other similar cases the last figure of the difference
was (i) 7, (ii), 9, (iii) 0. What were the figures in each case ?
44. Express (i) (at 2 + bf 2 + ct + d) - (dfi + ct- + bt + a)
and (ii) (at* + bt* + ct 2 + dt + e) - (et 4 + dt* + ct' 2 + bt + a)
each as the sum of two products.
45. Use No. 44 to show that the rule of No. 40 holds good
for numbers containing four and five digits.
46. I write down a number containing five digits and a
second number whose digits are those of the former taken in
reverse order. I subtract the larger of the^e numbers from
the smaller. I cross out one of the figures of the remainder.
The other figures are 3, 0, 9, 7. What figure did I cross
out?
47. I do the same with a number containing six figures.
The other figures in the remainder are 1, 3, 6, 3, 5. Show
that the figure crossed out must have been one of two.
48. Write down any sum of money (a bs. cd.) less than
12, the numbers of pounds and pence being different.
Obtain a new sum by reversing the order of the figures (c
bs. ad.). Find the difference between the two sums. Re-
verse the order of the figures of the difference and so obtain
a fourth sum. Add together the third and fourth sums.
Show that the result will always be 12 18s. lid.
49. Show that (5 + a) (5 + b) = 10 (a + b) + (5 - a)
(5 - b).
50. The following is an example of the Regula Stultorum
(The Fool's Rule) used in the Middle Ages for multiplying
two numbers between 5 and 10. Assign the numbers 6, 7,
8, 9, 10 in order to the fingers of each hand, counting the
thumb as the first finger. To multiply 7 by 9 bring together
the tips of the fingers which bear these numbers. Including
these two themselves there are now six fingers to the front
90 ALGEBRA
of the point of contact. Therefore the first figure of the pro-
duct is 6. Also there are behind the point of contact three
fingers on one hand and one on the other. Therefore the
second figure in the product is 3 x 1 = 3.
Use No. 49 to prove this rule.
D. GRAPHIC EEPRESBNTATION.
51. The following table gives the height of the spring tides
at points on the Kiver Thames, at specified distances from the
mouth of the river. Draw a graph based upon the table.
Use it to determine :
(i) The height of the spring tides at Barnes Bridge, 70 '5 miles
from the mouth of the river ,
(ii) The places where the height is 10 feet and 5 feet re-
spectively ;
(iii) How far from the mouth of the river the tidal influence
would be felt if the locks did nob intervene ;
(iv) The probable height of the spring tide at Greenwich, 50
miles from the mouth.
Place. Distance Height.
Miles. ft. in.
London Docks . 60 18 10
Putney
Kew
Richmond
Teddington
67'5 10 2
73 71
76 3 10
79 1 4J
52. The following table gives the times of sunrise and sunset
in London on certain dates in May, June, and July. Con-
struct from it a graph showing the length of the day for all
dates between 1 May and 31 July. (The day may be taken
to begin one hour before sunrise and to end an hour after
sunset.) When is the longest day and how many hours does
it last ? During how many days in the year is the night not
more than six hours long? (Bead the note above, Ex. II,
No. 17.)
May.
Rises. Sets.
4.20 7.32
4.4 7.48
3.52 8.2
June.
Rises. Sets.
3.45 8.13
3.44 8.18
3.48 8.18
July.
Rises. Sets.
3.56 8.14
4.7 8.5
4.21 7.51
10
20
30
53. The following table gives the temperature at different
times of a quantity of mercury which had been heated and was
EXERCISE XVI 91
then allowed to cool. Draw a graph representing the probable
history of its cooling. Why do you think that this history is
more likely to be described by the curve you draw than by
any other ?
Time in min. : 012345678
Temperature (F.) 425 296 230 188 160 136 120 108 100
54. Draw a horizontal " time-line " on the same scale as
the one in No. 53. Midway between the divisions 0-1, 1-2,
2-3, etc., erect perpendiculars to represent the fall of tempera-
ture of the mercury during the first, second, third, etc., minutes.
Through the tops of these perpendiculars draw a smooth curve.
Deduce from this curve how much the mercury cooled (i)
between !- and 2 minutes, (ii) between 3*2 and 4*2 minutes
after the beginning of the observations. In which minute
did it cool 20 ? Confirm these deductions by means of the
graph of No. 53.
55. Booth's " Life and Labours of the People in London "
gives the following particulars about the number of men and
boys (per 10,000 of the male population whose ages are be-
tween 10 and 80) who are employed between different ages.
Exhibit the information in a column-graph. Draw a con-
tinuous curve to show the probable distribution of employ-
ment at the intervening ages. How many persons are employed
between (i) 14 and 18, (ii) 40 and 45, (iii) 45 and 50, (iv) 60
and 65 ?
Ages. Number Employed.
10 and under 15 193
15 20 880
20
25
35
45
55
65
25 933
35 1636
45 1201
55 830
65 434
80 192
56. A small dog is swimming across a pond along the path
ABC (fig. 26) with a stick in his mouth. When he is at B
a larger dog jumps into the pond at D in order to capture the
stick. The big dog swims from moment to moment directly
towards the other. While the smaller dog swims a distance
P the larger swims a distance Q. Trace the path of the
larger dog and find whether he will overtake the smaller one
before he lands at C. (The figure may be traced through thin
paper or by holding it against a window.)
92
ALGEBRA
Bank
- P
Bank
f 'B?^;r. ...
57. Draw a long rectangle to represent the complete value
ofl/(l - a) when a = ^. Indicate the lengths which represent
the first and second approxima-
tions to the value of the fraction.
Do the same with regard to
1/(1 + a).
58. A square measuring 10
cms. each way is made into one
measuring (10 + h) cms. each
way by the addition of a
"gnomon" consisting of two
equal rectangles, E and B' and
a square Q. Draw a series of
strips representing the area of
the gnomon in the cases when
h is 1 cm., 0-8 cm., 0*6 cm.,
0*4 cm., and 0*2 cm. respec-
tively. (The width of the strip
should be the value of h in each
case.) Shade the rectangles
and leave the square Q un-
shaded so as to indicate the
amount of the error committed
in each case by taking the increase in the original square as
equal to 20 h.
59. A cube measuring 10 cms. each way is made into one
measuring (10 + h) cms. by the addition of solids to three of
its faces as in Ex. XI. Draw a long strip whose area shall
represent the total volume of the solids added when h has the
values 1, 0-8, 0'6, 0*4, 0'2. (The height should be the value
of 10 h.) Mark off lengths representing the volumes which
involve h, fo 2 , and W respectively, so as to indicate the relative
importance of their contributions to the whole.
60. The bearings of an aeroplane were taken every minute
at two stations, A and B, situated 2000 yards apart on an
east and west line. A record of the height of the aeroplane
was kept by a passenger. The observations are given below.
Draw a " bird's-eye view " of the track of the aeroplane. By
measuring the length of the line drawn determine the hori-
zontal distance covered by the flight of the aeroplane. On a
horizontal line of the proper length mark verticals representing
the recorded heights of the aeroplane. Draw a smooth curve
Fia. 26.
EXERCISE XVI 93
showing the probable height at any other time. Cut along
the curve and the base line. Fix the strip so obtained along
the plan of the track so that the curve may represent the
actual path of the aeroplane through the air. What was the
average horizontal speed? When did the aeroplane pass
between the two stations, at what distance from each, and at
what height ? When was the aeroplane nearest to A and to
B ? What were its distances from A and B (measured hori-
zontally) and what was its height at each of these moments ?
Minute*,
1
2
3
4
5
6
7
8
9
10
11
E. FACTORIZATION, ETC.
Write formulae in the form most convenient for calculating
the numbers specified in Nos. 61-68. (The linear measure-
ments indicated in the figures may be taken to be centi-
metres.)
61. The volume (V) of a block of wood of height h and of
uniform cross-section represented by fig. 27.
62. The density (d) of the wood if the block of No. 61
weighs W grams.
63. The volume (V) of a block a by a by 3b with a hemi-
sphere of radius b scooped out of the top and bottom faces.
(Fig. 28 would be a section across the middle of this block.)
64. The volume of the figure obtained by rotating fig. 27
about its axis PQ. (This figure can be thought of as repre-
senting a boiler with hemispherical ends.)
65. The volume of the figure obtained by rotating fig. 28
about the line PQ.
66. The volume (V) of a column of uniform cross-section
represented by fig. 28 and of height h.
Height above
Ground in feet.
Searing from A.
Bearing from B.
dueN.
76 W. of N.
120
31 E. of N.
79 W. of N.
390
83* E. of N.
88 W. of N.
420
76* K of S.
75 W. of S.
570
75* E. of S.
42J W. of S.
360
81 E. of S.
38 E. of S.
180
89 E. of N.
86 E. of N.
270
77J E. of N.
22 E. of N.
480
67* E. of N.
33 W. of N.
480
54 E. of N.
53.4 W. of N.
300
35 E. of N.
64J W. of N.
10 E. of N.
71 W. of N.
ALGEBRA
Fia.31.
FIG. 32.
EXERCISE XVI 96
67. The weight (W) of the column of No. 66 if the ma-
terial weighs w grams/cm. 3
68. Write down formulas expressed in the form most
suitable for calculation which shall give the total surface of
(i) a cylinder ; (ii) a four-sided pyramid on a square base ;
(iii) a cone. [Use the following symbols : (i) r, h ; (ii) s, I ; (iii)
r,l]
69. Write down a formula to give in the most convenient
form the surface area of the boiler of No. 64.
70. Write a formula for the area of a plate like fig. 29,
pierced with a circular hole.
71. Figs. 30 and 31 represent discs pierced respectively
with nine and sixteen equal holes. Find expressions for the
area of the suiface of each (A, E, r).
72. Fig. 32 represents a disc pierced by thirteen circular
holes, the radius of the large hole in the middle being double
the radius of the smaller holes. If the radius of the disc is R
and that of one of the small holes r, find the area of the surface
of the disc.
73. Find the formulae for
(i) The area of the upper surface of a plate of radius a pierced
with any square number (n a ) of holes of radius b ;
(ii) The weight of such a plate given its thickness (c) and the
weight (w) of a cubic unit of the material.
Why does the question concern only square numbers of
holes ?
74. Find formulae for the area of fig. 24 (p. 41), (i) calling
the radius of the hole a and the middle of the ring b ; (ii)
calling the width of the ring w and the radius of the hole r ,
(iii) calling the outer radius of the ring r and the width w.
75. Find formulae
(i) For the total internal and external curved surface of a pipe
whose external and internal radii are R and r and whoso
length is I ;
(ii) For the total surface of such a pipe counting in the top and
bottom ring-surfaces.
76. Find two formulae corresponding to those of No. 75,
given the internal radius (r) and the thickness (t) of the pipe.
77. Find two formulae corresponding to those of No. 75,
given the external radius (r) and the thickness of the pipe.
96 ALGEBRA
78. (i) The top of a ladder 15 feet lon^ rests against a wall at a
point 12 foot abovo the ground. How far is the foot of
tho ladder from the wall ?
(ii) Write down a formula for finding the distance (d) of the foot
of tho laddor from tho wall, given tho length of the ladder
(I) and tho height (h) of tho point against which it rests.
79. A kite at the end of a string I feet long is immediately
over the head of a boy who is standing d feet from the boy
wbo is flying it. Give a formula for calculating the greatest
possible height of the kite above the level of the latter boy's
hand.
Why are you not asked to give a formula for the actual
height of the kite ?
80. After walking due east for t hours at the rate of 3 miles an
hour I find I am due south of a tower on a hill d miles from
my starting-point. How shall I calculate my distance from
the tower ?
81. Each of the following expressions describes the factors
of a product, a being a symbol for any number you please.
Write the expressions which describe the products.
(i) (a + 3 )(a + 2). (ii) (a + 3 )(a - 2).
(iii) (a - 8)(a - 5). (iv) (a - 9 )(a + 4).
(v) (a + 3 )(a + 3). (vi) (a - 7) a -
(vii) (2a + x)(3a + 2). (viii) foa - 3)(sa - 4).
(ix) (7a - io)(3a + 5). (x) (ya - io)(3a + 4).
(xi) (xsa - 7)(i2a + xx). (xii) (2a + 3)*.
(xiii) (sa - 6)\ (xiv) ( 2 a 2 - 5 )(Sa 2 + 2).
Verify your answers to (iv), (viii), (ix), and (xiv) by substi-
tuting for a any numburs you please.
82. In the following expressions a and b are symbols for
any numbers. Write symbolic expressions of the products
of the factors described. Test the accuracy of the descrip-
tions in any three cases you please by substituting numbers
for a and b,
(i) (a + 2b)(a - 7b). (ii) (a - 4b)(2a - b).
(iii) (sa + 2b)(2a + sb). (iv) (7a - 4b).
(v) (loa - 4 b)(7a + 3 b). (vi) (i 3 a - sb)(2a + fb).
(vii) (a + 2b) a . (viii) (2a - b)'.
(ix) (2a - 3b) 3 . ^ ^ (x) ( 4 a + 3 b) 3 .
83. Complete the following identities :
(i) (*? + 5 a + 4 )/(a + i) - .
(ii) (a - 5a + 4)/(a - 4) -
(iii) (a 2 + 3a - 4)/(a - x) - .
EXERCISE XVI 97
(iv) (a 2 - 2a - io)/(a - 5) = .
(v) ( 4 a 2 + 8a + 3 )/(aa + i) - .
(vi) (35a 2 + i3a - i2)/(7a - 3) - -
(vii) (6a 2 - 5ab - 6b a )/(3a + 2b) = .
(viii) (2a 2 - nab + I5b a )/(a - 3b) = .
(ix) (6a 2 - ab - 35b 2 )/(2a - sb) = .
(x) (49a a - 28ab + 4b 2 )/(7a - 2b) - .
84. Factorize the following expressions :
(i) a 2 - SSL + 6. (ii) p 2 - 5? - 6.
(Hi) p 2 + 5P ~ 6. (iv) a 2 - a - 132.
(v) a 2 - 6a + 9. (vi) p 3 - I4p -f 49.
(vii) a 2 - 3ab - 4b 2 . (viii) 4a 2 -h 3ab - b 2 .
(ix) 4a 2 + I2ab + 9b' J . (x) 6a 2 - I3ab - sb a .
(xi) 2a 2 + 5ab - i2b a . (xii) ssa a + ?ab - 6b 2 .
(xiii) ab + 33 - b - 3. (xiv) ab - 3ap -I- bq - 3pq.
(xv) 2ap - 4aq + 3bp - 6bq.
85. Complete the identity (a + b)(c + d) = . Illustrate it
by a diagram showing how from four adjacent rooms with
certain dimensions a large room (a- + b) long and (c + d)
wide can be formed.
86. Complete the identity (a - b)(c d) . Illustrate
it by taking a thin cardboard or paper rectangle, a long and
c wide, and reducing it, by certain subtractions and additions,
to a rectangle (a - b) long and (c - d) wide.
Note. The expression a 2 + 2ab describes a figure like fig.
19, in which AF = a = CD and FE = b = ED. The figure
may be made into the square (a 4- 6) 2 by filling in the
square FD = b 2 . The addition of b' 2 to a 2 + 2ab is said to
complete the square. The complete expression a 2 + 2ab
+ b 2 is called a perfect square.
Next let AB = a - b and FE = b. Produce BA to G and
BC to H, making AG = CH = b and so obtain the square a 2 .
In this case fig. 19 will be represented by a 2 - %ab, and to
complete the square (a - b) 2 it is again necessary to add b 2 .
87. What additions (or subtractions) are necessary to make
the following expressions perfect squares (or multiples of
perfect squares) ?
(i) a 2 + 6a. (ii) a 2 - 8a.
(iii) a a -H I2a. (iv) a 8 - 22a.
(v) a 2 + 3a. (vi) a 2 - 7a.
(vii) a 2 + I4a + 20. (viii) a 3 - 8a - 3.
(ix) p 2 + 13? + i. (x) p 2 - lop + 42.
(xi) a 2 - 4ab. (xii) a 2 + 6ab + 2ob 2 .
(xiii) 9a a + I2ab. (xiv) i6p 2 + 40pq.
7
98 ALGEBRA
(xv) 4m* - i6mn -f 7n a . (xvi) 3a 2 + I2ab.
(xvii) 55* - 3osh - 7h 2 . (xviii) 6p 2 - I2p + 17.
(xix) 3a 2 -f i Sab. (xx) 2a 2 ~7ab + b 2 .
88. In the following equivalences s = " the side of a certain
square ". Find (by completing the square) what is the length
of the side in each case.
(i s 2 + 2s = 8. (ii) s 2 -f i2s = 13.
(iii s 2 - I2s = 13. (iv) s 2 -f 8s + 65.
(v 60 - 143 + s 2 = 36. (vi) s 2 - 33 1375.
(vii) s 2 - 2a = 3a 2 . (viii) s 2 + 6a = 7a 2 .
(ix) s 2 - i8a -f 27 = 46. (x) s 2 - Js -f i = 5.
89. Let the volume of the complete cube model (Ex. XI)
be a 3 and that of the inner cube b' 3 ; then the additions have
a volume a 3 5 3 . Lay these additions side by side on the
table so as to show that
a 3 - 6 3 - (a - b) (a 2 + ab + b 2 )
Prove by multiplication that the result holds good for all
numbers.
90. Call the volume of the inner cube a? and that of the
small cube Z? 3 . Stand the latter upon the former and note
what further additions would make a column having a 2 for its
base and (a + b) for its height. Hence show that
a 3 + 6 3 = (a + b) a 2 - (a 2 - 6 2 ) b
= (a + b) (a 2 - ab + 6 2 )
Prove the universal validity of the identity.
F. APPBOXIMATIONS.
91. Find the difference between the distances of the
horizons visible from the bottom and top of a tower 60 feet
built on the side of a hill 666 feet above the sea
= 8-6). (See Ex. IX, No. 29.)
92. A pendulum consists of a small heavy ball at the end
of a light thread of length L When wetted the thread
stretches by a small amount h. Write formulae for calcu-
lating approximately (i) the new time of swing of the pen-
dulum (t r ), and (ii) the increase (i) in the time of swing. (See
Ex. VII, No. 11.)
93. Draw a straight line, AB ; of length a. At B erect a
perpendicular of length b, small compared with a. Write
down a formula for calculating the hypothenuse c to a first
approximation.
EXERCISE XVI 99
94. Verify the formula of No. 93 either by drawing a figure
on a large sheet of paper or by arranging pins and strings on
the floor or a large table. Choose your own values for a and b.
95. The centre of the bob of a pendulum is 30 inches below
the nail to which it is attached and is being held 6 inches
out of the vertical. Calculate the approximate length of the
pendulum from nail to centre of bob.
96. As measured on a map the summit of a hill is 3 miles
from a certain house at its foot. Between these points the
difference of level is of a mile. By how many yards does
the direct distance between the house and the summit exceed
the horizontal distance ?
97. A boy on a cliff 325 feet above the sea descends a dis-
tance of 40 feet. What difference does the descent make to
the distance he can see? ( Vl3 = 3*61.) Prove the formula
you use.
98. What difference is made to the time of swing of a
pendulum 4 feet long if it is shortened 3 inches ? Prove your
formula.
99. The radius of the earth may be taken as 4000 miles and
was at one time rather greater. Assuming that it was once
10 miles greater than at present, calculate approximately
(i) the superficial area, (ii) the volume which it has lost.
(The formulae for the area and volume of a sphere are
A = 47rr 2 and V = ^Trr 3 ).
100. A quantity a is increased by a small amount h and it is
desired to know the approximate increase in a' 2 and a 3 . Show
that the error in taking the increase in a 2 to be %ah is not
more than h/2a of the whole increase, and the error in taking
the increase in a 3 to be 3a' 2 h not more than (h/a + 7& 2 /3a 2 )
of the whole increase. Apply these results to the calculations
of No. 99.
G. CHANGING THE SUBJECT, ETC.
101. Take the formula of No. 1 (ii) and change the subject
to m. What information does the new formula give ? Is it
correct ?
102. Change the subject of the formula of No. 2 (ii) to b.
What does the new formula describe ?
103. In No. 5 change the subject of (iii) to s 1 and of (iv)
to 5 2 .
7*
100 ALGEBRA
104. In No. 7 change the subject of (ix) to s , and of (x)
to s r
105. Deduce a formula for^ from the one given in No. 11.
Use it to find the greatest speed allowable when a train goes
round a curve of 1600 feet radius, the gauge being 5 feet
and the outer rail raised 9 inches above the inner rail.
106. Change the subject of No. 15 to H. Calculate the
horse-power that may be transmitted safely by a 4-inch shaft
revolving 195 times a minute.
107. Change the subject of No. 16 (i) to D, (ii) to d. Inter-
pret the new formulae.
108. Change the subject of No. 13 to s. Calculate the
speed at which the steamer in No. 13 (i) would travel if the
engines were working at 220 horse-power. [ ^10 = 2-154.]
109. A square lawn measures 6' feet each way. The owner
wishes to enlarge it into a square of area A square feet by
adding a feet to each side. Change the formula A = (s + of
into one for calculating a, given s and A.
110. A flower bed shaped like fig. 14 (p. 35) is made by
adding triangles in which b = 6 to the sides of a square. Find
the length a so that the bed may have a total area of 64
square feet. (See Note to No. 87.)
111. Give a general formula for calculating a when b and
A (the total area) are given.
112. In the case of fig. 15 obtain a formula for calculating
a when b and A are given.
113. By the addition of a feet to both its length and its
breadth a lawn 30 feet by 20 feet becomes 875 square feet in
area. Calculate a.
114. Obtain a general formula for calculating a when the
original length and breadth and the final area of the lawn
are given.
115. A lawn 40 feet by 30 feet is to be reduced to 1000 square
feet of turf by cutting a strip from the width and a strip
twice as wide from the length. Calculate to the nearest
tenth of a foot the final dimensions.
116. In fig. 27 (p. 94) a is 11 cms. and the area 88 cms. 2
Calculate b to the nearest millimetre.
117. Give a general formula for solving problems like No.
116 given a and A.
Note. The numbers n - 10 and 10 - n yield on squaring
identical results, n 2 - 20n 4- 100 or 100 - 20n + ri*. If one
EXERCISE XVI 101
of the last two expressions is given it is impossible, therefore,
to say whether it is the square of a number greater or less
than 10. That can be told only from the context. Thus
n 2 - 20^ = 44 gives, on completion of the square,
w 2 - 20n + 100 = 144 . . . (i)
n - 10 = 12
n = 22
It is evident here that n - 10 is the required square root, for
10 - 7i = 12 would be impossible. On the other hand, if we
are told that
n a _ 20n + 100 = 36 . . . (ii)
both n - 10 = 6 and 10 - n = 6
or n = 16 and n = 4
are possible.
Unless, therefore, the statement of the problem shows (as
in Nos. 110-117) that it has only one answer, two must be
sought. Can you tell (by considering lines (i) and (ii) above)
when there will be two and when only one is possible ?
118. Calculate (to two decimal places) the numbers which
comply with the following conditions :
(i) n 2 - 8n + 7 = o. (ii) n 2 - I2n + 4 = 49.
(in) 3n 2 -f I2n = 180. (iv) 2n 2 - 711 = 3.
(vj 2'2n 2 + nn = 20. (vi) n 2 - 140 = 32.
(vii) n 2 - 2on + 91 = o. (viii) 150 - 5211 + n 2 = 6.
(ix) (311 - 4)(zn - x) 20. (x) (zn - i) (30 - 2) = 40.
119. Calculate the numl ers which comply with the follow-
ing conditions:
(i) n(n - i) = (n - i)(n + 2) - 10.
(ii) 3(n + 2)(n - 8) - (n - 4)(n - 6) - 2(n - 2)(n - 7).
am J 4. 2 - 3
(U1) n + n-i~n-2'
(iv) 1LZJ . !LI_4 + nrJ.
5 10 40
(vii) 3 (n - i) + s(n - 2)
J
15 16 20
_ 2 3H + 3 = 7JLJ
102 ALGEBRA
n
if I _l\ + 3/! _ *} _ 5/i _ i\
2\n 4 7 + 4 Vn 3) - 6\n 5)'
120. Trace the steps by which you would determine the
value pf the numbers, which comply with conditions of the
following form :
(i) a a n - c = b 2 n + d.
,.. x n a n - b
(iii)
_ __ .
I.I 2
n-p n-q n
a _ a -
(V ) ^ _ = 5c b
v 7 bn 2 an 2
( v n E_tJP + n - q = P + q
q P q
/.. x n
(vii)
-
n-q n n + q '
(viii) a(2n - a) + b(2n - b) = 2ab.
(ix) (n + a)(n - b) + (n -f a)b = r
I
EXEECISB XVII.
DIRECT PROPORTION.
A.
1. Construct a ready reckoner for the cost in shillings (0)
of given lengths (L) of picture moulding of three patterns.
The first costs 4d., the second 9d., the third Is. 3d. a foot.
Give the formulae which correspond to the three graphs.
Find by the graphs how many feet of the first and second
kinds of moulding cost as much as 16 feet of the third kind.
Test your answers by the formulae.
2. Draw (on one sheet) the graphs of the formulae (i) y =
\x, (ii) y = 1'2#, (iii) y = 2*5rc. Find from the graphs the
values of y corresponding to x = 12 in (i), x =* 4 in (ii), x =
2*4 in (iii). Compare the results with those obtainable from
the formulae.
3. A piece of cardboard containing 16 sq. cms., cut out of
a uniform sheet, weighs 8 -4 grms. Draw a graph giving the
weight of pieces of the card of given area.
How should the graph be held so as to be a graph showing
the area of pieces of the card of given weight ?
Write formulae giving (i) the weight (W) in terms of the
area (A), and (ii) the area in terms of the weight.
4. It is found that 12 sq. cms. of another uniform sheet of
cardboard weigh 7*8 grms. Write formulae (i) for the weight
of a given area, and (ii) for the area of a given weight.
5. The speed of a marble allowed to roll down a smooth
slope is proportional to the time it has been rolling. In a
given instance the speed was 12-5 feet per second after the
marble had been rolling for 5 seconds. Write a formula
giving the speed (v) in terms of the time (). Calculate the
speed after 12 seconds.
6. Two variables are in direct proportion. When the inde-
pendent variable has the value 14 '8 the value of the dependent
variable is 3*7. Write the formula connecting them.
103
104 ALGEBRA
7. Two variables are connected by a relation of the form
y = kx, and, when x = 7'2, y = 10*8. Find the value of k.
What would the relation be if the independent variable be-
came the dependent variable, and vice versa ?
8 Write a formula showing that W, the weight of a quantity
of liquid, is directly proportional to V, the volume of the
liquid.
Given that 10 cubic feet of water weigh 625 lb., adapt the
formula for calculating the weight of given quantities of
water.
9. Adapt the same formula for calculating the weights of
given quantities of sea water, given that any quantity of sea
water weighs 1*025 times as much as an equal quantity of
fresh water.
B.
10. A bath already contains 24 gals, of water when a
tap delivering !- gals, a minute is turned on. Draw a
graph showing the number of gallons in the bath at subse-
quent times. Also write the formula corresponding to the
graph (Q, t).
Change the subject of the formula to t. How should the
graph be held so as to correspond to the changed formula ?
11. A bath had in it 24 gals, of water when the waste
pipe was partially opened. It then began to empty at a con-
stant rate of J gal./min. Draw (on the sheet used for No. 10)
a graph showing the amount of water left in the bath at given
subsequent times. Write the formula corresponding to the
graph.
Change the subject of the formula to t. Use the new
formula to find when the bath will be empty. Compare the
answer with that obtained from the graph.
12. Write down formulae corresponding to the straight
lines of Nos. 10 and 11, but not referring to any particular
variables.
What do these formulae become when the dependent vari-
able replaces the independent variable, and vice versa ?
13. The increase in the length of a vertical rubber cord is
directly proportional to the weight hung at the end of it.
When there is no weight the length is 16 inches. When a
weight of 8 oz. is added the length becomes 17 '2 inches.
EXERCISE XVII 106
Write a formula (i) for the increase in length in inches (i)
due to a given weight in pounds (W), and (ii) for the total
length of the cord (L) when supporting a given weight.
Change the subject of the second formula to W. What is
the use of the new formula ?
14. Draw (on one sheet) the graphs corresponding to the
formulae (i) y = 20 + 3x ; (ii) y = 20 - 3x ; (iii) y = 4-7
+ 5-3x.
15. Water runs into an empty bath at the rate of 3 gals./min.
After seven minutes the tap is turned on further and the rate
of flow increases to 5 gals./min. Draw a graph showing the
amount of water in the bath at various times.
Write a formula for the amount of water at a given time
(i) during the first seven minutes, (ii) after the first seven
minutes ; the time being measured in each case from the
moment when the tap was first turned on. Calculate the
amount at the end of thirteen minutes. Compare the result
with that obtained from the graph.
16. A motor runs from a point A at a constant speed of 24
mls./hr. for fifteen minutes when its speed is suddenly reduced
to 17 mls./hr. Write formulae for the distance it has covered
in a given number of minutes since it passed A. Calculate
its position (i) after ten minutes, (ii) after twenty-seven
minutes. Compare the results with those obtained from a
graph.
17. Change the subject of the second formula of No. 16 to
t. Use the new formula to find when the car will have
travelled 23 miles from A.
18. Water is running into a bath at a constant rate. After
three minutes the bath contains 20 gals., after eight minutes
30 gals. Draw a graph showing the amount of water in the
bath at different times. How would you calculate (i) the
amount of water run in every minute (r), (ii) the quantity origin-
ally in the bath (Q ) ? Give the formula corresponding to the
graph.
0.
Note. When the graph showing the connexion between
two variables is a straight line there is said to be a linear
relation between them. What is the difference between
saying this and saying that they are directly proportional ?
106 ALGEBRA
19. Two variables x and y are connected by a linear rela-
tion. When x = 8, y = 17 ; when x 18, y = 30. Find the
formula connecting them.
20. The same, given that when x = 7, y = 43 ; and when
x = 20, y = 4.
21. A motor-car after running for some time at 18 mls./hr.
suddenly changes its speed to 24 mls./hr. Five minutes after
the change it has travelled altogether 8 miles. Find (i) by
a formula, (ii) by a graph, how long it ran at the lower speed.
22. In No. 18 the rate of flow is at a certain moment
suddenly changed. After the water has been flowing for
seventeen minutes altogether the bath contains 52 gals. ; at
the end of the twenty -third minute it contains 70 gals. De-
termine (i) by a graph, (ii) by a formula, the moment of the
change. Give formulae for the quantity of water in the bath
at subsequent times, counting the time (iii) from the moment
of change, (iv) from the beginning.
23. At the end of an entertainment the people leave a hall
in a uniform stream. After one minute there are 1800 people
left in the hall ; after three minutes there are 1440. Shortly
after this another door is thrown open with the consequence
that seven and a half minutes after the performance there are
only 360 people left, while at the end of eight and a quarter
minutes altogether the hall is empty. Calculate the number
of persons present at the end of the entertainment, the number
who left per minute through each of the two doors, and the
time when the second door was opened. Also write formulas
for the number of people in the hall at different times.
How many people were in the hall when the second door
was opened ? When was the number exactly 600 ?
24. Two variables, x and y, are connected by a linear rela-
tion. When x reaches a certain value they become connected
by a different linear relation. When x = 4, y = 37 ; when
x 7, y 61 ; when x 14, y = 65 ; when x = 20, y = 35.
At what value of x does the relation change ? Find the for-
mulae of the two linear relations. For what value of x does
y = ? Confirm by drawing a graph.
EXERCISE XVII 107
TABLE OF TANGENTS.
To be read thus : Tan 28 = 0-532, etc.
Tan
1 2 3 4 5 6 7 8 9
o-oo 0-017 0-035 0-052 0-070 0-087 0-105 0-123 ' I 4 I 0-158
Tan
10 11 12 13 14 15 16 17 18 19
0-176 0-194 0-213 0-231 0-249 0-268 0-287 0-306 0-325 0-344
Tan
20 21 22 23 24 25 26 27 28 29
0-364 0-384 o 404 0-424 0-445 0-466 0-488 0-510 0-532 0-554
Tan
30 31 32 33 34 35 36 37 38 39
0-577 0-601 0-625 0-649 0-675 0-700 0-727 0-754 0*781 0-810
Tan
40 41 42 43 44 45 46 47 48 49
0-839 0-869 0-900 0-933 0-966 i-ooo 1-036 1-072 I'm 1-150
Tan
60 51 62 63 54 55 56 57 58 59
1-192 1-235 1*280 1*327 1*376 1-428 1*483 1-540 i -600 1-664
Tan
60 61 62 63 64 65 66 67 68 69
1-732 1-804 i'88i 1*963 2-050 2-145 2-246 2-356 2-475 2*005
Tan
70 71 72 73 74 75 76 77 78 79
2-747 2-904 3-078 3-271 3-487 3-732 4-011 4-331 4-705 5-145
Tan
80 81 82 83 84 85 86 87 88 89
5-671 6-314 7-115 8-144 9'5I4 "*43 14*30 19*08 28-64 57*29
EXERCISE XVIII.
THE USB OF THE TANGENT-TABLE.
Note. Distances should be calculated to the nearest tenth
of the unit, angles to the nearest half -degree. Neat diagrams
should accompany solutions, but need not be drawn to scale.
A.
1. At a point (on level ground) 120 feet from the foot of
a fir-tree the elevation of the summit of the tree is 35. The
observer's eye is 5 feet above the ground. How high is the
tree?
2. What, to the same observer, would be the angle of ele-
vation of the summit of the tree at a point (i) 100 feet, (ii)
200 feet from its foot ?
3. At what distance from the tree would the angle of ele-
vation be 61?
4. Early this morning the shadow of an upright me ire
rule was 205 cms. long. What was then the angle of eleva-
tion Cor "altitude ") of the sun?
5. What would be the length of the shadow when the
sun's altitude is (i) twice, (ii) three times as great as in
No. 4?
6. Find the altitude of the sun when the shadow is (i) one-
half, (ii) one-third of its original length in No. 4.
7. From an upper window in a house 160 feet from a
church tower the angle of elevation of the top of the tower
is 41 and the angle of depression of the bottom 15. How
high is the tower ? How high is the point of observation ?
8. Standing 60 feet away from a house I find that the alti-
tude of a window-sill on the first floor is 27 and the altitude
of the top of the wall 38. How far is the window-sill below
the roof ?
9. Lying on a cliff 425 feet above sea-level I observed two
boats both due west of me. The angle of depression of the
108
EXERCISE XVIII 109
more distant was 12, that of the nearer 18. How far were
they apart ?
10. From the battlement of a castle tower 200 feet high I
note that the line from my eye to the foot of the gateway
makes an angle of 65 with the vertical face of the tower.
How far is the gateway from the central tower ?
11. I note also that the angle between the face of the
tower and the line from my eye to the top of the gateway is
68. How many feet is the top of the gateway below the
level of my eye ? How high is the gateway ?
12. From the top of a vertical cliff I observe that the angle
of depression of the summit of a lighthouse is 33 and that
of the foot of the lighthouse is 37. The lighthouse is 320
feet from^the cliff. What is its height ? How many feet is
my eye above its summit ?
13. A flagstaff stands on the roof of a building 115 feet
high. Standing in the street some distance from the building
I observe that the elevation of the bottom of the flagstaff is
37 and that of the top 46. What is the height of the flag-
staff? (My eye is 5 feet above the ground.)
B.
14. AB is a straight line of length L From a point O the
perpendicular OP falls between A and B and is of length p.
The distances AP and BP are a and b respectively. The
angle AOP is a and the angle BOP p. Find formulae for a,
b, and I in terms of p, a, and fi.
15. Change the subject of the last formula of No. 14 to p.
16. Find a formula for p when P falls on AB produced
beyond B.
17. AB and CD are two vertical lines of length H and h
respectively, H being the longer. The straight line BD
joining their bases is horizontal. The angles BAC and BAD
are a and ft respectively. Show that
h SB H (tan a - tan /?)/tan a.
(Consider how you did Nos. 11 and 12.)
18. Change the subject of the last formula to H. Make up
a problem which could be solved by this formula.
19. ABC and CD are two straight lines at right angles.
Let AB - d, CD = h, and let the angles ADC and BDC be
a and fi respectively. Find an expression for calculating d.
110 ALGEBRA
(Consider No. 9.) Change the subject to h. To what prob-
lem does the second formula correspond ?
20. In the figure of No. 19 let DEC = a and DAC = (3.
Show that d h (tan a - tan /3)/tan a tan ft. Change the
subject to h.
21. The esplanade in a certain seaside town on the south
coast lies precisely east and west and is 1500 yards long. A
yachtsman wishing to know his distance from shore observed
(with his compass) that the lamp at one end of the esplanade
bore 11 east of north and the lamp at the other end 6 west
of north. How far was the yacht out at sea ?
22. On another occasion the bearings were, as before, 11
and 6, but both were to the east of north. Where was the
yacht?
23. Upon the top of a hill there is a flagstaff 42 feet high.
From where I stand the angle of elevation of the bottom of
the staff is 10, that of the top 12. How far away is the
flagstaff? What is the difference of level between my eye and
the top of the hill?
24. From the top of a church tower I look due north to-
wards a river flowing east and west. The angle of depression
of the further bank is 34, that of the nearer bank 37. The
river is at this point 20 feet wide. How high is the tower
(to the level of my eye) ? What is the distance of the river
from the tower ?
25. A boat is sailing towards a cliff. At a certain point the
angle of elevation of the cliff is found to be 8. When the boat
has come 200 yards nearer the angle is 13. Find the height
of the cliff.
26. A church spire is due north of a point A. From a
point B 300 feet west of A the spire bears 14 east of north.
What is the distance from A to the point on the ground below
the top of the spire ? The elevation of the top of the spire
as seen from A is 9. How high is the spire ?
27. Make a paper or cardboard model to illustrate problems
like No. 26. Let PN be the height of the spire. Draw the
two right-angled triangles PNA, BAN. Fold the figure about
AN so that the triangle PNA is vertical.
Let PN = h, AB = d, ANB = a, PAN = ft. Write formulae
for calculating (i) NA, (ii) h.
28. An iceberg was seen due east of a ship and had an
elevation of 27. After the ship had sailed a quarter of a
EXERCISE XVIII 111
mile north the bearing of the berg was 47 east of south.
What was its height ?
TABLE OF SINES AND COSINES.
To be read thus : Sm 7 = 0-122, cos 53 = 0-602, eto.
Sin
Cos
0*00
90
1
0-017
89
2
0-035
88
a
0-052
87
4
0*070
86
5
0-087
85
6
0-105
84
7
0-122
83
8
0-139
82
9
0-156
81
Sin
Cos
10
0-174
80
11
0*191
79
12
0*208
78
13
0*225
77
14
0*242
76
15
0-259
75
16
0*276
74
17
0*292
73
18
0-309
72
19* 6
0-326
71
Sin
COB
20
0-342
70
21
0-358
69
22
0-375
68
23
0*391
07
24
0-407
66
25
0-423
65
26
0-438
64
27
0-454
63
28
0-469
62
29
0-485
61
Sin
Cos
30
0-500
60
31
0-5I5
59
32
0-530
58
33
0*545
67
34
0-559
66
35
5f
36
0-588
64
37
0*602
63
38
0-616
52
39
0*629
51
Sin
COB
40
$ 3
41
0-656
49
42
0-669
48
43
0-682
47
44
0-695
46
45
0-707
46
46
0719
44
47
0731
43
48
0*743
42 U
49
0755
41
Sin
Cos
50
0766
40
51
0777
39
52
0788
38
53
0799
37
54
0-809
36
55
0-819
35
56
0-829
34
57
0-839
33
58
0*848
32
59
ffl 7
Sin
Cos
60
0-866
30
61
0-875
29
62
0-883
28
63
0-891
27
64
0-899
26
65
0*906
25
66
0-914
24
67
0*921
23
68
0-927
22
69
0-934
21
Sin
Cos
70
0-940
20^
71
$ 6
72
0-951
18
73
0-956
17
74
0*961
16
75
0-966
15
76
0*970
14
77
0-974
13
78
0-978
12
79
0*982
11
Sin
COB
80
0-985
10
81
0*988
9
82
0-990
8
88
#s
84
0-J95
85
0-096
5
86
0-998
4
87
0'999
3
88
0-999
2
89
roo
1
EXEEGISE XIX.
THE USE OF THE SINE- AND COSINE-TABLES.
Note. Distances should be calculated to the nearest tenth
of the unit, angles to the nearest half-degree. Neat diagrams
> v v\ud accompany solutions, but need not be drawn to
ocale.
A.
1. Calculate the amount of northing or southing, and of
easting or westing made by a ship on each of the following
occasions :
Course.
Distance run.
(i)
23 K. of N.
15 milos.
(ii)
42* W. of N.
17 miles.
(iii)
68 W. of S.
22 miles.
(iv)
7 E. of S.
35 miles.
2. After sailing 26 miles on a course E of N. a ship is 8*1
miles farther north than at starting. Find (i) the course and
(ii) the amount of easting.
3. Find the course of a ship which after sailing 18 miles
between west and south, has made 14 '6 miles towards the
west. Calculate the southing.
4. A ship sailing 34 W. of N. has reached a point 12
miles farther north than her point of departure. Find the
distance she has run and the amount of westing.
5. A ship which has been sailing 28 W. of S. has made
9-4 miles to the west. Find the distance run and the amount
of southing.
6. A smack after leaving Yarmouth harbour has sailed in
succession on the following courses. Find how far she is
north and east of the harbour mouth :
Courses. Distances run.
82 E. of N. 5 miles.
27 E. of N. U miles.
72 E. of S. 18 miles,
112
EXERCISE XIX 113
7. Use the tangent table to find the final bearing of the
ship in No. 6 from the harbour mouth. Calculate also the
distance.
8. A ship took her departure from a point in the Channel
where the Lizard bore 23 W. of N. and was 15 miles away.
She sailed, 45 E. of S., 34 miles, and then, 79 W. of S., 16
miles. Calculate the final bearing and distance of the Lizard.
9. A ship was steered for 3 hours 40 W. of N. in a current
running 60 W. of S. at 2 miles an hour. According to the
log the distance run was 17 miles. Find the actual course
and the actual distance run.
10. A Channel swimmer estimates that he has swum S.E. 6
miles from his starting-point at Deal, but that the tide has
carried him 40 W. of S. 14 miles. How far is he now from
Deal?
Note. In the figures illustrating Nos. 1-5 let A represent
the ship's starting-point, B the point at the other end of the
distance run, AC the northing or southing, CB the easting or
westing. It is convenient to represent the length of CB (i.e.
the side opposite A) by a, the length of AC (opposite to B) by
b, the length of AB (opposite to C) by c, and the course (i.e.
the number of degrees in the angle CAB) by a.
11. Write in symbols the sailor's rules for finding :
(i) The northing or southing, given the course and the distance
run ;
(ii) The easting or westing, given the same ;
(iii) The distance run, given the northing or southing, and the
course ;
(iv) The distance run, given the easting or westing, and the
course.
12. Give the rules for finding :
(i) The easting or westing, given the northing or southing, and
the course ;
(ii) The northing or southing, given the easting or westing, and
the course.
13. Give the rules for finding :
(i) The course, given the northing or southing, and the distance
run ;
(ii) The course, given the easting or westing, and the distance
run ;
(iii) The course, given the northing or southing and the easting
or westing.
8
114 ALGEBRA
B.
14. A boy scout is walking along a straight road towards
the west. He leaves the road at A by a straight footpath on
the left, in order to examine a tree at C, 32 yards along the
path. He then returns to the road by a second straight path
at right angles to the former, striking it at B, 73 yards from
the place where he quitted it. Find the angles between each
footpath and the road, and the length of the second path.
15. Between two points A and B in the footpath across a
field there is a piece of swampy ground. To avoid it a lady
leaves the path at A and walks 42 yards along a straight line
to a point G. Here she turns again towards the path and
rejoins it at B, 54 yards in a straight line from C. Given
that AG and CB make respectively angles of 47 and 35
with AB, calculate the direct distance from A to B. [Draw
a perpendicular from C to AB.]
16. In a case similar to that of No. 15, it is known that
AC 52 yards, the angle A = 39, the angle B = 51, and
the direct distance AB = 67 yards. Calculate the distance
along GB.
17. A man is walking in a north-easterly direction across
a common. At a point A he turns 22 towards the left and
proceeds 150 yards to read a sign-post at C. Here he turns
towards the right and walks 180 yards farther to a rock B
where he sits down. From his present position the angle
between A and G is 24, and A is 280 yards away. Calculate
(i) the angle between AB and the north-easterly line upon
which he was originally walking ; (ii) the number of degrees
through which he turned at C.
Note. In a diagram intended to illustrate completely any
one of the last four problems it would be necessary :
(a) to draw the lines to scale to represent the given
distances ;
(b) to draw them in the proper directions, i.e. making
the given angles with one another;
(c) to mark them with arrow-heads to show which way
along them the movements were supposed to take
place.
The movements actually taken along AC and CB may be
called component movements : the direct movement AB,
which would lead from the same starting-point to the same
EXERCISE XIX 116
final point, may be called the resultant movement. Straight
lines drawn to represent such movements completely are
called vectors. AC and CB are component vectors, AB is
a resultant vector. Eemember that the resultant vector like
the component vectors should always carry an arrow-head.
Care should be taken in naming a vector by letters at its ends
to give the letters in the order which represents correctly the
" sense " of the movement, i.e. the way which it takes along
the line it follows. Thus the second component vector in
Nos. 14-16 is called CB, not BC ; the resultant vector is
called AB not BA. The operation of finding the resultant of
two or more component vectors is called compounding" the
vectors. When a single vector is replaced by two or more
vectors of which it would be the resultant it is said to be
resolved into components.
18. There are two vectors at right angles, a and b. The
length of a is 5'6 cms., and it makes an angle of 37 with the
resultant vector c. Calculate the lengths of b and c.
19. Eesolve a vector, c, of length 14*8 inches into two
component vectors, a and b, at right angles, so placed that
a makes an angle of 52 with c.
Note. The difference of direction between two vectors, AC,
CB, is the angle through which you would turn at the moment
of passing from AC to CB. It is not the angle ACB but its
" supplement ".
20. The difference of direction between two vectors is 69,
their lengths are 10*6 inches and 23*5 inches respectively, and
the latter makes an angle of 20 with the resultant. Calculate
the length of the resultant.
21. Find the resultant of two vectors of lengths 17*2 cms.
and 14*6 cms. respectively, given that the former makes an
angle of 48 with the resultant, and that the difference of
direction between them is 109.
22. In No. 20 resolve the vector CB = 23-5 into a vector
CP along the line of AC and a vector PB at right angles to
it. That is, let a point travel from A to B along the lines
AP, PB, at right angles to one another, instead of along the
original AC, CB. Calculate the length of PB and of CP and
hence of AP. From AP and PB calculate the angle A and
hence the angle B. Calculate the length of AB. Compare
the results with those obtained in No. 20. What information
given in No. 20 was, strictly speaking, superfluous ?
8*
116 ALGEBRA
23. Show that No. 21 could have been solved in a similar
way if the angle of 48 had not been mentioned.
24. Two vectors of length 8*6 cms. and 13*2 cms. differ in
direction by 47. Calculate the angles which they make with
the resultant, and the length of the latter.
25. Calculate the resultant of two vectors of length 8 '6
inches and 4 -3 inches respectively when their directions differ
by 142.
C.
26. By means of his range-finder an artillery officer dis-
covers that one of the enemy's guns is 3200 yards and an-
other 2700 yards away. The angle between them is 114.
Calculate their distance from one another.
27. Two hostile warships are firing at one another. An
observer on shore judges by the interval between the flash
and the sound of a gun, that at a certain moment one ship is
1T3 miles and the other 8*4 miles distant from him. He
judges also that the angle between them is about 20. About
how far are the ships apart ?
28. Two straight high-roads, AC, CB, meet at C. A lane,
also straight, cuts across from A to B, making an angle of
53 with AC and of 49 with CB. AB is 800 yards long.
How much distance does a cyclist save by taking the short
cut from A to B ?
[Draw the vector diagram. Eeplace AB, as before by a
vector AP along the line of AC and PB at right angles to it.
The length of PB can be calculated in two ways from
the triangle APB and the triangle BPC. Hence show that
a sin 78 = 800 sin 53. Calculate a from this relation. To
find b replace AB by AQ, QB at right angles, Q being on BC.]
29. Solve a problem similar to No. 28, substituting 63 and
42 for the angles and 1120 yards for the length of the lane.
30. Two railway stations, A and B, are 15 miles apart in a
straight line, but the line deviates from its direction at A in
order that trains may call at a third place C. Supposing
that the lines AC and CB are straight and that they make
respectively angles of 23 and 34 with AB, find how much
longer the journey is from A to B than it would be if the line
did not deviate to C.
31. Two lighthouses are exactly 8 miles from one another
on a north and south line. The master of a ship who wishes
EXERCISE XIX 117
to fix his position observes (with the compass) that the
northern light bears due west and the southern light 50 west
of south. How far is the ship from the two lighthouses.
32. An hour later the northern lighthouse bears 80 W. of
N. and the southern lighthouse 64 W. of S. How far is the
ship now from the two lighthouses. (Indicate the two posi-
tions of the ship in the same diagram.)
33. A certain seaside town has a straight esplanade a mile
long. From the northern end a flagstaff on an island in the
sea bears 43 W. of N. From the southern end the flag-
staff bears 32 W. of N. From the southern to the northern
end the esplanade itself points 12 E. of N. Find the dis-
tance of the flagstaff from the northern end of the esplanade.
34. I want to find the distance from my house to a distant
church whose spire is visible from my window. I measure
the angle between the church and the last telegraph post I
can see on the road which runs before my house, and find it
to be 64. Standing by the post I find the angle between
the church spire and my window to be 51. The distance
along the road between the post and my window is, by my
cyclometer, 0*6 mile. How far is the church from my house ?
EXERCISE XX.
SOME NAVIGATION PKOBLEMS.
A.
Note. The word " miles " in this Exercise always means
nautical miles. To find sin 24 20' add to sin 24 one-third
of (sin 25 - sin 24). To find cos 24 y 20' subtract from
cos 24 one-third of (cos 24 - cos 25).
1. A steamer left the Island of Ascension (lat. 7 56' S.)
for Tristan d'Acunha and sailed 1600 miles due south. Find
its latitude.
2. Another vessel sailed due north from Ascension with the
object of making the African coast near Sierra Leone. Find
its latitude after a run of 1200 miles.
3. A ship sailing from Halifax (Nova Scotia lat. 44 36' N.)
feo the Bermudas (32 20' N.) follows practically a southern
course. Find the distance.
4. Auckland (lat. 50 40' S.) and Bering Island (lat. 55 N.)
are on the same meridian. How far are they apart ?
5. When Captain Scott's Antarctic expedition left Christ-
church, New Zealand (lat. 43 30' S.) in 1911, how far was it
from the South Pole?
6. An exploring expedition circumnavigated the Antarctic
seas along the sixtieth parallel. What was the distance
travelled ?
7. Ndw York (long. 74 W.) and Oporto (8 31' W.) are both
very nearly on the forty-first parallel (N.). What is their
distance apart along this parallel ?
8. A ship sailing from Sydney (long. 151 12' E.) to Val-
paraiso (71 30' W.) could follow the thirty-third parallel (S.)
for practically all the way. Calculate the length of the
voyage.
Note. A sailor can determine his latitude wherever he is
by simple observations of the sun or the stars. To find his
longitude he needs, as a rule, a chronometer which will tell
him at any moment what the time is at Greenwich. Before
118
EXERCISE XX 119
Harrison's invention of the chronometer (1736) determina-
tions of longitude were troublesome. Consequently the sailor
seeking a distant port preferred, if possible, to sail due north
or south until he reached the latitude of the port and then
to sail east or west along the parallel until he came to it.
This method of navigation is called parallel sailing".
9. A ship leaves Boston (42 25' N., 71 W.) for Barbados
(13 N., 59 45' W.) and uses parallel sailing. Calculate the
southing and the easting.
10. Find the southing and westing of a ship which goes
by parallel sailing from Bombay (18 55' N., 72 54' E.) to
Cape Town (33 40' S., 18 30' E.).
B.
Note. In Nos. 11-13 no allowance need be made for change
of latitude.
11. A ship in lat. 54 N., long. 34 16' W., sails 24 miles
38 W. of N. Find its new latitude and longitude.
12. A ship sails from lat. 63 S., long. 43 W. to lat. 63
40' S., long. 42 18' W. Find the southing, easting, course
and distance run.
13. A ship sails from lat. 46 N., long. 164 18' E., and
reaches the 47 parallel in long. 163 30' E. Calculate the
course and the distance run.
Note. Nos. 14-18 are to be taken as examples of middle
latitude sailing.
14. Find the course and the distance between Cape Clear
(lat. 51 25' N., long. 9 29' W.) and Brest (lat. 48 23' N., long.
4 29' W.).
15. Find the latitude and longitude of a ship after it has
sailed from Brest, 67 W. of S., 200 miles.
16. From a place in lat. 48 N., long. 25 W., a ship sails
(roughly) south-easterly 215 miles until her departure from
the meridian (i.e. her easting) is 167 miles. Find the course
steered and the new latitude and longitude.
17. Another ship sails from the same place and in three
days reaches lat. 52 N., having made a departure from the
meridian of 260 miles to the west. Calculate her course, dis-
tance run, and longitude.
18. A ship sails north-easterly from lat. 50 S., long. 80 E.,
330 miles, and finds herself in lat. 46 S. Find the course
steered and the new longitude.
EXEKCISE XXI.
RELATIONS OP SINE, COSINE, AND TANGENT.
A.
1. Use the relation tan a = sin a/cos a to test the concord-
ance of the ratios given in the tables on pp. 107, 111 for the
angles 12, 36 and 64.
2. Test by means of the relation sin 2 a + cos 2 a = 1 the
concordance of the values given in the table on p. Ill for
20, 30, 53.
3. The sine of an angle is -%. What is its cosine ?
4. I am told that the sine and cosine of a certain angle
are respectively 0*18 and 0*88. Is the information correct?
5. Use the relation tan a = sin a/ ^/(l sin 2 a) to test
the agreement of the values given for sin 35 and tan 35.
6. The sine of an angle is ^|-. Calculate its tangent as a
vulgar fraction.
7. Use the relation tan a = ,/(! - cos 2 a)/cos a to test the
values given for cos 56 and tan 56.
8. The cosine of an angle is ^. Calculate its tangent.
9. Apply the test cos a = l/^/(tan 2 a + 1) to the values
given for tan 32 and cos 32.
10. The tangent of an angle is ^%. Calculate its cosine.
11. Apply the test sin a = tan a/ ^/(tan 2 a + 1) to the
values given for tan 66 and sin 66.
12. The tangent of an angle is f. Calculate the sine.
B.
13. Prove by a figure that (i) tan a = sin a/cos a,
(ii) sin a = cos a tan a, (in) cos a = sin a/tan a.
14. Prove that tan a/(l - tan a) = sin a/(cos a - sin a).
Verify by substituting the values given in the tables for 28.
Why does this formula apply only to angles less than 45?
120
EXERCISE XXI 121
Does it hold good for 45 itself ? How must the formula be
written for angles greater than 45?
i c -o xi_ A 1 ~ * an a cos a ~ sm a t 11
15. Prove that = : for angles less
1 4- tan a cos a -f- sin a
than 45. Verify when a = 24. Eewrite the formula to
suit angles greater than 45.
16. Prove in two ways that cos 2 a + sin 2 a 1. Deduce
formulas for calculating (i) cos a when sin a is known ; (ii)
sin a when cos a is known.
17. Prove that cos 2 a - sin 2 a = 2 cos 2 a - 1 = 1-2 sin 2 a.
18. Prove that (cos a - sin a) (cos a + sin a) = 1 - 2 sin 2 a.
Verify by substitution when a = 0, 15. Modify the for-
mula to suit angles greater than 45.
19. Demonstrate the following equivalences :
(i) tan a = sin a/ ^/(l - sin 2 a).
(ii) tan a == ^/(l - cos 2 a) /cos a.
(iii) cos a = I/A/(! + tan 2 a),
(iv) sin a = tan a/ ^/(l + tan 2 a).
20. Calculate to three places of decimals the values of the
sine, cosine, and tangent of 30, 45, 60.
EXEKCISE XXII.
LINEAR RELATIONS.
Note. The expression (5, 13) means that the value of x is
5 and the corresponding value of y is 13.
A.
1. The following pairs of values of x and y are connected
by linear relations. Apply a test to each pair to find the
form of the relation.
(i) (5, 13), (12, 27). (ii) (3-5, 10-5), (67. 20-1).
(iii) (8, 27), (15, 6). (iv) (4, 2-1), (10, 87).
2. Find the full form of the relation in each of the fore-
going examples. Draw (on one sheet of paper) the graphs
in (iii) and (iv).
3. Find th linear relations connecting the following pairs
of values of x and y, using the composition method and the
substitution method alternately.
(i) (3, 14), (7, 7-2). (ii) (5-4, 3-0), (9-6, 15-6).
(iii) (07, x-6), (87, 12). (iv) (5, 2-3), (12, 20-5).
4. Two variables are connected by a relation of the form
y = a -t- b/x 2 . When x = 2, y =* 3 ; when x 4, y 2-1.
Find the relation. Calculate the value of y when x = 1 and
when x = 10.
5. Two variables are connected by a relation of the form
y = a *Jx - b. When x = 4, y = 8*5 ; when x = 25, y = 25*3
Find the relation and the value of y when x = 64.
6. The relation between two variables has the form :
= 1 a b
y ^ + 1 + x + 1 + x r
When x = 1, y 5 ; when x 2, y = 3. Find the relation.
What is the value of y when x 3 ?
7. Explain the tests which you apply to determine whether
a linear relation is of the form y = bx, y = a - bx, y = a + bx
or y ^ bx a.
8. A linear relation is of the form y = a + bx. Describe
122
EXERCISE XXII 123
in symbols (i) the composition method, (ii) the substitution
method of calculating b in a given case. Give also a formula
for a. Use (P, Q) and (p, q) as symbols for the given pairs of
values of x and y.
B.
9. Find by the substitution method the simultaneous values
of x and y which are common to the following pairs of linear
relations :
(i) y - 9 - 3x, y - 4x - 5.
(ii) y = i'8 + 5x, y = xx'4 - 3*.
(iii) y = 2x - 3-4, y = 2'3x - 6*4.
(iv) y = 24-8 - 2x, 3x - 27 5.
(v) y = Jx + 7, Jx - Jy =* i.
Confirm your answers to (i), (ii), arid (iv) by graphs.
10. A cyclist leaves his home which is 20 miles from Char-
ing Cross at 9 a.m. and rides towards London at 10 mls./hr.
A motorist starts for London at the same moment from
a place 30 miles from Charing Cross, and travels 18 mls./hr.
At what time does the motorist overtake the cyclist and
where ?
11. The water in a certain reservoir is 6 feet deep, but the
level is sinking 4 inches per day. The water in another
reservoir is 3 feet deep and is rising 5 inches per day. When
will the depth in the two reservoirs be the same, and what
will that depth be ?
12. Two vertical spiral springs hang side by side. When
weights of 5 grms. are hung at their ends the first is 26 cms.
and the second 20 cms. long. When the weight of 5 grms.
is replaced by one of 10 grms. their lengths are 30 cms. and
28 cms. respectively. What weight will cause them to have
the same length and what will that length be ?
13. Calculate by the composition method the simultaneous
values of x and y which are common to the following pairs
of linear relations :
(i) 4x + 3y - 25, sx - 3y - ix.
(ii) 2x - 3y = 5, 3y - x = 8.
5x - 2y - xx, x + 4y - 33-
iv
(vi
y 20, 2x + 3y = 20*1.
67, 4x - 3y = 6.
7x -f ny = 36, I3x + xoy = 46.
(vii) 2'3x - o'7y = 3-4, O'3y - o^x = 0-3-
(viii) 4*6x - i'3y = 2*2, 6'Qx - 2'7y = 1*8.
124 ALGEBRA
Your results should be tested in each case by substitution.
14. Illustrate by graphs your answers to No. 13, (i) and
(vii).
15. Solve No. 13, (ii), (iv), and (vii) by the substitution
method.
16. Two delicate spiral springs hang side by side. When
they are loaded with weights of 5 grms. their lengths are 31*3
cms. and 40*8 cms. respectively. When weights of 10 grms.
are substituted the lengths become 46 -8 cms. and 64'3 cms.
Find (i) by drawing a graph and (ii) by a calculation whether
any weight will stretch them equally assuming a linear re-
lation between length and load.
17. A man said: "I am thinking of two numbers; four
times the first added to five times the second gives 7 ; twice
the first subtracted from six times the second leaves 22 ". Is
it possible to find two such numbers ?
18. Find the values of x and y which satisfy simultaneously
the following pairs of relations :
(i) i* 2 + y 2 = 6, 5* 2 - 6y 3 = 21.
/. N 12 I 6 I
(lv) i? - & = I >** + 3 T 2 = '
(v) 1 x 2 - y 2 = 16, x + y = 8.
(vi) 4x 2 - py 2 = 63, 2x - 37 = 3.
(vii) x 2 + 2xy + y 2 = 49, 2x - 3y = 4.
(viii) 4 x 2 - I2xy + 9 y 2 = 4, 4x - 77 = 3.
0.
19. Eliminate the variable z from each of the following
pairs of relations :
(i) 3x - 47 + 2z = i, x + y - z = o.
(ii) 7x + 3z = 2,
(iii) \L - ix - y
(ii) 7x + 3z = 2, 2z + 57 = i.
ii)
(iv) - + - + - = -, + = o.
v x y z V 3x 2y z
1 If x* - -j/ 2 = 16 and x + y = 8 what is the value of x - y ?
When you know the value of x + y and x - y you can find the value
of x and y.
EXERCISE XXII 125
20. From your answer to No. 19 (i) find (i) the value of y
when x = 2 ; (ii) the value of x when y = 12. Find the
corresponding values of z. Do both relations give the same
value for z in each case ?
21. In No. 19 (iii) what value of x is associated with
y = 1^? Verify that these values of x and y are associated
with the same value of z.
22. Eliminate a from the three relations :
x - y = sin a, x + y == cos a, y/x tan a.
23. From the following pairs of relations derive statements
involving x and y only :
(i) x = 3 sin a, y = 4 cos a.
(h) a x sin a, b = y cos a.
(iii) ^(x + y) sin a =* I, ^/(x - y) cos a = I.
24. In No. 23 (i) find the value of y associated with
x = 2 ^2. What value of a is associated with these values
of x and y ?
25. In No. 23 (iii) what is the value of y when x 4 ?
26. Eliminate a from the relations :
,J(%x - 3y) tan a, >J(3x - %y) cos a = 1
27. In the last example find the value of x when y = J,
Find the corresponding value of a.
EXERCISE XXIIL
INVERSE PROPORTION.
A.
1. A man and his son both determine to save money regu-
larly the former to buy a motor- bicycle costing 40, the
latter to buy an ordinary bicycle costing 10. Draw on the
same sheet two graphs exhibiting the average savings, in
shillings per week, necessary to provide the required sums
in a given number of weeks. (Choose your scales so as to
include a saving by the boy of sixpence a week.)
2. Let the formulae corresponding to the graphs in No. 1
be S =* kjt and S = k 2 /t. What will be the value of & x and & 2
if S represents the savings (i) in shillings per week, (ii) in
pounds per month of four weeks, (iii) in pence per day ?
3. Two variables, x and y, are in inverse proportion. When
x = 2 ? y = 10*8. Write down the relation between them.
What is the value of y when x 0*018 and the value of x
when y = 0'0002 ?
4. If the graph of the relation of No. 3 were drawn what
would be (i) the co-ordinates of the vertex, (ii) the length of
the axis ?
5. What pair of values of x and y satisfies simultaneously
(i) the relations y = 2# and xy = 200 ; (ii) the relations y =*
lxa,udxy = 800?
6. Illustrate your answer to No. 5 by drawing across the
graphs of No. 1 the straight lines corresponding to y = 2a? and
y = &
7. Show that the pair of values (p/q, pq) is common to the
relations xy = p* and y = (fx. At what point would the
corresponding graphs cross one another ?
8. Prove by algebra that two inverse proportion curves can
never intersect. [Take xy = a and xy = b as the corre-
sponding formulas and show that they allow no common
values of x and y.]
128
EXERCISE XXIII 127
9. Kegard the curves in No. 1 as corresponding to the
relations xy = a and xy = b. At a certain point on the
z-axis let the ordinates of the two curves be P and Q. At
a certain point on the 7/-axis let the abscissae be p and q.
Show that wherever these points are taken, P/Q = a/b = pjq.
10. What is the ratio of the axes of these curves?
B.
11. A number of cylindrical tins of different shapes are to
be made, each to hold a pint of liquid (34*7 cubic inches)
when filled to 1 inch from the top. Describe in words the
relation between the height and the area of the bottom of the
tins. Express the relation in a formula (h, A). Calculate
the height of the tin in which the bottom contains 6*94 square
inches.
12. In the case of another set of tins I know that each is
to carry 100 cubic inches but I do not know how much free
space is to be left above the liquid. Taking up one of the
tins T find that its height is 10 inches and the area of cross-
section 12 square inches. Up to what point is the vessel
intended to be filled? The sectional area of another of the
set is 20 square inches. What is its depth ?
13. Write formulae descriptive of the relation between x
and y :
(i) When (y - 3) is inversely proportional to x ;
(ii) When y is inversely proportional to (x + 4*7) ;
(iii) When (y + 2) is inversely proportional to (x - 5).
Use k in each case as the symbol for the unknown constant.
14. Rewrite the formulae of No. 13, replacing k by its
numerical value calculated from the following information :
In (i) when x 4, y = 10.
In (ii) when x = 5'3, y = 0'36.
In (iii) when x 13, y = 7.
15. A variable y is inversely proportional to (x - a).
When x = 12, y = 10 and when x = 10, y = 15. Find the
formula for the relation.
16. The following relations between x and y may be
regarded as expressing inverse proportion between certain
numbers. Describe the numbers.
128 ALGEBRA
(i) xy + 5x + 2y = 20. (ii) xy - 3x + 77 == 121.
(iii) 6xy -f 4x + gj = 132. (iv) 5xy - loox - 2y == o.
17. In No. 16 (i) find the value of y when x ~ 3. In (iii)
find x when y 10. In (iv) find x when y = 30.
18. What pair of values of the variables is common to the
relations y = x + 2 and xy = 8? [After substitution use
the method of Ex. XVI, No. 118.]
19. Find the values of x and y which satisfy simultaneously
the relations y = 4# - 4 and xy = 8.
20. Illustrate your answers to Nos. 18 and 19 by draw-
ing on one sheet the graphs corresponding to y = x + 2,
y = 4:X - 4, and xy = 8.
EXERCISE XXIV.
PROPORTION TO SQUARES AND CUBES.
A.
1. Obtain a graph showing the breaking load of a hempen
rope (Ex. IV, No. 19) by first drawing the straight line?/ = 0'6#
and then transforming it into the curve 1 y = 0'6# 2 . (Gradu-
ate the tf-axis from to 30.) From the graph determine (i)
the breaking load when the circumference of the rope is
2 -5 inches, (ii) the circumference of the rope that will just
support 7 '8 tons.
2. Obtain a graph showing the distances visible from a
given height (Ex. IV, No. 6) by transforming the straight
line y 1*22# into the curve y ~ 1*22 *Jx. (Graduate as
before.) Use it to find (i) the distance visible from a height
of 2000 feet, (ii) the height at which you can see 50 miles.
3. Describe in words the relation between p and L in Ex.
IV, No. 5. Obtain a graph of the relation by transforming
the line y = Q'lx into the curve y = 0*lo; 3 . From the graph
find (i) p when L = 2*8, (ii) L when p = 2.
4. The diameter in centimetres of a certain kind of spheri-
cal bullets is given by the formula d 0'8 l/w, w being the
weight of the bullet in grammes. Obtain a " ready -reckoner "
of the diameters of these bullets by transforming the line
y = Q-Sx into the curve y = 0'8 */#. Find (i) the diameter
of a bullet weighing 35 grms., and (ii) the weight of the
bullet with a diameter of 1*7 cm.
5. Write formulae descriptive of the following relations :
(i) y is directly proportional to the square of x and when
x = 7, y = 4-9;
(ii) y is directly proportional to the square root of x + 4 and
when x = 0, y = 2*8 ;
(iii) y - 3 is directly proportional to the cube root of x 4- 7 and
when x = 20, y = 18.
1 " The curve y = 0'6x 2 " is, of course, a shortened expression for
" the curve corresponding to the relation y 0'6x a ".
129 9
130 ALGEBRA
Note. Suppose you have two statements such as :
16 = ^(30 + *>) . (i)
8-aV(20+&) .... (ii)
They can be reduced to a single statement containing b but
not a by dividing the two sides of (i) by the corresponding
sides of (ii) :
2-^(30 + 6)7^(20 + 6) . . (iii)
From (iii) b can be calculated, while a can then be found
from (ii) or (i).
6. It is known that in a certain relation y is directly propor-
tional to the square root of x - a, that, when x ~ 10, y = 6
and that, when x 17, y 8. Find the relation.
7. A variable y is directly proportional to the square of
x -h a. When x = 0, y = 36, when x = 1, y 64. Find
the relation.
8. What simultaneous values of x and y are common
(i) to the relations y = 6x' 2 and y 3x
(ii) to the relations y = Q'&f/x and y = 0'2sc ;
(iii) to the relations y = 3 ^x and y = |2 /v/(. + 10) ;
(iv) to the relations y = 2(x + 3) 2 and y = 2(x - 7) 2 ?
Illustrate your answers to (i) and (ii) by means of the
graphs of Nos. 1 and 4.
9. Draw a graph to illustrate your answer to No. 8 (iii).
B.
10. Describe in words the relation of P and D in Ex. IV,
No. 30. Construct a " ready-reckoner " for the formula by
transforming the curve xy = 18 into the curve y = 18/^/ic.
Find (i) the loss of steam-pressure with a cylinder 20 inches
in diameter, and (ii) the diameter of cylinder that gives a loss
of 3-6 lb./in. 2
11. When a long straight magnet is held vertically above
a piece of iron (e.g. a nail on a table) its lifting power is in-
versely proportional to the square of the distance between
the iron and the bottom of the magnet. A certain magnet
can just lift a nail weighing 12 grins. at a distance of 1 cm.
Write down the relation between w and d.
12. Exhibit the relation of No. 11 in a graph obtained by
transforming the curve xy = 12 into the curve y = 12/# 2 .
13. Express in formulae the following relations :
EXERCISE XXIV 131
(i) y is inversely proportional to the square of x - 3 and when
x = 5, y = 3 ;
(ii) y - 4 is inversely proportional to the square root of x - 10
and when x = 11, y = 6 ;
(iii) y is inversely proportional to the cube of x and when
x == 2, y = 3 ;
(iv) y is inversely proportional to the cube root of x and when
x = 64, y = 6.
14. On the same sheet of paper transform the curve xy = 24
into the curves which correspond to the relations of No. 13
(iii) and (iv).
15. If the curves corresponding to the relations of No. 13
(i) and (ii) were drawn what lines would be, respectively, their
asymptotes ?
16. Find the relations that satisfy the following condi-
tions :
(i) y is inversely proportional to the square of x + a ; when
x = 0, y 12 and when x = 2, y = 3 ;
(ii) y is inversely proportional to the square root of 2x a ;
when x = 2, y = 21 and when x = 6, y = 7.
17. Find the pairs of values of x and y which are common
(i) to the relations y = 12/ <Jx and y = 18/ *J(x + 5) ;
(ii) to the relations y 3 *J(x - 6) and y = 24/ *J(x + 6) ;
(iii) to the relations y = 4/ N /(x 2 - 100) and y = I/ J(x + 10).
18. Illustrate your answer to No. 17 (ii) by drawing the
graphs of the relations.
EXERCISE XXV.
JOINT VARIATION.
1. Describe in words the relations between the variables in
the following examples from Ex. IV : (i) No. 17, (ii) No. 18,
(iii) No. 22, (iv) No. 25 (i), (v) No. 32.
2. Express the following relations in formulae, putting k for
the unknown constant :
(i) z varies directly as x 2 and inversely as y;
(ii) z varies directly as the square root of x and inversely as y 2 ;
(iii) z is directly proportional to fyx when y is constant and in-
versely proportional to y when x is constant ;
(iv) z is inversely proportional to x when y is constant and to
the cube root of y when x is constant.
3. Supply the values of k in No. 2 from the following in-
formation :
(i) when x = 1-2, y = 0'2 and z = 7'2 ;
(ii) when x = 25, y 4 and z = 15 ;
(iii) when x = 8, y = 07 and z = 6 ;
(iv) when x = 1'4, y 1 and z = 0*5.
4. The east and west distance between two points on a
globe which are situated on the same parallel of latitude is
directly proportional to their difference of longitude (Z), to the
cosine of the latitude (X), and to the radius (r) of the globe.
Two points on the earth's surface whose latitude is 52 and
whose difference of longitude is 13 are 553 miles apart. How
far apart are two points on the surface of the moon whose
latitude is 37 and whose difference of longitude is 24? The
radius of the earth is 3959 miles ; of the moon 1080 miles.
5. To measure the strength of a bar-magnet the electrician
lays it on a line east and west pointing to the middle of a small
compass needle. He then notes the number of degrees (a)
132
EXERCISE XXV 133
through which the needle is deflected and the distance (d) of
the middle of the bar-magnet from the middle of the needle.
The rule is that the strength of the magnet (M) varies directly
as the cube of the distance and directly as the tangent of the
angle of deflection.
When a magnet of strength 6444 is placed with its mid-
point 80 cms. from a needle the deflection is 4. What is the
strength of a magnet which when placed 60 cms. from the
needle produces a deflection of 9 ?
6. Kepler (c. 1610) discovered that the square of the time
(T) taken by a planet to revolve about the sun varies as the cube
of its distance (d) from the sun. The earth (whose revolution
takes, of course, a year) is distant from the sun about 93
millions of miles. Find (i) the distance from the sun of
Jupiter whose revolution takes twelve years ; and (ii) the time
of revolution of Neptune whose distance is about 2780 millons
of miles. (Take a million miles as unit.)
7. A variable y is the sum of two parts of which the first
is directly proportional to x and the second is inversely pro-
portional to the square of x. Give a formula for y, using a
and b as symbols of the two constants needed.
8. Replace a and b in No. 7 by their numerical values
obtained from, the knowledge that when x = 4, y = 5 and
when x = 10, y = 5-08.
9. A variable y is the difference between two other quantities.
Of these the first varies directly as sin a and inversely as
x - 1 ; the second varies directly as cos a and x conjointly and
inversely as (x 2) 2 . Express the relation in a formula.
10. Eeplace the symbols of the constants in No. 9 by their
numerical values given that, when a = 26 and x = 3, y
1-683, and that, when a = 20 and x = 4, y = T34.
11. If a current of electricity is flowing round a circular
coil of wire fixed vertically with its plane north and south and
a small compass needle is brought near to it along the axis of
the coil the needle is deflected from its usual north and south
position. The tangent of the deflection (a) varies directly as
the strength of the current (C), the area of the coil (A) and
the number of turns of wire (ri) in the coil conjointly. It also
varies inversely as the cube of the distance of the needle from
the plane of the coil. When a current of 5 units (" amperes ")
flows round a coil containing 100 turns and offering an area
of 300 sq. cms. a needle at a distance of 30 cms. is deflected
134 ALGEBRA
through 5-. What deflection will be produced at a distance
of 60 cms. by a current of 8 amperes flowing in a coil con-
taining 300 turns and having an area of 200 sq. cms. ?
12. Write proofs of the following propositions :
(i) If z oc x (y constant) and z oc y (x constant) then zee xy;
(ii) If 2 oc cc 2 (y constant) andz oc 1/y 3 (x constant) then z oc ;e 2 /z 3
EXERCISE XXVI.
SUPPLEMENTARY EXAMPLES.
A. TEST PAPER 1.
1. The speed with which waves travel across the sea is given
by the formula :
s = speed in ft./sec. ; d depth of water in feet; h^ height
of wave in feet.
(i) Do high waves or low waves travel more rapidly in water
of the same depth ?
(ii) What formula would be sufficient to calculate s for waves
moving on very deep water ?
(iii) Calculate the speed of a wave 2 feet high where the sea
is 10 feet deep ;
(iv) Calculate the speed of a wave 4 feet high where the sea
is 24 feet deep.
2. From the formula of No. 1 obtain a rule for calculating
the depth of the sea by observing the speed of waves of a
known height.
Waves 2 feet high are observed to travel across the sea at
a speed of 32*2 ft./sec. What is the depth of the sea at that
point ?
3. Factorize :
(i) ab - aa + 2b - 6 ; (ii) p(x - 27) + sq(x - 27) ;
(iii) p 2 + 2pq + q a - p - q ; (iv) 9 - (2 - p) 2 .
4. Find the values of n which satisfy the following rela-
tions :
(iii) (n - 2)(2n - 3) = 2(n - i) a ; (iv) n 2 - 6n = 16.
5. Show (i) that the square of an odd number is always
odd, and (ii) that the product of any two odd numbers is
always odd. [Expressions such as 2?i + 1, 2m - 1, describe
odd numbers whether n and m are themselves odd or even.]
135
136 ALGEBRA
6. When 2-3 has been subtracted from a certain number
the square root of the residue is 3. What is the number?
7. The sides of an oblong table are a feet and b feet long.
In damp weather the wood swells, but unevenly in the two
directions. Each inch in the longer side of the table becomes
(1 4- p) inch and each inch in the shorter side (1 + q), p and
q being, of course, very small. Write down an expression
for the increase of area of the table to a first approximation.
8. Find the linear relations between x and y which are
satisfied (i) by the values (4, 5) and (5, 7) ; (ii) by the values
(0, 3) and (1, 1). Have the relations a common pair of values
of x and y ? Illustrate your answer by graphs.
9. The road from one village A to another B bears 27 E.
of S. for 2- miles. It then turns 12 further to the east and
continues in this direction for 2 miles. Find the direction
from A to B as the crow flies.
10. A ship in latitude 54 N., longitude 17 W., received
by wireless telegraphy a message of distress from a ship in
longitude 20 W. on the same parallel. How far was the
distressed ship due west of the other ?
B. TEST PAPER 2.
1. When a submarine mine explodes near a ship it pro-
duces a tremendous pressure on its side. (To produce a fatal
injury to an ironclad the pressure should be > 12,000 lb./ in. 2 )
The following formula shows how to calculate the pressure
produced at a point A on the side of the ship, the centre of the
charge being at a point B. Let d = distance AB, a = the in-
clination of the line AB to the horizontal, C = the weight of
the charge in lb. K and k are constants (or coefficients) which
depend on the explosive used. For No. 1 dynamite (the
best explosive for the purpose) K = 100, k = 20 (also for gun-
cotton). For gunpowder K = 25, k = 35.
d \ ePJ\ 9<V
(i) What does the formula become when the points A and B
are on the same level ?
(ii) If AB is horizontal, what number of pounds of gunpowder
is as effective as 1 lb. of dynamite ?
(iii) Would this quantity of gunpowder produce a greater or less
pressure than the 1 lb. of dynamite when the point A is
at a higher level than B ?
EXERCISE XXVI 137
(iv) A charge of 100 Ib. of No. 1 dynamite is exploded near a
warship. Find the pressure at a point on the same level
as the mine and 5 feet away. Will it be sufficient to do a
vital injury to the ship ?
(v) Calculate the pressure produced by the same charge at a
point 10 feet away and 20 above the level of the charge.
2. The following formula is taken from an electrical
engineer's pocket-book. Change the subject (i) to C v (ii) to
C 2 , (iii) to C 3 . How can you determine the third formula
when you know the second ?
3. The Egyptians used the fact that 3 2 + 4 2 = 5 2 as a
means of drawing a right angle. Show that if a = 4w and
b = 4n 2 - 1 (n being any whole number), then a 2 + & 2 is
always the square of a whole number. Use your proof to
construct a table of the first four triads which have the same
property as 3, 4, and 5.
4. Transform each of the following expressions into a shape
more convenient for computation :
(i) (2a - 3b) a - i6c 3 ; (ii) (sp - 2) a - (2q + 3 ) a ;
,. i 2a. - I
TTi a* - 6a + 8'
5. Find values of n which fit the following statements :
(i) j(n - 2) + J(5 - n) + f (n - i) - o ;
n + a n - a n 2 - a 2 '
6. On a windy day a boy cycled to a place 24 miles away
and returned by the same road. He estimated that the wind
added 2 miles an hour to his speed on the outward journey
and retarded him by the same amount on the way home.
He took an hour longer to return than to go. At what rate
does he normally ride ?
7. The relations y = 3x - 5 and y 1 + 9/x include a
common pair of values of x and y. Find it. Illustrate your
138 ALGEBRA
result by drawing the graphs of the relations (from x = to
x = 5) on one sheet.
8. Show that
(P - g)s ^ 1 ___ 1
1 - (p + q)x + pqx* 2 1 - i)X 1 - qx
Use this equivalence to find a formula for the value of
(P - g)s
1 - (p + q)x + pqx 2
to the second approximation when # is a small fraction.
9. The relation between two variables is known to be of the
form y = a - b sin x. When x = 14^, y = 8 ; when x = 30,
y = 6. Find the relation and calculate (i) the greatest value
of y ; (ii) the least value of y ; (iii) the value of y when
x = 56.
Illustrate your answer by sketching the graph of the rela-
tion.
10. A church spire is in the form of a pyramid 24 feet high,
standing on a square base which measures 14 feet each way.
Calculate to the nearest degree the angles (i) between two
opposite faces ; (ii) between the two sloping edges of one of
the triangular faces; (iii) between two opposite sloping
edges of the pyramid.
[Either construct a paper model or draw sections of the
pyramid to help you in your reasoning.]
0. TEST PAPER 3.
1. The speed with which water is carried along a pipe
depends partly on the size and length of the pipe and partly
on the " head " of the water. By the head is meant the
vertical height above the pipe of the water surface in the
cistern or reservoir. The number of gallons discharged per
minute is given by the formula :
a - 29-4
H = head in feet ; d = diameter of pipe in inches , I = length of
pipe in feet.
(i) Calculate the rate of discharge from a 2-inch pipe 3200 feet
from the reservoir under a head of 100 feet ;
(ii) The same, with an 8-inch main 5 miles long aud a head of
132 feet,
EXERCISE XXVI 139
2. Change the subject of the formula of No. 1 (i) to I, (ii)
fco H, (iii) to d.
[You are given that 1/864-4 = 0'001157 and that the fifth
root of this number is 0*2586.]
3. If a = 2w- + 1 and b = %n(n + 1), n being any whole
number, show that a 2 + b 2 is always the square of the number
6+1.
What triads are obtained by putting n = 1, 3, 10, in suc-
cession ?
4. Factorize (n 2 + 3n - 2) 2 - (n 2 - 3n + 2) 2 , and use the
result to find two distinct values of n which reduce the value
of the original expression to zero. Confirm your conclusion
by substitution in the original expression.
5. Find values of n which comply with the following con-
ditions :
(i) J(7n - i) + i(sn - i) - K4n - x) = o ;
(11) p(n - p) - q(n - 2p) = q 2 ;
n - 4 n - 3
6. One customer buys 14 Ib. of tea and 10 Ib. of coffee for
2 3s., and another buys 11 Ib. of tea and 15 Ib. of coffee for
2 4s. 6d. Find the prices of tea and coffee per Ib.
7. The weight (w) of a given body in the neighbourhood of
a heavenly body (such as the earth or the moon) is propor-
tional directly to the mass (M) of the heavenly body and in-
versely to the square of the distance (d) of the given body
from the centre of the heavenly body. Express this relation
in a formula.
What value must be given to the constant in this formula
so that the latter may give the weight of a body which weighs
1 Ib. on the surface of the earth ; the mass of the earth being
taken as 1 and its radius as 4000 miles ? (A. thousand miles
is to be taken as the unit of distance.)
What would this body weigh on the surface of the moon ?
(The moon's radius may be taken as 1080 miles, and its
mass as y^ of the earth's mass.)
8. In Jules Verne's "Journey to the Moon," bodies were
found to lose their weight altogether at a certain point between
the earth and the moon. Calculate (in thousands of miles) the
distance of this point from the earth's centre, assuming the
distance between the centres of the earth and moon to be
240,000 miles,
140 ALGEBRA
9. I have a wooden rectangular box 12 inches long, 9 inches
wide and 5 inches deep. A sheet of glass is fixed inside it so
that one edge lies along the bottom edge of one end of the box
and the other along the top edge of the other end of the box.
Find the angle between the plane of the glass and the bottom
of the box. Find also the angle between a diagonal of the
glass sheet and the bottom of the box.
10. The diameter of the circular tower of a castle is 40 feet.
Starting from the wall the roof rises at an angle of 35^ with
the horizontal plane until it encloses a circle 12 feet in dia-
meter. It now rises at a steeper slope to a level 20 feet above
the top of the wall. Finally it is capped by a cone 5 feet
high with a semi-angle of 31. Sketch a vertical section of
the roof. Calculate (i) the slope of the middle section ; (ii)
the slope of the edges of the cone ; (iii) the vertical height of
the middle section ; (iv) the diameter of the base of the cone ;
(v) the total vertical height of the roof.
D. STATISTICS.
1. The folio wing numbers give the heights in inches of seven
boys measured in each case on the fourteenth birthday :
62, 58, 59, 63, 58, 59-5, 60-5.
Arrange the heights in a column in order of magnitude, and
indicate the middle number of the series. This is called the
median. Calculate also the arithmetic average or mean of
the heights.
2. Set beside the column of No. 1 a second column showing
how much greater or less each height is than the median
height. Take the average of these differences. [Note that
you must divide the sum by 7 not 6.] This average is called
the mean deviation from the median.
3. Add a third column giving the difference between each
height and the arithmetic mean of the heights. Obtain the
average of the differences. This number is the mean devia-
tion from the mean. Which of the two mean deviations is
the less ?
4. Find (a) the median, (b) the arithmetic mean, (c) the
mean deviation from the median, (d) the mean deviation from
the arithmetic mean in the case of each of the following sets
of numbers ; -
EXERCISE XXVI 141
(i) 4, 13, 10, 6, 7 ;
(ii) 24, 16, 19, 9, 17, 10, 17, 8, 15 ;
(iii) 17, 14, 11, 6, 21, 17, 10, 12, 13, 12, 15, 14, 7.
Note, The results of Nos. 2-4 illustrate the following
facts : (i) the median may either be equal to, greater than, or
less than the mean ; (ii) whether the median is greater or less
than the mean, the mean deviation from the median is always
less than the mean deviation from the mean. It is, in fact,
less than the mean deviation from any other of the measure-
ments. For this reason when we speak of " the mean devia-
tion " without further specification the mean deviation from
the median is the number intended.
The terms " median " and " mean deviation " are often
used to express the general result of a group of measurements.
The median fixes the position of the middle of the group,
the mean deviation measures the degree of dispersion of the
measurements on either side of the middle one.
5. Two marksmen, A and B, fire five rounds each at a
target. The following figures give the distance in inches of
each "hit" from the centre of the bull's eye. Express nu-
merically the value of each performance by means of the median
and mean deviation . Which do you think is the better shot ?
A 15, 8, 17-5, 10-5, 9.
B 10-5, 13-5, 16, 9, 6.
6. When there is an even number of measurements there
is, strictly speaking, no median. The name is, however,
generally given to the number half-way between the two
middle measurements. What is the median of the following
measurements ? Prove without calculation that the mean
deviation would be the same if 13 or 14 were assumed as the
median. Calculate the mean deviation, using one of these
numbers instead of the number which you give as the
median :
15, 12, 12, 14, 13, 24.
7. Find the arithmetic mean of the measurements given in
No. 6 and the mean deviation from the mean. Is it greater
or less than the mean deviation from the median ?
8. Out of a certain form eleven pupils succeeded in solving
one of the more difficult problems in this book. Their ages
(in years and months) were :
142 ALGEBRA
12 : 5, 12 : 8, 13, 13 : 2, 13 : 5, 13 : 8, 13 : 11, 13 : 11, 14 : 2, 14 : 4,
14 : 9.
Describe the general position and dispersion of this group.
Note. In the foregoing group of measurements 13 (which
is a quarter of the way up the series) and 14 : 2 (a quarter of
the way down the series) are called the lower and upper
quartiles. The difference between them (1 : 2) is called the
interquartile range. The semi-interquartile range,
here 0:7, is often taken, instead of the mean deviation, as a
measure of the dispersion of the measurements. It may be
called, more briefly but less accurately, the quartile devia-
tion.
Observe that if there were 12 measurements the quartiles
would be taken to lie half-way between the third and fourth
terms from each end of the series.
9. Find the median, the quartiles, and the quartile devia-
tion in the case of each of the following groups of measure-
ments :
(i) 3-8, 4-1, 4-1, 47, 5-2, 5*5, 6'3, 6*3, 6'9, 7, 7% 7'5, 77,
8-1;
(ii) 137, 14-2, 14-2, 15-6, 1S'9, 16'3, 16'5, 17, 171, 17'4, 17 '6,
18'2, 187, 18-8, 19*2, 20'1, 207, 21% 22'8, 23.
10. Two sets of measurements of a quantity were made,
each set containing a dozen measurements. The median was
in each case 24'8. The dispersion of the measurements, as
measured by the mean deviation, was 4*8 for the one set and
2 -3 for the other. Which set was the more trustworthy and
why?
11. The medians of two sets of twenty measurements of a
length were 30'7 cms. and 25-6 cms. respectively, while the
quartile deviation was in each case 4*2 cms. Which was the
better set of measurements ?
12. Two surveyors, A and B, determined, each the same
number of times, the height of a certain hill. The median of
A's results was 824 feet, and the mean deviation 2 -3 feet.
The median of B's results was 822 feet and the mean devia-
tion 2*1 feet. Which surveyor obtained the most consistent
results ?
13. Measurements ot a number of wooden discs were made
in order to find the ratio of the circumference of a circle ta
the diameter. The results are shown in the table. Plot OD
EXERCISE XXVI
143
squared paper the points which represent the various measure-
ments. Let P be any one of these points, PN the perpen-
dicular upon the horizontal axis, and O the origin. Then if
PO be joined the value of the ratio given by the correspond-
ing pair of measurements is PN/NO i.e. tan PON. The
ratios given by the other pairs are obtained similarly by join-
'ing the corresponding points to O. The fan of lines has a
median and quartiles which correspond to the median and
quartile values of the ratio. The best way to describe the
general result of the measurements will be to state the median
and the quartile deviation of the measured ratios.
In practice the median should be drawn as a thin but firm
line, and the two quartiles as fine dotted lines. The median
line is to be regarded as the "graph" of the measurements
while the closeness of the quartiles serves as an ocular in-
dication of its trustworthiness. The other lines need not be
drawn.
Exhibit graphically by this method the general result of the
measurements given in the table and calculate the median and
quartile deviation of the ratios.
Diam.
Circf.
27
8-1
3-5
11-4
4-0
127
4-5
145
5-01 5-8
15-2 17 9
6-01 6*6
18-8J 20 -5
7-3
23-4
7-5
8 I cms.
25-0 cms.
14. A variable P is known to be directly proportional to
another variable Q, i.e. P = kQ, k being a constant. The
following measurements were made in order to determine k.
Exhibit graphically and express numerically the general result
of the measurements.
Q1-5I2-OI3-0 3-9 61 6*2 6-81 7'2| 8-6 911 9-9|10-4|ll-0|ll-5|12
P 3-6|5-3|7-9 9-4 12-2 16-0 17'2|l77l21-822-6|25'6|26-o|277|27'8|297
15. A variable P is known to be inversely proportional to
another variable Q, i.e. P = k/Q, k being a constant. A
number of corresponding measurements of the variables are
given in the table. Plot the values of P against the recipro-
cals of the values of Q so as to exhibit graphically the various
values of k given by the different measurements. What is
their median and quartile deviation ?
Q12345678910
P 1-9 0-9 0-67 049 0-46 0'37 0-31 0'27 0-18 0-17
144
ALGEBRA
16. Draw on a single sheet the inverse proportion curve
corresponding to the median value of k in No. 15 and also
the curves corresponding to the quartile values. Draw the
median curve with a firm line and the quartile curves with
dotted lines.
17. A variable P is known to vary directly as the square of
a second variable Q. The table gives a series of correspond-
ing measurements of the variables. Plot P to Q 2 in order to
exhibit the values which the measurements give for k in the
formula P = &Q 2 . Calculate the median and the quartile
values.
Q
i-o
0-55
1-5
0'95
2-0
1-8
26
2-91
3-0
3-78
3-5
6-62
4-0
8-27
4-6
10-42
5-0
12-6
5-5
14-23
60
19-78
18. Draw as in No. 16 the median and quartile curves
which correspond directly to the measurements of P and Q
given in the table of No. 17.
19. The relation between two variables is known to be
given by the formula P = k JQ. Exhibit graphically the
values of k derived from the following measurements, indicat-
ing as in No. 14 the median and quartile values.
Qil-0
2-0
15-0 |8-5llO'Oil3-5
I
P|l-352-35|3-05!4'5| 4'6|
10-0|17-6
5-6 6'85
20-0
715
23-5 128-0
7 -461 7'85
31-0 135-540
8-15 911016
20. Draw as in No. 16 the median and quartile curves
which correspond directly to the measurements given in No.
19.
B. SORDS.
Note. In carrying out calculations with surds it is generally
best to follow the opposite plan to that of Exs. V and VI.
Instead of replacing an expression by its factors the factors
should be replaced by their product before the values of the
surds are substituted. In this way troublesome multiplica-
tions and divisions are avoided.
Assume in these examples that ^2 1*414, J3 ** 1*732;
JZ - 2-236, ^6 = 2-449, JW = 3-162.
1. Find to three decimal places the value of
s/(3~
EXERCISE XXVI 145
2. Find to three decimal places the value of
(i) (i + ^2X1 + 5v/3) ; (ii) (7^5 - 2X^2 ~
(ill)
3. Evaluate
(i) (i + j*y ; (ii) (2 -
4. Evaluate
(i) (i + ^2)* ; (ii) (2 - V3) 8 5 (i") (a>/3 - N/2) 1 -
5. Find the value of the following products :
(i) U5 + V2XV5 - v/2); (ii)
(>/i7 - 4)(<s/i7 + 4)
(7 + 3s/aX7 - 3N/2);
(ui) (^17 - 4)(Vi7 + 4) ; ( iv ) ( 2 Vs + I X 2 > / s - ;
" '>; (vi) (3\/5) + '
6. Complete the identities :
(i) ( \/a N/b)( \/a
(ii) (2 Ja - 3 \/b)(2 ^/a + 3 \/b) =
^iv) (>/pq -
(vii) (a -
(viii) (a - 3<x/b)(a+
-The fraction -^--
This foim of the fraction is much easier to evaluate than the
original form. The expressions ( ^5 - ^2) and ( ^5 4- */2)
are said to be conjugate binomial surds. We have,
therefore, the rule that the binomial surd in the denomi-
nator of a fraction may be removed by multiplying both
numerator and denominator by the conjugate surd.
7. Convert the following fractions into the form most
suitable for evaluation :
in
8. Simplify the following practical expressions, giving the
final denominator in a rational form :
(i) i ~ ^T72 ' w pVs-4 " pVs '
9. Find values of n which accord with the following state-
ments :
(i) (n ~ i)v/2 = ^5 + 2 J (") n ( J + >/3) = 2n + 4^/5 ;
(iii) n 2 - 6n = i ; (iv) n 2 - n - i = o ;
(v) s/(2n - x) = N/S - 2.
10. A straight line AB of length a is divided at H so that
(AH) 2 = AB.BH. Show that
Calculate the position of H when the line is 20 cms. long.
Verify the correctness of your calculation.
F. TEST PAPER 4.
1. If the course of a river is obstructed (e.g. by a sudden
narrowing of its bed as at "the Strid," near Bolton Abbey)
the level of the water in front of the obstruction is higher than
it would otherwise have been. The rise in feet is given by the
formula :
ggy + 0-05) (p. - 1).
v = velocity of the river (in feet per second) before the obstruc-
tion is reached ; p = cross-section of river before the obstruction
divided by cross- section at obstruction.
(i) A river flowing with a velocity of 4 feet per second has
suddenly to pass between rocks where the cross-section
of the bed is reduced to of its amount just before the
obstruction. How much will the water rise at this point ?
(ii) Repeat, substituting 6 feet per second and for the numbers
given.
EXERCISE XXVI 147
2. Write down a formula for calculating the normal speed
of a river from the rise of level caused by a sudden constric-
tion of its bed. (See No. 1.)
3. Take any number of three different digits, a, &, c.
Obtain a second number by reversing the order of the digits.
Find the digits in the difference between the two numbers.
Obtain a third number by reversing the order of the digits in
the difference. Add this third number to the difference
between the first two. Show that the sum will always be
the same.
4. The tangent of a certain angle is ^, 7v . Find an
^ & 2n(n+l)
expression (i) for the sine, (ii) for the cosine of the same
angle.
5. Show that a 4 + a 2 6 2 + Z> 4 - (a 2 + fc 52 )' 2 - a 2 6 2 . Hence find
the factors of the former expression.
6. (i) Simplify the fractional expression :
7 6a-5 3a-f4
(a-l)(2a-3) " 2a^~3 + ~a^T'
Use your result to find (ii) the value of the expression when
a = 21-5 ; (iii) the value of a for which the expression has the
value 2.
7. When 3 is added to the numerator and also to the
denominator of a certain fraction its value becomes 0*7.
When the 3 is subtracted its value is 0'25. Find the frac-
tion.
8. What can you deduce from the following statements
about the value of n in each case ?
(i) 711 - 3 > 4fc + i) ;
(ii) (sn + 3) 2 - (3n - 5) 2 1
(*> ^T
9. A certain number of persons promised to subscribe
equally to a gift which cost 20. Three of them failed to pay
their subscriptions. Each of the others had, therefore, to in-
crease his subscription by 18s. How many persons originally
undertook to 'subscribe, and what was the amount which they
promised ?
10. Where I stand the top of a tree is seen to have an
altitude of 11. I walk 100 yards towards it along a slope
10*
148 ALGEBRA
which rises at an angle of 10 with the horizontal. The alti-
tude of the top of the tree is now 12. Calculate the vertical
distance of the tree-top above the level of the eye, and the
horizontal distance between the vertical through the tree- top
and the vertical through the eye.
G. TEST PAPER 5.
1. The rate of movement of coal-gas is described by the
formula :
Q = 1000
t
Q = quantity of gas in cubic feet per hour ; I = length of pipe
in yards ; d = diameter of pipe in inches ; g = the specific
gravity of the gas, i.e. the weight of any bulk of it divided
by the weight of an equal bulk of air ; H = pressure of gas
supply, measured by the water gauge as a " head " of water
(in inches).
(i) At what rate can gas be supplied by a 3-inch main, 300
yards long under a pressure of inch of water ? (The
specific gravity of the gas may be taken as 0'45.)
(ii) The same, the pipe being 1000 yards long and 4 inches in
diameter, and the head being 0'8 inch of water.
2. From the formula of No. 1 obtain formula) for finding
(i) the pressure of the gas supply, (ii) the diameter of the
supply-pipe.
[The fifth root of 01 may be taken as 0*63.)
3. A and B are to choose any two numbers with the con-
dition that one is to be even and the other odd. A's number
is multiplied by 2 and B's by 3 and the products are added.
Show that if A's choice was an even number the sum will be
even, if an odd number the sum will be odd.
Show also that these consequences will follow if A's choice
be multiplied by any even number and B's by any odd
number.
4. Find the factors of
(i) i6a 4 + s6a*b 2 + 8ib 4 ;
(ii) a a + i + -\ ; (iii) p 2 - q 2 + 2q - I.
EXERCISE XXVI 149
5. Given that m = 2 ^3 - 1 and n = 2 ^3 + 1, find the
value of
(i) JL + JL ; (ii) JL - ^.
6. Show that the expressions , + r- and
fa b\ /a 3 6 3 a 3 + 6 3 \
always have the same value. Find their value when
a = J2 + 1 and 6 = 72-1.
7. Find values of n which satisfy the following condi-
tions :
(i) n - i = iHsn +i) + ;
(v) (n - i) a + (n + i) 2 -f (2n + 3 ) 3 = 29
8. I have a large number of match boxes. I wish to set
them side by side on a table so that they shall form a rect-
angle in which the number of rows of boxes is the same as
the number of boxes in a row. The first time I arrange them
in this way there are 24 boxes over. I try to arrange them,
therefore, with one more box in a row. This time the number
is 29 short of that required. How many boxes are there ?
9. Two small bodies are moving along the same straight
line and are 3000 feet apart. Their velocities are such that
if they were going the same way the faster would catch the
slower up in thirty seconds, while if they were moving
towards one another they would meet in twenty-five seconds.
Find the velocities.
10. A hollow cone of paper rests on a geographical globe in
such a position that it touches the globe along the 50th
parallel of latitude. The radius of the globe is 20 cms.
Calculate (i) the semi-angle of the cone; (ii) the distance
between the apex of the cone and the centre of the sphere.
150 ALGEBRA
H. TEST PAPER 6.
1. In a water supply a certain " head " is necessary merely
to overcome the friction of the pipes (i.e. if there were no
friction the water would be delivered more quickly). The
head (in feet) used up in this way is given hy the formula :
s = speed of the water-flow in feet per second ; I = length of
pipe in feet ; d = diameter of pipe in inches.
(i) Calculate the head required to overcome the friction in a
2-inch pipe 1000 feet long in which the water is running
4 feet per second.
(ii) The same, the length being a mile and a half, the diameter
of the main 6 inches, and the speed of the water 9 feet per
second.
2. The following formulae are copied from an electrical
engineer's pocket-book. Obtain from the first of them a
formula for B and from the second a formula for G.
(Does your ignorance of the meaning of the formula} cause
any difficulty ?)
(i) E = d(B + G G + i- + R);
(ii) R = ( r + ^_ H.
3. A person, A, having taken any number he pleases out
of a heap of counters, another person, B, is told to take p
times as many. The person who conducts the game specifies
p but does not know how many counters A took. A is now
told to hand to B a certain specified number, q, of the counters
which he holds, and B is told to give in exchange to A p
times as many counters as A has left. Show that B will have
at the end (p -f- l)q counters. Give a numerical illustration.
4. The sine of a certain angle is 4w/(4w 2 + 1). Find ex-
pressions for (i) the cosine, (ii) the tangent of the same angle.
5. For what value of p does the expression
P 2 + zp - 3 ^ P 2 + P - 6 = ?
p 2 - 2 p - 8 ' p 2 - 3 P - 4 5
6. Find values of n which comply with the following
conditions :
EXERCISE XXVI 151
(l)25 ~3 i6n + 4 j 23
b - c n
(iii) n 2 + ion + 3 = 2n 2 - 50 + 53 ;
fiv^l ( 2n + 5)" 6
V ' 2(n - x)( n + 2) - s'
7. A circular disc is to be made by cutting two sectors
from sheets of gold and of lead of the same thickness and
putting them together. How many degrees will there be in
each sector if 19 sq. cms. of the gold plate weigh as much as
11 sq. cms. of the lead plate ?
8. The position in which the lens of a magic lantern must
be fixed so as to throw on the screen a clear image of the
slide is given by the formula (x - a) (y - b) = c.
In this formula x = the distance from the front lens to the
screen, y = the distance from the same point to the slide,
while a, 6, and c are constants. Find the value of the con-
stants in the case of a certain lantern from the following
data :
Distance of screen: 3 feet 3 inches; 5 feet 11 inches; 11
feet 3 inches.
Distance of slide : 14 inches ; 13 inches ; 12-J- inches.
What should be the distance of the front of the lens from
the slide when its distance from the screen is 16 feet 7 inches ?
9. Write down a formula for calculating, to a first approxi-
mation, the volume of a sphere whose radius is I'Ol r, when
the volume of a sphere of radius r is already known.
An error of 1 per cent is made in measuring the radius of a
sphere whose radius is actually 5 inches. What will be, to a
first approximation, the error in the calculation of the volume ?
4 22
[Assume V = ^ Trr 3 where TT -=- .]
10. Find the values of the angle a which satisfy the fol-
lowing conditions :
(i) sin a - (4 - 2 sin a) = 7(sin a - i) ;
(iii (tan a - x)(tan o - 2) = (tan a - 3)(tan a - 4) ;
(iii) 9 sin 2 a - 24 sin a cos a 4- 16 cos 2 a = O i
[Deride by cos 2 a.]
(iv) tan a + = -.
v ' tan a 2
PRINTED IN GREAT BU1TATN BY
THE UNIVERSITY PRESS, ABERDEEN
SECTION 11.
WHBOTED NUMBERS.
EXERCISE XXVII.
THE USE OF DIRECTED NUMBERS.
A.
Use directed numbers to solve the following problems :
1. A lift in a large hotel starts at ten o'clock on the fifth
floor. During the next hour it makes the following journeys :
Up three floors, down five, down two, up four, down one, up
five, down eight, up six, up one, down three, down two.
Where is it at the end of the hour?
2. On January 18, 1907, the highest temperature during
the day was 37 '1. On successive days after that date the
highest reading of the thermometer went down 2-9, up 5'9 tJ ,
down 7 '2, down 8 <J , down 1*3, up 3*4, up 3 -4, up 6 '3, up
9-4. What was the highest temperature (i) on the 24th ; (ii)
ou the 27th ?
3. The following table gives the rainfall in London during
each of the months of 1907. The numbers in the second row
are the average amounts of rainfall for each month recorded
during the preceding fifty years. In a third row put a di-
rected number to show the difference between the numbers
for 1907 arid the averages. Was the total fall for 1907 above
or below the average ?
Jan.
Feb.
Mar.
Apl.
May
June
1-09
1-27
0-9
3-48
1-46
2-64
1-18
1-48
1-46
1-66
2-00
2-02
July
Aug.
Sep.
Oct.
Nov.
Dec.
0-9(i
2-33
063
o-44
413
274
2-47
2-35
2-25
2-81
227
2-13
4. A boy puts 10s. into the Post Office Savings Bank on
January 2 and 3s. on February 18 ; he draws 15s. on March
23, puts in 25s. on his birthday (April 19), draws 4s. on June
15, 6s. on August 4, puts in 5s. on October 10, and draws two
guineas on December 18. During the year his deposit earned
165
156 ALGE13KA
7s. 8d. If his balance at the beginning of the year was 16
4s. 7d., what was it at the end of the year?
5. The following table gives the tirno by the clock when
the sundial registers noonday at different periods of the
year. Make a table of directed numbers that will show the
correction necessary to turn sundial-time into clock-time at
those periods.
Jan. Feb. Mar. Apl. May Jimo
1 12.3 1 12.13J 1 12.12J 1 12.4 1 11.57 1 11.571
15 12.9 14 12.14J 15 12.9 15 12.0 15 11.56 15 12.0
July Aug. Sop. Oct. Nov. Doc.
1 12.3J 1 12.0 1 12.0 1 11.50 1 11.43 1 11.49
15 12.6 15 12.4J 15 11.55 15 11.45 15 11.44 15 11.55
6. Make up two examples like any two of the foregoing.
Work them both out.
B.
7. Find the total value of each of the following sets of
directed numbers :
(i) + 7 + 12 - 9 - 8 + 17 - 24.
(ii) - 3-2 - 4-7 + 2-8 - 0'6 - 12 + 9-7 - 2-3.
(iii) + 0-7 - 23 - 1-8 + 73 - 0-03 + 5-43 + 81-5 -
143-72.
(iv) - 3 14s. + 6 7s. 5d. + 3 Os. 7d. - 16s. 8d. -
7 12s. dd.
(v) + 7| lb. - 4 Ib. - | Ib. + 6| Ib. - 12 Ib. + 7^
Ib.
(vi) - 2 hrs. 14 miiis. + 3 hrs. 7 mins. - 5 hrs. 37 mins.
- 43 mins. + 1 hr. 2^ miris.
(vii) 64 - 2-3 - 0-6 + 4-1 - 2 + 5-05.
8. Make up problems to fit one of the foregoing sets of
numbers.
Note. The numbers 38 and 26 can be written 32 + 6 and
32-6 respectively or, more concisely, 32 6. The number
32 is in this case called the arithmetical mean of 38 and
26.
9. Express each of the following pairs of numbers in terms
of its mean, as in the foregoing example.
(i) 27 and 13 ; (ii) 94 and 66 ; (iii) 81 and 38 ; (iv) 12-C
and 7-7 ; (v) 0*43 and 2-05.
EXERCISE XXVII 157
10. (i) 136 12s. and 79 6s. ; (ii) 23 7s. and 64 13s. ;
(iii) 8 15s. and 1 14s. 8d. ; (iv) 2 cwt. 17 Ib. and
4 cwt. 5 Ib. ; (v) 14 hrs. 45 mins. and 11 hrs.
9 mins.
11. The first of each of the following pairs of numbers is
the mean between the second and another number not given.
Find in each case the other number.
(i) 31 and 18 ; (ii) 4-7 and 6-8 ; (iii) 154 6s. and 137
15s. ; (iv) 14 cubic feet and 21| cubic feet ; (v) 59'3
and 53-2.
12. Use a diagram to find the mean of the following pairs
of directed numbers. When you have discovered the rule for
calculating the mean, you can use it instead of the diagram.
(i) + 7 and - 17; (ii) + 12 and - 6; (iii) -f 4 and
- 10 ; (iv) + 16-7 and + 14'9 ; (v) - 13 and - 27 ;
(vi) - 7-3 and - 19'4 ; (vii) + 3'8 and - 46-3.
13. The first of each of the following pairs of directed
numbers is the mean between the second number and another
number not given. Find the other number in each case,
using a diagram to obtain or to check your results.
(i) - 6 and 16 ; (ii) + 3 and - 41 ; (iii) + 4-8 and
- 17-4; (iv) - 5-3 and -f 1-6; (v) - 5-8 and
+ 14.7 ; ( v i) _ 41 and - 53 ; (vii) - 7'5 and - 8'2.
G.
Note. Suppose that the maximum (i.e. the highest) tem-
perature on a certain day was 70 F. and the minimum (i.e.
the lowest) temperature was 50, so that the mean temperature
was 60. Wo can conveniently express these facts by writing
that the extreme temperatures were 60 10 (which is read
"60 plus or minus 10 ").
14. The coldest weeks of 1907 were the fourth week in
January and the second in February ; the hottest was the
third in July. The following table gives the maximum and
minimum readings of the thermometer for each day in those
weeks. Express the daily temperature in terms of the mean
daily temperature as in the above example.
Jan. Max. Min. Feb. Max. Min. July Max. Mia
o o o o o o
20 38-1 317 3 32*9 29*9 14 71 '9 57'3
21 44-0 34-2 4 33'8 30'1 15 78'5 53'1
22 36-8 26-8 5 36'0 310 10 770 54'G
158 ALGEBRA
Jan. Max. Miu. Feb. Max. Min. July Max. Min.
23 28 8 23 (J 6 39 30'3 17 72'1 53'5
24 27'2 22-4 7 388 23'5 18 75-8 52'6
25 307 24-1 8 38'2 23*6 19 78*9 487
26 341 23-3 9 41 -2 27 '0 20 73 '3 50'9
What facts about the temperatures are brought out by this
way of writing them ?
15. During a week iu June I found the temperature in a
shady corner of my garden daily at ten o'clock a.m. The
readings were 64, 67, 62, 51, 53, 57, and 52 respec-
tively. Looking at these numbers I guessed that the average
temperature was 58. Express the given temperatures in
terms of this guessed (or " trial ") average, and use the results
as a quick way of finding out whether I was right.
16. During another week the temperatures were 54, 50,
48, 43, 40, 45 and 49. I guessed the average or mean
temperature for the week to be 48. Was I right ?
17. During yet another week the temperatures were 49,
48, 53, 51, 58, 56, 49. What do you guess the average
to be ? Test your guess. If you were wrong use your test to
find the correct mean in the quickest way.
18. The following table gives the times of departure and
arrival of trains from London (Paddington) to Gloucester.
Calculate how many minutes each train takes, and de-
termine by the use of a " trial average " the mean length of
the journey :
Depart 1.0 5.40 7.30 9.0 10.50 11.40 3.15 4.45 5.15
Arrive 3.56 9.13 10.30 12.4 1.43 3.2 5.54 7.59 8.40
Depart 0.10 9.15
Arrive 8.50 12.26
19. Find by use of a trial average the mean of the follow-
ing numbers :
(i) 115, 112, 104, 101, 109, 87, 79, 93,
(ii) 16-5, 16-1, 18-9, 14*7, 13-1, 12-5.
(iii) +11, +7, +19, +9, -3, -7, -1.
(iv) +4, +3, +2,0, -5, -11, -9, -8.
20. Use the results of your answer to No. 14 to find the mean
temperature during each of the three weeks mentioned in
that question. Which was the coolest week in 1907 ?
21. Find the mean daily variation in temperature during
each of the same weeks.
EXERCISE XXVII 159
22. Draw graphs to show the variation of the daily mean
temperature about the weekly mean in each of these weeks.
23. In January, 1907, the moon was new on the 3rd, and
full on the 18th. The following table gives the time at which
it was south during this half of its monthly period. Find the
mean value of the time interval between one " southing " and
the next.
Day. Moon ftth. Day. Moon 8th.
3 11.38 a.m. 12 7 A3 p.m.
4 - 13 8.30
5 1.47p.m. 14 9.17
6 2.47 15 10.6
7 3.43 16 10.55
8 4.35 17 11.44
9 5.24 18
10 0.11 19 12.32a.m.
11 6.57
24. Draw a graph to show the variation between the suc-
cessive intervals and the mean interval in the last question.
25. The following table gives the times of high tide at
London Bridge on the days mentioned in No. 23. Find (to
the nearest minute) the mean interval between one high tide
and the next. Compare the result with that of No. 23.
Day.
Morn.
Aft.
Day.
Morn.
Aft.
Day.
Morn.
Aft.
3
1253
1.18
9
5.52
6.17
15
11.59
4
1.44
2.9
10
6.44
7.9
16
12,27
12.53
5
2.33
2.58
11
7.35
8.3
17
1.16
1.39
6
3.23
3 47
12
8.32
9.7
18
1.59
2.19
7
4.13
4.38
13
9.42
10.17
5.2 5.26 14 10.53 11.27
25. Add to the graph which you drew in answer to No. 24
a graph showing the variation of the tidal interval from its
mean value. (It is convenient to draw the second graph in
red ink or with dotted lines.)
27. The leader of a squad in "figure-marching" takes
the following movements in succession : 21 paces north, 17
paces east, 19 paces south, 14 paces west, 11 paces north, 3
paces west. Draw a diagram to show his final position and
then show how you could have determined it by calculation.
28. The following is l the description of the movements made
by a boy who is trying to get out of a maze : 35 yards N., 52
E., 36 N., 21 W., 18 8., 13 W., 18 N., 26 W., 44 S., 12 W.,
32 N*., 41 W., 47 S., 30 E., 10 N., 20 W., 28 N., 28 E.,
160 ALGEBRA
53 S., 23 E. Where will he be after the movement marked
u * " and at the end ? Check your calculation by a diagram
of the maze.
29. A dirigible balloon begins a testing flight by rising
vertically through 120 feet. It subsequently carries out the
following movements : 1560 feet N., 780 feet W., 1470 feet S.,
840 feet E., 90 feet upwards, 900 feet E., 650 feet S., 140 feet
downwards, 700 feet W., 640 feet N. Where will it now be
with regard to its original position ?
30. Invent another problem like No. 15 and solve it.
31. Invent another problem like No. 16 and solve it.
EXEECISE XXVIII.
ALGEBRAIC ADDITION AND SUBTRACTION.
A.
1. The following statements have reference to the move-
ments of a lift in a large hotel, a movement from one floor to
the next being taken as unit. The symbol n stands for
" number of floors ". State in words what problem is meant
in each case and solve it.
(i) n = ( + 2) + ( + 3).
(ii)n = (+ 2) - (- 3).
(iii) n = (- 5) + (+ i).
(iv) n = ( - 5) - ( - i).
(v) n = ( + 3) + ( - 5).
(vi) n = (+ 3)- (+5).
(vii) n = (- 4) + (+ 4).
(viii) n = (- 4) - (+ 4).
(ix) n = (- 3 ) + (- 5).
(x) n -(- 3) - (- 5).
(xi) n = (+ 4) + (- 5).
(xii) n = (+ 4) - (+ 5) - (- 3).
+ (+ 3) + (- i).
(xiv) n = (- 7) - (+3) - (- i).
(xiii) n = (- 7) + (+ 3)
(xv) n = (- 2) + (+ 3) - (- 4).
(xvi) n = (- 2) - (- 3) + (+ 4).
Note. Examples Nos. 2-9 are to be set down in such a
way as to show whether the problem is one of calculating an
unknown resultant from given components, or of calculating
an unknown component, given the resultant and the other
component (or components). Use the symbol R for an un-
known resultant and the symbol p for an unknown com-
ponent.
2. A lift starts with a passenger from the ground floor of
a hotel and carries him up to the seventh floor. There he
finds that his room is three floors lower down, so the lift
descends with him to that level. On which floor is his
room?
ifil 11
162 ALGEBRA
3. A visitor in an hotel left his room and entered the lift in
order to descend to the dining-room on the ground floor.
He thought, of course, that the lift was about to go down, but
it was actually going up and carried him up three floors and
then descended seven floors to the dining-room level. On
which floor was his room ?
4. A motorist is about to start from the village of X in
order to go northwards to Y. Noting that his supply of
petrol is insufficient he drives south 5 miles to the nearest
town Z, obtains additional petrol and then travels 52 miles
to reach Y through X. What is the direct distance from X
to Y?
5. A motorist starts from X to go to Y. Twelve miles
from X he overtakes a friend who is cycling to a farm-house
some distance beyond Y. He takes his friend and the bicycle
into the motor-car and the two drive on for 38 miles till they
reach the village of Z. On making inquiries here they find
that they passed the farm-house 4 miles before entering Z.
They drive back to it and the cyclist leaves the car. The
motorist now has to drive 6 miles back to Y. How far is it
from X to Y ?
6. A cyclist starts for a distant place. Five miles from
home he has to stop to mend a puncture. Having ridden
1 i miles farther on he discovers that he has missed a turning
which he should have taken. He rides back 3 miles to the
sign-post and finds that he has now 4 miles to go to his
destination. How long would the ride have been if he had
not made the mistake?
7. A party of people start from a seaside town A to sail
to another town B, 9 miles away. The tide carries their boat
past this town to a place, C, 3 miles farther on. They effect
a landing here and sail back to their original destination when
the tide turns. How many miles did they sail from A to C ?
8. A cyclist started for a place 19 miles away. He stopped
to buy a repair outfit at a town 6 miles from home and stopped
again for tea 9 miles farther on. He was told here that he
ought to have taken a turning some distance back. He rode
back to this turning and found that he was still 7 miles from
his destination. How many miles had he gone beyond the
turning ?
9. A cyclist starting from A to ride to B was told that the
distance was 14 miles, and that he must look out for a narrow
EXKROISE XXVTTI 163
lane on the right which he was to follow. After some time
he made inquiries and found that he had ridden 2 miles
beyond the turning He went back, found the lane, and after
6 miles reached B. How far is it from A to the beginning
of the lane?
10. Explain the meaning of the following problems and solve
them, given a = -3, b = +5, c = - 7 :
(i) R = a -f b. (ii) R = a + 2b.
(iii) R = sa + 2b H- c. (iv) d = a - c.
(v) d -- a 4- b - c. (vi) d = a - b + c.
(vii) k = 43 - 2b - 30. (viii) m = b + 30 - 6a.
B.
11. A clerk occupies a post in which his salary increases 7
10s. per annum. His present salary is 150. Write down a
formula which will give his salary t years hence.
Plow could you use the same formula to calculate his
salary t years ago? Use your formula to find (i) what his
salary will be in four years' time, and (ii) what it was six
years ago.
12. A man inherited a sum of money many years ago, but
he has been withdrawing 12 a year from it to pay his life
insurance premiums. Its present amount is 243. Write
down a formula by which the amount at other times may be
calculated (A, t).
Use your formula to find (i) how much he had fourteen
years ago, (ii) how much will be left in twenty years' time if
he lives so long.
13. A butcher owns two shops. The receipts from one
are improving at the rate of 14 a month, from the other
they are falling off at the rate of 19 a month. This month
the two shops yielded together 1317. Find, by the use of
a single formula his whole weekly takings (i) twelve months
ago, (ii) eight months hence (T, t).
14. A liner crossing the Atlantic consumes 4 tons of coal
every hour. When a certain distance out from Liverpool she
has 620 tons in her bunkers. Write down a formula by
which the amount of coal in her bunkers may be calculated
for every hour after or before this moment (C, t).
How much coal (i) will she have in two days 7 time, (ii)
had she two days ago ?
15. Draw a graph to illustrate the preceding question.
164 ALGEBRA
How will you arrange your graph so as to distinguish times
before the present moment from times after ?
[It will be convenient to take 20 hours to the inch for the
time-scale and 200 tons to the inch for the coal-scale.]
Use the graph to check the calculations you made with
the formula and also to answer the following questions :
(i) How much coal will the linor burn in 40 hours?
(ii) When will she have 500 tons left ?
(iii) When did she have 800 tons ?
(iv) She had 812 tons when she left Liverpool. How long has
she be on out 'i
(v) She will reach New York with 236 tons. How long does
the crossing take her ?
16. A cricket club begins the year with a balance of 33,
but the subscriptions have for several years been 7 less than
bhe annual expenses. Give a formula for the financial posi-
tion of the club at the beginning of each year, and use it to
find (i) what the position was four years ago, and (ii) what it
will be in six years' time if the present conditions continue
(B, t).
17. A rival club to the one in the last question begins the
year with a deficit of 8, but its subscriptions have for several
years exceeded the year's expenses by 4. Determine, by
bhe use of a formula, the financial position of the club (i) five
years ago, (ii) five years hence if the present conditions
continue.
18. A baker who opened a new shop some years ago esti-
mates that it has returned him his capital and, in addition,
las yielded a total profit of 144. The receipts have always
3xceeded the expenses by 16 a month. Write a formula
showing the financial position of the enterprise at the end of
3ach month. What was it (i) two years ago, (ii) what will it
:>e in four and a half months (P, t) ?
19. The same baker estimates that another shop has still to
^ring him in 120 profit before it will have returned 1he whole
>f the capital originally invested in it, but that the receipts
jxceed the expenses at the rate of 24 a month. Write a
brmula showing the financial position of the shop at the end
>f each month, (i) What was it a year ago ? (ii) What will
t be in three years' time ?
20. Illustrate the last two questions by drawing on the
\arne pieca of squared paper two graphs, one for each of the
EXERCISE XXYIII 166
shops. Use the graphs to find (i) when each of the shops
had just paid, or will just have paid back the whole of the
capital expended on it, (ii) when they will both have earned
the same total profits.
Check each of these results by the formulae.
21. The shop of No. 18 was opened three years ago, that
of No. 19 two and a half years ago. Find by the graphs how
much capital was invested in each. Check the results by the
formulae.
22. Why is it impossible to illustrate Nos. 16 and 17 by
graphs similar to those just employed ? What kind of graphs
would be suitable ?
23. A railway rises for several miles at a regular rate of 6
feet for every 100 feet of the slope. Just outside a certain
station it is 300 feet above sea-level. Use a formula to find
its height above sea-level (i) 750 feet farther up the slope,
(ii) 450 feet farther down (h, d).
24. A long gallery in a coal mine beneath the sea slopes
downwards from the foot of the shaft at a uniform rate of 1
in 7. When a certain side gallery branches off it is exactly
1000 feet beneath the sea. Write a formula that shall give
the depth (D) of points at a given distance (d) from this
place. Use it to find the depth at a place (i) 490 yards before
you reach the place in question from the shaft, (ii) 1400
yards farther on.
25. It is said that after a certain depth the temperature
of the earth beneath England increases 1 F. for every 50
feet that you descend towards the centre of the earth. At a
depth of 3000 feet it is 110. Write a single formula which
shall give the temperature (T) at a point distant d feet below
or above this point.
Use the formula to calculate the temperature at a depth of
(i) 4000 feet, (ii) 1800 feet below the surface.
26. Write another formula expressing the temperature at a
depth d below the surface.
Use this formula to calculate the temperature at the depths
mentioned in the last question. Do your results agree ?
27. Write a third formula by which the depth from the
surface (d) can be calculated when the temperature (T) is
given.
Use it to calculate the depth where the temperature is (i)
160, (ii) 85. .
166 ALGKWIA
28. Draw a graph corresponding to the formula of No. 27.
Show how you can use it to answer the questions of No. 25.
Hdw would the graphs corresponding to the formulas of
Nos. 25 and 26 differ from this one?
29. It may be taken that the harometer falls practically (Ml
inch for every 100 feet you rise so long as the total ascent
does not much exceed 2000 feet. At a village on the side
of a Welsh mountain 750 feet above the sea the barometer
stands to-day at 29'8 inches. Write a formula giving the
heights of the barometer (B) at places on the mountain d feet
higher or lower than the village.
Use the formula to calculate the height of the barometer
(i) at the sea-level, (ii) at the top of the mountain which is
1820 feet high.
30. Wiite a formula giving the height of the barometer (B)
at different heights (Ji) above sea- level.
Calculate (i) and (ii) of the preceding question by this
formula.
31. Write a formula giving the height above sea-level (Ji)
corresponding to a given height of the barometer (B). Use
it to find the height where the barometer records (i) 29*1
inches, (ii) 30'3 inches.
0.
32. Find the value of P as given by each of the following
formulae : (1) by direct substitution ; (2) by first simplifying
the formula and substituting afterwards. Take a = - 2,
b = 4- 5, c = - 7. The meaning of the original formula is
to be explained in each case and also the meaning of the
simpler formula to which it is reduced :
(i) P = (a - 2b) + ( 3 a - b).
(ii) P - (3a + b) + (2b - a),
(iii) P = (a - 4b) - (2a + 6b).
(iv) P = ( 5 b - 2a) - (a - 6b).
(v) P = (a + b) + (2a - b + c).
(vi) P = (sa - 2b + 30) - (4a + 2b + 30).
33. Reduce each of the following formulae to the form most
suitable for substitution. Find the value of the subject when
p = + 10, q = - 6, r = - 2 :
(i) A - (2p - q 4- r) - (p - 3 q + 4 r).
(ii) C - 2(3? + q) + 4(p - 2q).
(iii) n = 3(P + q + r) - 2(p - q) - 3(q - r) - 4(r - p).
KXEROISK XXVIII 167
(iv) E - .\(2p - q) - i(3P - 2q),
(v) B = K3P - 4q) + L>(4q - ST) - A(5f - 6p).
(vi) M = i-2(p + q - 2r) + 2'3(p - 2 q + r) - 3'4(q + r - *?)
34. Reduce the following expressions to the forms most
suitable for substitution :
(i) {2m + n - 3(m - n)} + {4(111 + 2n) - 3(m - 2n)}.
(ii) {2'5(a - 2b) + 37(2a - b)} - {2'i(a + 2b) - O'6(2a + b)].
(iii) {2(i'3p + 2) - 3(27q - i)} - {4'2(p - q + i) + 2'8}.
(iv) 2{ 4 (tja - |b) + 5 (jb - |c) + 6(^c -ja)}.
_
(v) 3{5(3a - 5b - a - 2b) - 4(Sa - 3b - 2a - b)}.
35. Subtract (of course algebraically) :
(i) 2a + 3b - 40 from 5a - 3b f- 2c.
(ii) sa - 3b + 2c from 2a + 3b - 40.
(in) a + b - c from 3(a - b -f c).
(iv) 6'2(p - q + 2) from 2'3(2p - 2q + i).
^v) i{2 P - 3 (q - i)} from }{4p - 3(r - i)}.
EXERCISE xxix.
DIRECTED PRODUCTS,
A.
1. York station is 32 miles to the north of t)oncaster
station on the Great Northern Eailway. Write a formula for
calculating the distance (d) of a train from Doncaster, given
the time (t) since it left York. The train may be supposed
to travel with a constant velocity v.
Use the formula to find the distance of the train from
Doncaster in the following circumstances :
(i) Tho train is travelling to the north at o5 mls./hr., and left
York two and a half hours ago ;
(ii) The same, except that the train is travelling Kouth ;
(iii) The train's velocity is 46 mls./hr. towards the north, but it
will not reach York for one and a half hours ;
(iv) The train is coining from Scotland at 44 mls./hr., and will
reach York in three-quarters of an hour ;
(v) The train is travelling southwards at 40 mls./hr., and left
York forty-eight minutes ago.
2. A steamer, which trades between Cape Town and
Melbourne, sails for many days along the 39th parallel (S.)
at a uniform rate of v mls./hr. At a certain moment it is
reported (by wireless telegraphy) to be at a given number of
miles (d ) from the island of St. Paul, which is on the same
parallel. Write a formula for its position with regard to St.
Paul (d) a given number of days (t) after or before this
moment.
Find the position of the steamer :
(i) Three days after it is reported to be 250 miles to the east of
St. Paul on its westward journey and travelling at 12
mls./hr. ;
(ii) Fifteen hours after it is 120 miles to the west of St. Paul,
pursuing a westward course at 15 mls./hr. ;
(iii) Ten hours before it is reported as in (ii).
3. The water in a reservoir has a depth of d feet which
is increasing h feet per day. Find the depth (d) after t days.
168
EXERCISE XXIX 169
Use the formula to calculate the depth twelve days ago,
given that the surface is sinking 3 inches daily and that the
present depth is 13 ft.
4. A ship taking soundings in a bay finds that the depth of
the sea at a certain point is d Q fathoms and that it increases
regularly by i ijathoms for every quarter of a mile towards the
weat. Write a formula giving the depth, d, m miles west of
the given point*
Use the formula to calculate the depth 4 miles to the east
of a point where the depth is 400 fathoms, assuming that the
soundings decrease in this direction at the rate of 2'4 fathoms
per quarter of a mile.
5. A road climbs up a valley at a uniform slope of a. A
certain inn on the roadside is /t feet above sea -level. Write
a formula giving the height (h) above sea-level at a point m
miles along the road farther up the valley.
Use the formula to find the height above sea-level at a
point (i) 2 miles before, (ii) 3 miles after the inn is reached,
when you are walking downhill at a constant slope of 7, and
the inn is 2500 feet above the sea.
G. A farm-house, A, is situated on a road that runs east and
west. Another house, B, lies p miles due north of A on a
straight road that crosses the former at an angle of a some
distance to the west of A. A third house, C, lies on the
second road q miles from B in the opposite direction to the
crossing. How far is C north of the road through A ?
Given that the two roads cross at an angle of 37, that B is
1 miles to the south of A, and that G is 2 miles from B in
the direction of the crossing, find by the above formula the
distance of G from the road through A.
Illustrate your solution by a diagram.
7. A charitable society has a balance of B at the bank.
Its revenue exceeds the expenses by an annual amount p.
What will be the balance, B, after t years?
Suppose that the society has overdrawn its banking account
to the amount of 130, but that it may be assumed that its
regular income always exceeds the expenses by 20 per
annum. Find by your formula the balance (i) 5 years hence,
(ii) 6 years ago.
8. A motor-car is going with a velocity of u mls./hr., and
this velocity is increasing regularly at such a rate that at the
end of every minute the car goes a mls./hr. faster than at the
170 ALGKBRA
beginning of that minute. Give a formula for v, its velocity,
after t minutes.
At the moment a motor-car passed me it was travelling at
27 mls./hr., but its speed had been diminishing and continued'
afterwards to diminish. Assuming that the rate of diminution
of speed may be taken as 3 mls./hr. every minute, use the
foregoing formula to calculate the speed of the car (i) in five
minutes' time ; (ii) four minutes ago ; (iii) in nine minutes'
time; (iv) in eleven minutes' time. What is the meaning of
the last result ?
9. Two motor-cars start in the same direction at the same
moment and travel with uniform velocities ?? x and u>. The
lirst reaches a place X after t L hours, the second a place Y
after t., hours. Give a formula for d, the distance between
X and"Y.
Two motor-cars pass one another on the road at exactly
twelve o'clock. One is travelling at 24 mls./hr., the other at 20
mls./hr. in the opposite direction. The first car left X at
10.30, the second reached Y at 2.30. Assuming that the
speeds were constant throughout, find the distance between
X and Y by means of the foregoing formula.
10. A motor-car is travelling at u mls./hr., but its speed is
increasing a mls./hr. every minute. In how many minutes
will it be v mls./hr. ?
(i) A motor-car is travelling at 28 mls./hr., but its speed is
diminishing 4 mls./hr. every minute. Apply the foregoing
formula to find how long it will be before its speed will be
reduced to 8 mls./hr.
(ii) Assuming that the conditions remain constant, in how
many minutes will it be travelling at 14 mls./hr. in the
opposite direction?
B.
11. What laws of succession are exhibited by the terms of
the following sequences of directed numbers ?
(i) ... +4, +7, + 10, +13, +16, . . .
(ii) . . . +18, +13, +8, +3, . . .
(iii) ... -8, -4, o, +4, ...
Continue each sequence three terms to the right and three
terms to the left.
12. Write a formula for the term in No. 11 (i) which lies
EXKftCTSE XXIX 171
n places to the right of 4- 10. Use it to calculate the 50th
term to the right of + 10.
13. Write a formula for calculating the nth term to the left
of -f 10 in the same sequence. What is the 60th term in this
direction ?
14. Show that by making n a directed number the formula
of No. 12 can also be used to answer the question of No. 13.
Use this formula to find (i) the 100th, (ii) the 200th term to
the left of 4- 10
15. Write a single formula that can be used to calculate the
Tith term to the right or left of + 8 in No. 11 (ii). Use it to
find (i) the 20th term to the right, (ii) the 12th term to the
left of + 8.
16. Write a formula for the nth term of No. 11 (iii), count-
ing forwards or backwards from 4. Calculate (i) the 1000th
term to the left, (ii) the 1000th term to the right.
Note. Sequences like those of No. 11 are called arith-
metic sequences. Their characteristic is that each term
is derived from the next term* to the left of it by adding (alge-
braically) a constant number. This number is called the
common difference of the sequence. What are the com-
mon differences of the sequences in No. 11?
17. The common difference of an arithmetic sequence is -f 3
and one of its terms is - 7. What are the four terms re-
spectively to the left and right (or before and after) this
term?
Give the formula for the ?ith term of the sequence to the
right or left of - 7. Use it to find the 40th term to the left
and the 15th to the right.
18. The common difference of an arithmetic sequence is - 24
and one term is - 72. Give the four terms immediately
preceding and following this term. Calculate the 500th term
to the left of - 72.
Note. The formula for the terms before and after (or to
the left and right) of a given term may be called the gener-
ating formula of the sequence. The term from which the
counting right and left proceeds may be called the Starting
term.
19. State the starting term and the common difference of
the sequences whose generating formulae are :
(i) T n = - 7-2 + 3'in. (ii) T n = + 67 - r6n.
(iii) T n + 2J - gn. (iv) T n = - J - ,\>n.
172 ALGEBRA
Calculate in the case of (i) and (ii) the 100th term to the
left of the starting term, and, in the case of (iii) and (iv), the
1000th term after the starting term.
20. Write the generating formulae of the arithmetic sequences
in which the common differences are respectively (i) + 18,
(ii) - 8'6, (iii) - 3|, (iv) + 0*07, and the starting terms are
respectively (i) - 3, (ii) 0, (iii) + 1|, (iv) - 0*9.
21. Give a generating formula that will apply to any
arithmetic sequence. (Let T = the starting term and d = the
common difference.)
22. Counting to the right of the starting term the 20th
term of an A.s. is -f 17 and the 30th term -\ 84. Find the
common difference, the starting term, and the generating
formula.
23. In an arithmetic sequence T 12 = + 8, T L , = - 32.
Find the generating formula.
24. The 14th term to the right of the starting term of an
A.S. is + 6-7, the 6th term to the left is - 18'3. Find the
generating formula.
25. The 17th term before the starting term of an A.s. is
+ 23 6 ; the 13th term after it is - 57'4. What is the gener-
ating formula ?
26. Find the generating formula of an A.s. in which T_ 24
= + 19 and T_ 3 7 = + 58.
27. In a certain A.S. T,, = u and T v = v. Express d, T ,
and the generating formula in terms of u t v, p, and q. Do
your formulae hold good for all possible values of p and q, u
and v ?
C.
Note. Select any term from an unlimited arithmetical
sequence. The series made up of this term and any number,
limited or unlimited, of consecutive terms which immediately
follow it is called an arithmetical progression (A.P.).
The series composed of this term and any number of conse-
cutive terms which immediately precede it is called an
arithmetic regression (A.B.). Thus a progression always
has a first term but may have no last term ; while a regres-
sion always has a last term but may have no first term.
The numbers which constitute an A. p. are often said to be
" in arithmetical progression ". The terms which lie between
EXERCISE XXIX 173
the first and last terms of an A. P. are called the arithmetic
means between those terms. Thus, if an A.P. has 12 terms
there are 10 arithmetic means.
28. There are 8 numbers in arithmetic progression, the first
being 3 and the last 31. "What is the common difference?
What are the arithmetic means ?
29. The first term of an A.P. is + 3, the last - 21, and
there are 5 arithmetic means between them. What are they ?
30. Given that the first and last terms of an A.P. containing
n terms are respectively a and I, write a formula for d, the
common difference.
31. There are n arithmetic means between two terms a and
I. What is the common difference ?
32. An A.P. contains n terms altogether. The first term is
a and the common difference is d. Write expressions for the
2nd, 3rd, pih, and last (i.e. ni>h) terms. (Note the important
difference between the first of a number of terms in A.P. and
the " starting term " of an endless A.S. In counting the
terms the former is included, the latter is not.)
33. The first of 21 numbers in A.P. is + 16*7 and the
common difference is - 3*2. Find the last term, the first
mean but one, and the last mean but two.
34. The third of 16 numbers in A.P. is 14, the thirteenth
is 44. What are the first and last terms?
35. The common difference of an A.P. is 3. The first
and last terms are respectively + 23 and - 25. How many
means are there ?
36. Are any of the numbers 4- 5, 0, - 1, - 13, - 20,
among the means in No. 35 ? If so, state their positions.
37. Is 72 a term of the A.P. 1, 3, 5, . . . If so, which
term is it ? If not, between which terms does it lie ?
38. I make a series by taking the 2nd, 5th. 8th, etc., terms
of an A.P. Will the new series be in A.P. ? If so, what will
be the common difference?
Show that the numbers obtained by taking every pth term
of an A.P. are themselves in A.P.
39. Between each pair of terms of an A.P. p arithmetic
means are inserted. Show that the whole series forms an
A.P. If there were originally n terms, how many terms are
there in the new series ?
40. Taking a seiies of numbers in A.P. (i) I add the same
number to each; (ii) I subtract the same number from
174 ALGEBRA
each ; (iii) I multiply each by the same number; (iv) I divide
each by the same number ; (v) I multiply each by the same
number and take another constant number away from the
product ; (vi) I square each ; (vii) I take the square root of
each. In which of these cases will the resulting numbers
be in A.P. ? Give full reasons. When tbe new series is in
A. P., what will dts common difference be?
EXERCISE XXX.
SUMMATION OF ARITHMETIC SE&IES.
A.
1. Draw diagrams to illustrate the following summations :
(i) 7 terms of the series 1 + 5 + 9 + . . . ;
(n) 6 terms of the series 24 + 21 + 18 + . . . ;
(Hi) 5 terms of the series - 3 + 1 + 5 + . . . ;
(iv) 8 terms of the series + 13 + 8 + 3 - . . .
2. Calculate the sum of :
(i) The series 1 + 2 + 3 + . . . + 1,000,000 ;
(ii) The series 1 + 2 + 3+.. ,+n;
(ni) The series 1 + 3 + 5 + ... + 191) ;
(iv) The first n odd numbers ;
(v) The first n even numbers.
3. Find the sum of the series:
(i) - 8 - 1 + 6 + ... + 104 ;
(ii) 92 + 80 + 68 + ... - 76 ;
(iii) 21-2 + 18-3 + 15'4 4- . . . - 33'9 ;
(iv) i + 5 + 1J + . . . + 14tf.
4. Sum the series :
(i) - 1 - 11 - 21 - ... to 50 terms ;
(ii) 17 + 11 + 5 - ... to 21 terms ;
(iii) - 3-4 - 17 + + ... to 8 terms;
(iv) - 188-8 - 185-6 - 182'4 - ... to 120 terms.
5. Find the sum of the A.S. in which the first and last
terms and the number of terms are respectively (i) 1, 101,
26; (ii) 1, 2, 101; (iii) -72, +36, 55; (iv) +17*3, -21-7,
51.
6. Write out the proof by symbols (i) that S = ? (a + I) ;
A
(ii) that S = {2a + (n - l)d}.
2
7. A man enters an office at a salary of 80, which is in-
creased annually by 5. How much will the firm pay him
in the caurse of twenty years ?
175
176 ALGEBRA
8. In a "block race" the first block is 3 yards from the
starting (and finishing) line; the others follow at regular
distances of 5 feet. There are sixteen blocks altogether.
What is the total distance that the blocks are carried?
9. The -first week a restaurant was opened the proprietor's
expenses exceeded his takings by 5 12s. ; the second week
the loss was 3 4s. If the improvement were maintained at
the same rate, now much profit would the proprietor make
altogether in 13 weeks?
B.
10. Three boys, A, B, and C, are apprenticed, each for six
years, under the following conditions, A is to receive no-
thing during the first year, 10 the second year, *20 the
third year, etc., and 50 the last year. B is to receive nothing
during the first six months. His wages are then to rise by
equal half-yearly steps so that during the last six months he
will be paid like A, at the rate of 50 a year. C is to serve
for nothing during the first three months. His wages will
then be increased uniformly every three months by such an
amount that during the last quarter of the sixth year he also
will be paid at the rate of 50 a year.
Draw (to the same scale) ''column graphs" exhibiting
the wages received by A, B, and C respectively at different
periods. Write out a proof by means of the graphs that dur-
ing the six years each apprentice will receive the same total
amount in wages. What is this amount? What were B's
half-yearly and C's quarterly increments?
11. For a minute after a tap delivering into a tank is
turned on no water flows. It then pours out at the rate
of 2 gals./min. At the end of a minute the tap is suddenly
turne 1 on farther so that it now delivers 4 gals./min. After
another minute the tap is again turned and the flow increases
to 6 gals./min. After one more minute the flow is increased
to 8 gals./min. At the end of the fifth minute the tap is
suddenly turned off completely. Draw a rectangle graph
exhibiting the quantity of water supplied in each minute.
(Make each rectangle 1 inch wide.) What is the total
quantity of water delivered ?
12. On another occasion the tap was turned on farther
every six seconds for five minutes, in such a way frhat the
KXKRCIRK XXX 177
flow increased by equal steps. The rate of delivery during
the last six seconds was 8 gals. /m in. During the first six
seconds no water was delivered. Erect upon the base line
of the former graph a column graph exhibiting the new con-
ditions of flow. What is the total quantity of water delivered ?
13. On a third occasion the tap was turned on gradually
and without a break, so that the flow increased constantly
and uniformly. At the end of five minutes when the tap was
suddenly turned off the rate of delivery had leached 8 gals./min.
Using the same base line as in Nos. 12 and 13, draw a graph
exhibiting the conditions of flow. How much water was de-
livered in five minutes? How much did the rate of flow
increase in the course of each minute ?
14. In Nos. 10, 11, 12, 13 what line measures the rate of
wages or the rate of flow at a given moment?
15. Show that in No. 13 the rate of flow at any moment
is given by the formula
r - 1-6 *
r being the rate of flow in gallons per minute and t the time
in minutes (and fractions of a minute) since the tap was turned
on. What was the rate of flow after 4'2 minutes?
16. Show that in No. 13 the number of gallons (Q) de-
livered in t minutes is given by the formula
Q - 0-8 &
How much was in the tank at the end of 3'3 minutes?
17. For a considerable time one morning the state of the
crowd passing over London Bridge was given by the
formula
r = 180 i
r being the rate of movement of the crowd measured by
the number of persons per hour who would have passed a
certain point if the rate had been maintained ; t being the
time m minutes since the observations began. Show that the
total number who passed a certain point in the first t minutes
is given by the formula
N = * P
18. The speed of a marble permitted to roll from rest down
a smooth slope increases uniformly. After ten seconds it is
3 feet per second. Draw a graph showing the speed at
different times. What does a unit of area in this graph
represent ? Find from the graph the distance the marble
rolls (i) in five seconds, (ii) in ten seconds. Write formulas
12
178 ALGEBRA
for the speed acquired (v) and the distance rolled (s) in t
seconds.
19. A railway train starts from rest and increases its speed
uniformly until after twelve minutes it is going at GO miles an
hour. Draw a graph showing the speed and the distance
travelled in a given time. What distance is represented in
this graph by the unit of area ? Write formulae for v, the
speed in miles per hour after t minutes, and for s the distance
in miles travelled in t minutes. What is the speed after eight
minutes, and how far has the train then travelled?
20. The speed of a moving body in feet per second is given
by the formula
v = at
t being the time in seconds since it began to move. What
formula gives the distance in feet travelled in t seconds ?
What does this formula become if t is measured in seconds
but v in feet per minute?
21. A motor-car is moving at the rate of 15 mls./hr. when
it reaches a descent down which its speed increases regularly
at the rate of 3 mls./hr. per minute. Draw a graph showing
its speed at any time, measured from the moment when it
began to increase. Write formula) for the values of (i) v
(measured in miles/hour), (ii) v (measured in miles/minutes),
and (iii) s (measured in miles) after a given time t (measured
in minutes).
22. The speed of a moving body is given by the formula
v = u + at
u and v being measured in feet per second and t in seconds.
Write a formula for the distance moved in t seconds. What
does the formula become if u and v are measured in miles per
hour and t in minutes?
23. Show by a diagram that if v = u - at, then s ut -
Jrafi, the same unit of time being used in measuring u, v,
and t.
24. A train is travelling at 40 mls./hr. when the brakes are
put on. The speed now decreases uniformly and in five
minutes the train is at rest. Draw a graph exhibiting its
speed and the distance travelled from the moment when the
brakes are put on. Give formulae for v and s, t being
measured in minutes.
25. A boy rolls a ball up a smooth slope with a speed of
18 ft./sec. It gets uniformly slower, stops, and then rolls
EXERCISE XXX 179
down again with uniformly increasing speed. The decrease
and increase of the speed are both at the rate of 3 ft. /sec.
every second. Draw a diagram showing the speed of the
ball and the distance travelled at any time for forty seconds
after it is thrown. [How will you distinguish between upward
and downward speed ? Between distance up and distance
down the slope?]
26. From the graph answer the following questions :
(i) When will the ball have exhausted its speed ?
(ii) How far will it then have travelled ?
(iii) What will be its speed twelve seconds after it is thrown 1
(iv) Where will it then be ?
(v) What will be its speed twenty seconds after it is thrown ?
(vi) Whore will it then be ?
27. Show that you can obtain the answers to No. 26 from
the formulae v = u + at and s ut + -a 2 , by substituting
directed numbers for the symbols.
28. A bullet is fired up into the air with a speed of
1600 ft./sec. Its speed falls olf as it goes up and increases as
it comes down at a regular rate of 32 ft./sec, every second.
Calculate (i) how long it will be rising ; (ii) how high it will
rise ; (iii) its velocity after twenty seconds ; (iv) its height
above the ground at that moment ; (v) its velocity after sixty
seconds ; (vi) its position then.
29. The following formulae give the velocities of three bodies,
A, B, and C, which may be supposed to be moving along lines
parallel to the lines across your paper. Write down the cor-
responding formulae for their distances at a given moment
from the point from which the measurements are taken. In
each case v is measured in centimetres per second, and t in
seconds.
(i) v = + 7'2 -i- G-4 ; (ii) v = - 10 + S'6t ; (iii) v = - 8'5 - 3'6.
30. Answer by means of your formulae the following ques-
tions with regard to the bodies A, B, and C :
(i) Which way and with what speed was each body moving ten
seconds ago ?
(ii) Where were they ten seconds ago ?
(iii) How was each moving one second ago ?
(iv) Where was each body one second ago ?
(v) Each body has been or will be motionless for an instant.
Find when this happened or will happen in each case. What did
or will each body do immediately after these moments ?
12*
180
ALUEBRA
Frc. 3.'3.
P
J
FIG. 35.
A
B A
FIG. 36.
EXEKCISE XXX
181
Km. 38.
182 ALGEBRA
C.
31. Determine by measurements taken at random whether
equidistant lines drawn across tig. 33 are in A.P. If they are
calculate the area of the figure in sq. cms. [See Ex. XXIX.,
No. 38.]
32. Do the same with fig. 34.
33. A number of square slabs 2 inches thick are piled on
top of one another so as to produce the solid figure shown in
elevation in fig. 35. A pin 2 inches high, but of negligible
volume, surmounts the pile. The squares in fig. 36 are the
outlines of the successive blocks as seen from above. Cal-
culate the differences between the areas of successive squares,
counting the pin as having zero area. Hence calculate the
volume of the solid.
34. Fig. 37 shows an elevation of a pointed solid standing
on a square base. Fig. 36 may be regarded as a " contour
map " of this solid, the squares being contour lines taken at
intervals of 2 inches measured vertically. What is the prob-
able volume of the solid ? Upon what assumption is your
calculation based?
35. Fig. 38 is a contour plan of a heap of gravel. Each
small square in the plan represents an area of 1 square foot.
The contours are taken at distances of 1 foot measured
vertically. The height of the heap is 7 feet. Calculate its
probable volume.
36. Fig. 39 is the side outline of the reflector of a search-
light drawn one- fourth of the actual linear size. Its cross-
section is everywhere circular. On AB choose, at random,
any number of equidistant points. Measure the radii of the
circular cross -sections at the points so determined. Find by
calculation whether their areas are in A.P. If they are, cal-
culate the volume enclosed by the reflector.
EXERCISE XXXI.
ALGKB1UIC MULTIPLICATION.
A.
1. Test by diagrams the identity (a - b)c = ac - be, a
and b being negative, c positive, and a numerically greater
than 6.
2. Test by diagrams the identity (a -f b) 2 = a 2 4- %ab + b 2
when a is negative, b positive, and a numerically greater
than b.
3. Draw a set of diagrams to illustrate and test the steps
by which the identity can be proved by multiplication, a and
b being limited as in No. 2. [Study the frontispiece.]
4. Test by diagrams the identity (a - by 2 = a' 2 - 2a6 -f 6' 2
when a and 6 are both negative and a is numerically less
than b.
5. Draw diagrams to illustrate and test the proof of the
identity by multiplication, a and b being limited as in No. 4.
B.
6. Complete the following identities :
(i) (2a + 3b)(3a ~ 2b ) = 5 () (4P - 3q)(2p - 3q) = J
(iii) (2x - 5y)(sx - 47) = ; (iv) (ax + by)(bx + ay) = ;
(v) (ax - by)(bx - ay) = ; (vi) (ax - byXbx + ay) = ;
x y/\x y/
7. Test your answer to No. 6 (i) by putting a = - 3,
6 = + 4.
8. Test your answer to No. 6 (v) by putting x = + 2,
2/=-4, a=-3, 6=+ 5.
9. Test your answer to No. 6 (viii) by putting a = -f 2,
6 = - 3, j? = - 8, = + 7, x = - 10.
10. Show that for a given value of n, positive or negative,
the expressions 2n + 1 and 2n - 1 always describe con-
183
184 ALGEBRA
secutivo odd numbers. What expression will describe the
product of two consecutive odd numbers?
Test the validity of the expression when n = 10 and when
n - -4.
Prove that the product is itself always an odd number.
11. Prove that the product of any two odd numbers,
positive or negative, is always odd. (Take 2p + 1 and 2(7+1
as expressions for the two odd numbers.)
12. Two numbers, N\ and N 2 , have respectively the forms
n 2 - n -{- 1 and n + 1. What is the form of their product ?
Make a table of the values of N x and N >2 , when n + 3,
+ 2, 4- 1, 0, - 1, - 2, - 3 respectively. Do the products
follow the calculated law?
13. Two numbers are respectively of the forms n 2 -f n + 1
and n - 1. What is the form of their product? Test the
result by putting n = - 2'3.
14. What is the form of the product of two numbers which
have respectively the forms n 2 + n + 1 and n 2 - n + 1 ?
Confirm in the cases when n = 0, + 3, and - 10.
15. Show that | ! (a - fe) 2 + (b - c)* -{- (c - a)* } -
a- 4- b- 4- c 2 - ab - be - ca. Test by putting a = - 2,
1) = +3, c 1. Select any three values you please for
a, b, and c and test again.
16. Why must a number of the form a 2 4- b 2 + c 2 ab-
be - ca always be positive? Test the statement by sub-
stitution.
17. Two numbers, N 1 and N 2 have respectively the forms
a 2 -f b 2 + c 2 - ab - be - ca and a 4- b + c. Find the form
of their product. Prove that the sign of the product is the
same as the sign of No.
18. Prove that ab (a b) + be (b c) + ca (c + a) -
J { (a - b)' 3 + (b - c) 3 + (c - 6) 3 }. Test when a = + 1,
b - - 1, c - - 2.
C.
Note. The meaning of a number of several digits, such as
2137, can be expresse 1 as follows :
2437 - 2 x 1000 + 4 x 100 4- 3 x 10 + 7
= 2 x 10 3 + 4 x 10' 2 + 3 x 10 + 7
= 2J 8 + 4iJ 2 + 3< + 7
the symbol i being substituted for 10 for the s^ke of concise-
ness,
EXERCISE XXXI 185
19. Write in ordinary notation tho following numbers :
(i) 3t< + 7t 2 + st f 6. (ii) 2t* + pt { -f 6t a -f 3t + i.
(lii) 3t' + 5t { H- 2t" 4- 6. (iv) t (i -f 2t 4 + 2t' J + i.
(v) t : < - 3t* + 21-5. (vi) t' - t 4 4- t 5 - t 2 -f t - i.
20. Find tho form of tho product of two numbers whose
forms are t' 2 + I 2t + 3 and 3t + 2 respectively. Compare the
process with that of multiplying 123 by 32.
21. Show that 123 may he expressed as 2' 2 - 8 + 3, and
32 as 4 - 8. Find the form of the product of the numbers
expressed thus, and find whether it expresses the arithmetical
result correctly.
22. Find the forms of the products of the pairs of numbers
whose forms are given by tho following expressions :
(i) 2t r> - 3t 4 + 4f - 3t 2 + 2t - i and 3t - 2.
(ii) 3t 2 - 2t + i and t 2 - 2t + 3.
(in) t ! + 2t + 2t 4- I and t 3 - 2t + I.
(iv) it l - 8t 5 + 4t 2 - 2t -I- i and 2t + I.
(v) t 4 + 3t' + gt 2 + 27t + 81 and t - 3.
(vi) 2t' - 3t 2 + i and 3t" - 2t - 3.
Confirm (iv) and (v) by arithmetic.
23. Obtain the algebraic product of the following ex-
pressions :
(i) a 4 - a?ti* + b 1 and a' 2 -f b 2 .
(ii) a' 5 -f a 2 b 2 + b' 1 and a' J - a 2 b y + b\
(iii) x' J - xy + x + y y + y + i and x -f y - i.
(v) 4P! + ?L + ! and ^ - 2p , + i.
} 9Q 4 3q y 9q 4 3T
24. A number has the form a + b + c. What is the form
of its square? Test in the case when a = + 3, b = - 2,
c- - 3.
25. Use the result of No. 24 to write down without
multiplication the squares of tho following expressions :
(i) a-
b + c.
(ii) a + b - c.
(iii) 2a -
f 3b + c.
(LV) 2a +
3b - c.
(v) p-
2q + 3-
(vi) 3P -
2q 4- i.
(vii) I +
2 + l
(viii) a 2 +
b 3 -f c 2 .
(ix) p< -f
q' 1 + t\
w|-
a r'
180 ALGEBRA
D.
26. Complete StifeFs Table as far as the row giving the
coefficients of (a + b) }(} .
27. Write down the expansion of (a + by. Deduce from
it the expansion of (a - b) 7 .
/ d\ rj / Q\f>
28. Write down the expansion of U + gj and of ( %P ~ ~J .
29. Demonstrate the following properties of the binomial
coefficients :
(i) The sum of the coefficients of (a + b) n is always 2".
(Put a = b 1) ;
(ii) The sum of the odd coefficients is always equal to the
sum of the even coefficients ;
(iii) The coefficients succeed in the same order counting
either from the first or from the last ;
(iv) If n is even there is one greatest coefficient, if odd,
there are two equal greatest coefficients,
30. Calculate:
(i) The fourth term in the expansion of (3a - i) s ;
(2 \
1 ~ o^ 2 ) l
(iii) The term involving a (5 in the expansion of (a - 3&) 9 ;
(iv) The middle term in the expansion of (p* 1 - qr) 6 ;
(v) The term containing no variable in the expansion of
(- + -V
V2 + J
(vi) The terms containing p and - in the expansion of
\7
EXEKCISE XXXII.
THE INDEX NOTATION.
Note. Approximate answers should (in the absence of other
instructions) be given in the standard form and should be
correct to the first decimal place of the unit.
A.
1. The acreage and population in 191.1 of the four largest
and four smallest counties of England are given in the follow-
ing table. Eewrite the table in the index notation, substitu-
ting approximate numbers and taking one rmllion acres and
one hundred thousand people as the unit. Two decimal places
of the unit should be retained.
County. Acres. Population.
York . . . 0,721,339 3.909,151
Lincoln . . 1,668603 '557,543
Devon . . . 1,663,467 701,981
Northumberland . 1,291,515 697,014
Bedford . . 307,338 197,660
Himtingdon . . 207,572 48,105
Rutland . . 108,700 21,168
London . . 74,816 4,522,961
2. From your table calculate approximately the total acreage
and population (i)- of the four largest, (ii) of the four smallest
counties. Calculate approximately the average acreage and
population of each of the two groups of counties.
3. Express the following numbers in the ordinary notation :
41-082 x 10 5 , 7-201 x 10*, 0-002871 x 10 8 , 4-20576 x 10 :5 ,
0-00034792 x 10 4 .
4. Express the following numbers in the standard form :
23,827; 26-28; 41,200,000; 928-4; 147-6 x 10 7 ; 1659 x
10 r> ; 0-976 x 10 7 ; 0-00325 x 10.
X87
188 ALGEBRA
B.
5. Light travels at the rate of about 186,000 miles per
second and takes about two and a half years to reach the
nearest fixed star. Calculate the approximate distance of that
star.
6. The Atlantic Ocean covers about 31,530,000 square
miles and has an average depth of about 12,000 feet. A cubic
foot of sea water weighs 64*2 Ib. Find the approximate weight
of the ocean in pounds.
7. The average population of the British Isles during the
last ten years may be taken to be 43,600,000. Each person
on the average breathes about 16 times a minute and at each
breath draws in about 25 cubic inches of air. Find roughly
how many cubic inches of air have passed in and out of British
lungs during the last ten years.
8. Express each of the factors of the following products in
the standard form and obtain the value of the products to
threo significant figures. Express the products also in the
standard form :
(i) 18,360 x 5,018.
(ii) 31,069 x 283 x 8,204,921.
(iii) 3-2 x 1,921 x 723,431 x 17-9.
(iv) 136 x io r> x 0*0372 x io r '.
(v) 18*03 x ! :i x 0*741 x ! 3 x 0*000503 x io*.
C.
9. The distance of the earth from the sun is about
93,000,000 miles. Plow many minutes does the sun's light
take to reach us? (See No. 5.)
10. Calculate roughly, from the result of No. 6, the number
of tons in the Atlantic Ocean.
11. Calculate roughly, from the result of No. 7, the number
of cubic miles of air breathed by the inhabitants of the
British Isles during the last ten years.
12. Throw the numbers in the following expressions into
the standard form and find the value of each expression to
three significant figures, giving it in the standard form :
(i\ 18324 x 927 x 349i
' 17024 X 8189 X 21 '
({[\ 182 x IP 4 x 9 -6 x io' J x 0-23 x io 3
^ 1209 x 0-048 x 10^
Ci m \ ii '3 ^ io!_ x _J>'08 X J^ 5
40-03 x io 3 x 565 x^crozlT jp**
EXERCISK XXXII 180
D.
13. Simplify the following expressions :
(i) a & b* x be 4 -f a'bc'.
x
bp : <
4 P
(v) (a>) 3 .
(vi) (a*)"
(vii) [{(p 2 ) 4 } 3 ] 5 -
(viii) [{(p') 4 }'T-
14. Simplify the following expressions :
(i) N/a 5 : (ii)
(iii) V(a 4 /b'^). (iv)
(vii) /p6. (viii)
(ix) VKaV) 3 x (ab 4 ) J x a s }.
15. Simplify the following expressions :
(i) (x';) 1 x (y*) -f (z=0 (ii) (xy) 1 . (yz) m . (zx)n.
/a m \ I /b n \ m
(iii) (ab)' . (b^c) m . (c%). (iv) - S _,
16. Find the value of the following expressions :
(\\ (ifrz) 4 x (i4'2) 3 x 3000
u (9-i) 4 x (7'i) J '
/,n <S'6) J x (o'Q2 4 ^ x (07)^
u; ^(o-28) 4 x (4-2)
(0-9)3 x (4-9)* x (27)2
^ ; "(6'3) 5 x 4'5 x 10' '
17. Express each number in the following expressions as
the product of powers of its prime factors. Eedtice each ex-
pression to its simplest form as a product or ratio of such
powers :
(i) 1890 x 24500 x 504.
. _____
_____ ^.
x... 2299 x 253 x 396
v 247 x 1472 x 11362*
190 ALGKBRA
18. Write out in full a justification of each of the following
equivalences (The symhols m and n represent integers
without sign) :
(i) x a m x a n -= x a
(ii) x a m -f a" - x a m ~ n if m > n,
x a m -r a" - -f a n ~ m if m< n,
(iii) x (a 01 ) = x a mn = x (a n ) n V
(iv) x ^/a p x aP ' n if p is an exact multiple of n.
EXERCISE xxxirr.
NEGATIVE INDICES.
A.
1. Express the following numbers in the standard form
by means of negative powers of ten : '0*0248, 0-000372,
0-00006781, |, T -V, ^V<j> 36 / 105 0-23/10 1 , l/(4 x IO 7 ),
1/(250 x 10).
Note. The answers to Nos. 2-5 are to be given in the
standard form and correctly to three significant figures.
2. From the table in Ex. XXXII, No. 1, calculate approxi-
mately the number of persons per acre in each of the four
smallest counties.
3. In ]850 the French experimenter Foucault measured
the speed of light by finding how long it took to travel 20
metres. Assuming the speed to be 300,000 kilometres/sec.,
what was the interval of time which Foucault measured ?
4. The weight of the column of air resting on a square inch
of the surface of the sea is about 15 Ib. The air is supposed
to reach to a height of about 200 miles. Calculate approxi-
mately the average weight of a cubic inch of the air above the
sea.
5. A thousand feet above the sea the weight of the column
of air resting on a square inch is about 14 Ib. Find the
average weight per cubic inch of the lowest thousand feet of
air above the sea.
B.
6. Find, in the standard form, the value of each of the
following expressions :
... 7'2 X IO 3
^' 180 x io 5 '
(ii) 87 x io 4 x 2*5 x io~ tt x 800.
x ...v I '4 X IO~ 3 X 2'*t X IO 2
M * x Tn - 7 x o. T x Tn r
4 2 x io x o i x io
(iv) O x IQ4 x 57 x IQ" 6 x _4J4 X_IO " *
^ 1*87 x io ~ 4 x 9-5 x IO' 1 x 3 x io~ 4 '
191
11)2 ALGEBRA
7. Express each of the following as a product of powers
of prime numbers :
/i\ 9 x 63 x 51
w 81 x 14 x 867'
(11) _J|3_x_ 3?J<_65_ _____
338 x 598 x 256 x 115*
(iii) 88 x 143 x 5071
2299 x 922 x 64*
8. Eewi'ite the following expressions with positive indices :
(i) a-'b-'c ij .
(ii) 4a~ 5 b ; c~ 4 -r 6a*b"*c- 4 .
(iii) i5pV * 9q 2 r~ : ' x I2r 2 p"- 1 .
(iv) a-'^bc- 1 - b-'c) + b-'^ca- 1 - c-'a) + c^^ab- 1 - a~ l b).
Give the last answer in a simplified form.
9. Express each of the following expressions in the form
AcrtV. . . . (All symbols used as indices represent whole
numbers) :
(iii) 6a-V -=- sv'a-V: (iv) g?)' .
10. Write out a justification of the equivalences :
(i) a' 11 - i, (ii) a - i
a n
n being a positive whole number.
EXERCISE XXXIV
FACTORIZATION,
A.
1. Each of the following expressions gives an algebraic
product and one of its factors. Find the other factor.
(i) (25p 2 - I2iq 2 )/(sp - nq).
(ii) (a 2 + 6ab + 9b 2 )/(a + b.
(iii) (i6m 2 - 2411111
J>
(vi) (p- + 7 P + i2)/( P + 3)-
(vii) (p - 25p 3 + i56)/(p a - 12).
(viii) (a 2 x 4 - 2ax 2 - 8o)/(ax" J -f 8).
(ix) (x 4 + 2ax 2 - 8oa 2 )/(x <J 4- loa).
(xi) (a' - 3ft* + 3a - i)/(a - x).
+ 5.-,,. J..,, t
(xiv) (8a 3 + 27)/(2a + 3).
<->(S + o/(;
(xvi) (a'p - q')/(ap -
(xviii) (a 2 + b' 2 + c 2 -- 2ab + 2bc + 2ca)/(a + b + c).
(xix) (p 2 + q 2 + r a - 2pq - 2qr + 2rp)/(p - q + r).
(xx) (a 3 H- b 3 + c- <{ - 3abc)/(a + b + c).
(xxi) (a 3 + b lj - c- J 4- 3abc)/(a + b - c).
(xxii) (x' - y' - z* - 3xyz)/(x - y - z).
2. Factorize the following expressions :
(i) a 2 + 9a + 20.
(ii) x 2 - px 4- 20.
(iii) a' 2 - isab + 56**,
193
194 ALGEBRA
(iv) p a x 4 4- 23px 2 4- 130.
(v) i2x 4 - 7px' 3 -\- p 2 .
(vi) 6a* 4- I3ab + 6b 2 .
(vii) 6a 2 - I7ab 4- i2b 3 .
(viii) 8x 2 - 38x 4- 35.
(ix) 4x 2 4- 28x + 49.
(xi) a 2 4- 2a - 8.
(xii) a- - 2a - 8.
(xiu) p 2 + 3pq - 28q 2 .
(xiv) a^'- 1 - 3ap - 28.
(xv) 2oa' J + a - i.
(xvi) 20p 2 - pq - q 2 .
(xvii) 6x- 4- 5x - 6.
(xviii) 6a a - 5ab - 6b 2 .
(xix) 2l- 4 4- 23 ., - 20.
p P J
(xx) 12 - 4p' J - 2Ip 4 .
(xxi) 39 - 46- - 8~.
(xxii) 8a J - 27.
(xxiii) x + -.
/ x 4 ^
(xxiv) 27a 3 4- 64b 3 .
UKV) p' - |.
V XXV1 ) 2 y a -6 + X ^
(xxvii) x 4 4- x 2 y 2 4- y 4 .
(xxviii) ~r 4- ^ 4- Pt.
v 7 16 4
(xxix) n 8 4- n 4 4- i.
(xxx) 8p (i 4- 63p* - 8.
B.
3. Write down by inspection the algebraic square root of
each of the following expressions :
(i) a 2 -f b 2 4- c 2 4- 2ab 4- 2bc 4- 2C'a.
(ii) a 2 4- b 2 4- c 2 - 2ab 4- 2bc - 2ca.
(iii) p 2 4- pq 2 4- 4r 2 4- 6pq 4- I2qr 4- 4rp.
(iv) 4a 2 + 9b' J 4- 25c 2 4- I2ab - 3Obc - 2oca.
, N a 2 b 2 c 2 i , 2 U i
(v) H 1 -ab - be 4- -ca.
4 9 25 3 15 5
(vi) ~p 4 4- ^q 4 4- ^r 4 4- ?p lj q 2 4- q' J r'- 4 ^r 2 p 2 .
(vii) a' 2 b 2 4- 4b ? c a 4- i6c 2 a'3 - 4ab 2 c - i6bc 2 a 4- 8ca 2 b.
EXERCISE XXXIV 195
Note. If we expand (a 2 -f a + 2) 2 by the formula we
have :
(a 2 + a + 2)' 2 ~ a 4 + a 2 + 4 -f- 2a :j -f 4a + 4a 2
= a 4 -4- 2a 3 + 5a' 2 + 4a + 4
In the second line (i) the two terms containing a 2 have heen
taken together and (ii) the terms have been rearranged in
descending powers of a. If, then, an expression like that in
the second line is given and its square root is required, the
terms must be rearranged and decomposed so as to fall into
the scheme of the first line.
4. Find the algebraic square root of the following ex-
pressions :
(i) a 4 - 2a- J -f 5a- -43 + 4
(ii) a 4 + 2a :{ - 3a 2 - 4a + 4.
(Hi) a 4 - 4a l + ioa 2 - i2a + 9.
(iv) 4a 4 -i- 2oa { 4- ^a 3 - a + l .
* 3 39
a 4 2
(v) - a >: -f 3a 2 - 6a + 9.
Note
/ 1\2 I 4
'. (a + 2 + - ) =a 2 + 4 + -, + 4a+-f2
V a/ a- a
= a- + 4a + 6 + - + ,.
a a"
The second line could be written
This way of writing it shows that the terms are arranged in
descending powers of a.
5. Find the algebraic square root of the following ex-
pressions :
(i) p* - 4 P + 6 - + 1 2 .
(ii) p' - 4 p + 2 + i + 1.
(iii) p a - 2 P + 3 - \ + *-r
.
+ 391 6b
b 240 sa asa*
13
EXERCISE XXXV.
ALGEBBAIC DIVISION.
A.
1. Obtain the algebraic quotients in the following cases :
(i) (a 3 - 6a* 4- i6a - 2i)/(a - 3)-
(ii) (3a 3 4- 29a 2 b 4- 64ab 2 - I2b**)/(a 4- 6b).
(iii) (2a t{ - I9a 2 x 2 4- I3ax 4 4- 55x h )/(2a - 5x 2 ).
(iv) (a 4 - 7a 3 4- i3a 2 - i9a 4- 2o)/(a - 5).
(v) (2 4- I3P 4- 20p 2 4- 5p' 4- 24p 4 )/(l 4- 3P>-
(vi) (9a 4 4- 6a 3 - 2ia 2 - 2a 4- 8)/(3a 4- 2).
(vii) (21 ~ 25x 4x ~ 27x* 4- 3^x )/\3 ~~ 4x)
(viii) (a 4 - 7a 2 4- 3a 4- 6)/(a - 2).
(ix) (6a r ' - 23a* 4- 4oa 2 4- 49)/(3* - 7)-
(x) (i - 2ia 3 - 54a 5 )/(i - 3a).
2. Divide :
(i) a 3 - a 2 - 9a - 12 by a 2 4- 3a 4- 3.
(ii) 6p 3 - p 2 q - I4pq 3 4- 3q 8 by 3p 2 4- 4pq - q 3 .
(iii) n 4 - n 3 - n 4- i by n 2 4- n 4- i.
(iv) 2a 5 4- 3a 4 - I2a 3 4- i5a a - na 4- 3 by 2a 2 - 3a 4- i.
(v) p 5 - 4p 4 4- 3p J 4-3p 2 - 3p 4- 2 by p 2 - p - 2.
(vi) 6a 5 - 2a 4 - sa 3 4- i8a' J - ua 4- 10 by 3a 2 - a 4- 2.
(vii) 2a e - 3a 8 4- 4a 4 4- 7a 3 - 9a 2 4- I4a 4- 21 by a' 4- 2a 4- 3.
(viii) a 5 - 3a :{ 4- 9a - 3 by a 3 - 3a 4- i.
(ix) 2 - 6x 4- x 2 4- I5x 3 - I2x 4 - 9x 5 4- 9x 6 by i - 3x 4- 3x 2 .
(x) i - i6x f) 4- 3x 6 by i - x 4- 3x' 2 .
3. Find the integral equivalents of the following fractions :
- 1 (a> :;!?. (iii> a ' + I
.-. .
a + b v ' a 4- i
^ ' i - a ^ ' i 4- a
Verify the results by putting a = -f 4 in (i), a = - 2 in (iii),
4. Find out why there are no integral equivalents of the
fractions (a 4 4- 1) /(a 4- 1), (1 + a 5 )/(l - a), (i 4- a)/(l - a).
5. Explain (i) why (i 4- a n )/(l + a) has an integral
equivalent when n is odd but not when n is even ; (ii) why
(1 - a w )/(l -f a) has an integral equivalent when n is even
196
EXERCISE XXXV 197
but not when it is odd ; (iii) why (1 - a n )/(L - a) always
has an integral equivalent; (iv) why (1 -f- a")/(l ~ a) never
has an integral equivalent. (It is assumed that n is a
positive whole number.) Write down the equivalents where
they exist.
B.
Note. The expression a*b - 3a 2 & 3 - 4c 2 is said to be of
the fourth degree in a, of the third degree in 6, and of the
second degree in c; the degree being measured by the
highest power of the variable named. The expression is also
said to be of the fifth degree in a and Z>, since in terms in
which both these variables occur together the sum of their
powers is five.
6. Find the values of P and Q in the following identities
with the restriction that the degree of Q is to be as low as
possible :
(i) a 2 - 4 a + 7 = (a + 2)P + Q.
(ii) 2a 2 + 3a - i = (a - 3)? + Q.
(iii) a 3 - 3a 2 + 2a - i = (a + i)P + Q.
(iv) 4a 3 - 7a + 3 = (2a - s)P + Q.
(v) a 4 - 3 a* - 4 = (a 2 - 7 )P + Q.
(vi) a 4 -f 2a 3 - a + i = (a + 4)? + Q.
7. From your result in No. 6 (i) fill in the values of P and
Q in the identity,
a* -4a+7 = Q
a -f 2 a + 2*
In other words, write down the integral expression and the
proper fraction (or complement) which are together equiva-
lent to the fraction (a 2 - 4a + 1)1 (a + 2).
8. Use the results of No. 6 to express each of the following
fractions as equivalent to an integral expression together
with a complementary proper fraction :
4*L=_7!L + J
v/ -*
....v
(in)
v '
x
a 4
4
9. What is the value (i) of (a - 3)P when a = 3 ; (ii) of
(a -f 2)P when a = - 2 ; (iii) of (2a - 5)P when a = | ?
Do the answers depend at all on the value of P ?
108 AT.OF.imA
10. Turn to your working of No. 6 (i). What is the value
of the right-hand side when a = -2? What, then, should
he the result of substituting - 2 for a in the left hand side ?
Find if it is so.
11. What rule would you give for finding Q without
previously finding P in problems like those of No. 6? Test
the rule by seeing whether it gives you the values for Q which
you have already obtained in (ii), (iii), and (v).
12. Find the complementary pi oper fraction in the following
cases without finding the quotient :
(i) (a ! - 4a'' -\- 6a + 2>/(a + 3).
(ii) (a* - 7a 2 + i 3 a - 4 )/(a - 4)-
(iii) (a 4 + a + i)/(a + i).
(iv) (a' - 2a + sa- - 4)/(a + i).
(v) (2a* - sa 2 - a + s)/(2a - 3).
13. Find the values of P and Q in the following cases, the
degree of Q being as low as possible :
(i) a 4 - a 3 -i- 2a + i - (a' 2 - 2a + 3)? -i- Q.
(ii) 3a 4 + a- { + 5a* - 5a + 2 - (3a 2 - 2a + i)P + Q.
(iii) 6a 4 - sa b + 4aV - sab { + 6b^ - (a' 2 - 2ab - 7b a )P + Q.
(iv) a 5 - i = (a 2 4- a - i)P + Q.
14 Use the results of No. 13 to exhibit each .of the following
fractions as equivalent to an integral expression together with
a proper fraction :
(i) (a. 4 - a j + 2a + i)/(a a - 2a + 3).
(ii) (3a* + a :{ 4- 5a 2 - 5a + 2)/(3a 2 - 2a + i).
(iii) (a :> - i)/(a 3 4- a - i).
15. Arrange a 4 - a :i - 2a~ - pa + 3 in the form
(a 2 - 3a + 1)P + Q.
16. For what value of p is a 4 - a 3 - 2a- - pa -f 3 exactly
divisible by a 2 - 3a + 1 ?
17. For what value of p is a 6 - a 4 -f pa -f 8 exactly
divisible by a- 4- a + 2 ?
C.
Note. When the integral part of the equivalent of a
fraction is calculated in ascending powers of the variable
(e.g. 1 + 2a - 3a 2 + 4a 3 + etc.) the complement cannot be
a proper fraction ; for the degree of the numerator will be
higher the greater the number of terms in the integral part.
The integral part is in these cases often called the expansion
of the fraction in ascending powers of the variable.
RXEROTSK XXXV 190
18. Find the integral expression and the complementary
fraction which are equivalent to the fractions
(i) (i + a 4 )/(i + a). (ii) (i - a')/(i + a).
(hi) (i + a")/(i - a). (iv) (i + a)/(x - a).
19. What is the value of the complementary fraction in
No. 18 (i) when a = - t V ?
20. What is the numerical value of the error that would
be made by assuming
(1 - a 7 )/(l + a) = 1 - a + a 2 - . . . + a Ct
when a = ? What is the ratio of the error to the whole
value of the fraction V
21. Within what degree of numerical accuracy can we
(1 + a 9 ) , ,
write -TJ- -- = 1 + a + a 2 + . . . + a 8
1 a
when a = - ?
22. Why are the questions in Nos. 19-21 asked with regard
to values of a less than 1 ?
23. Show that
1 1 -, A &
--- = 1 + a + a 2 4- a* -f a 4 -f .
1 - a I - a
Show that the ratio of the complement to the whole value of
the fraction is a :> . Putting S = the complete value of the
fraction, show that the error involved in neglecting the com-
plement after the term a 1 is less than S x 10 ~ 3 when a = .
24. Show that, in general,
a + a 2 + . . . + a n
-, - | VV | IV I . I LV | -.
1 - a 1 - a
and that the complement, after the fraction has been expanded
to n terms, is a u .S; S being the complete value of the frac-
tion. Show also that if a is numerically less than 1 the
complement can be made as small as is required by taking n
sufficiently large.
25. Expand the following fractions to the number of terms
indicated, adding in each case the complement after the ex-
pansion :
(i) i/(i + a) to a 5 .
(ii) i/(i + a 2 ) to 7 terms.
(iii) i/(i - 2a) to 6 terms.
(iv) i /( i + J to 5 terms.
200 ALGEBRA
(vi) (i
- a)/(i 4- 2a) to 5 terms.
(vii) (i
4- a)/(i 4- a) to 7 terms.
(viii) (i
- a)/(i- 4- ^a) to 4 terms.
3 1
\r/i//
Q >x - 1
a.YOttJ.
vv e nave
1 - a * * 1 - a
= 3(1 + a + a 2 +
_1 1
Als 2 - a ~ 2(1 - ia)
26. Give the specified number of terms of the expansion
and the complement :
(i) 2/(x - 3a) to 4 terms.
(ii) 3/(i 4- a) to 5 terms.
(iii) i/(2 - a 2 ) to 6 terms,
(iv) i/(3 4- 2a) to 4 terms.
(v) (i - 2a)/(2 4- a) to 5 terms.
(vi) (i + ia)/(4 - 3a) to 4 terms.
27. Show that
1 - a 4- a 3 - a 4 + . ~ r> .
1 + a + a 2
28. Expand 1/(1 + 2a a-) as far as the fourth term, and
show that the complement is (29a* 4- 12a r> )/(l + 2a - a 2 ).
29. Find the first four terms of the expansion of
(1 + 2a)/(l - a + a 2 )
and the complement after the expansion.
30. Find the first five terms of the expansion of 1/(1 + x) 2 .
What is the complementary fraction ?
EXERCISE XXXVI.
GEOMETRIC SERIES.
A.
1. A lamp hanging at the end of a chain is pulled to the right
1 metre out of the vertical and is then released. It swings
to a point 0'9 in. to the left of the vertical, then to a point
0'81 m. to the right of the vertical. The succeeding swings
diminish in accordance with the same law. Find the greatest
distance the lamp could travel before coming to rest (i) in-
cluding the first movement of 1 m. to the right, (ii) excluding
this movement and counting only the free swinging of the
lamp. Show in case (i) that when the lamp passes through
the vertical for the seventh time after being withdrawn it has
accomplished more than half its greatest total movement.
Illustrate by a diagram, showing the total space travelled after
successive swings.
2. A ball is lifted 10 feet from the floor and is then dropped.
It bounces to a height of 8 feet. Each subsequent bounce
carries it four-fifths as high as the preceding one. Find the
greatest distance the ball could travel before coming to rest
(i) including and (ii) excluding the distance through which
it was lifted in the first instance. After how many bounces
will the ball have travelled altogether one-half of the maxi-
mum distance ? After how many bounces three-quarters of
the maximum distance?
3. A weight is hanging at the end of a vertical rubber
cord. I pull it down 10 inches and release it. It swings
up and down for a considerable period, the amplitude of each
semi-vibration being 0'7 of the preceding one. (The ampli-
tude is the extreme distance the weight rises or sinks above
or below the position of rest. Each vibration includes two
amplitudes one below and one above.) Calculate the dis-
tance the weight would travel if it vibrated according to this
law for ever. Show that after four semi- vibrations it has
actually travelled more than 75 per cent, of this distance.
201
202 AtGEBUA
When will it have completed 90 per cent ? Illustrate you!*
solution by a diagram.
4. A man undertook to pay 1000 to a charity one year,
750 next year, three-quarters of this sum in the third year,
and so on until his death. What is the outside limit of the
expectations of the charity? If the man died after making
twenty donations, how much would their total fall short of
this sum? [(2)'-' - 0-003171.]
). Another subscriber to the same charity promised to give
100 the first year and to increase his donation by one-tenth
every year as long as he lived. What would be the total of
his donations if he also lived for twenty years ? What would
be the amount of his last donation? [(l*i)- = fr727.]
Illustrate Nos. 4 and 5 by two paiallel strips in which the
first and last donations are mtxiked off on the same scale
and the limiting sum is indicated.
B.
Note. A series of terms each of which is obtained by
multiplying its predecessor by a constant factor is called a
geometric sequence. The constant factor is called the
common ratio of the sequence. It is evident that, unlike an
arithmetic sequence, a geometric sequence can be continued
without end both ways even if the terms are non-directed
numbers.
A series of numbers made by taking any term of a
geometric sequence and any number, limited or unlimited, of
consecutive terms which immediately follow it is said to bo
in, or to form, a geometric progression (a. P.). [Cf. Ex
XXIX, G, Note.]
We have seen that when the common ratio is positive and
numerically less than unity the sum of a G.P. never goes
beyond a certain value however many terms are taken, al-
though it can be made, by increasing the number of terms,
to approach, and thereafter to keep, as near to that value as
we please. This value is best called the limiting sum of
the G.P., but is more generally known as " the sum to in-
finity ". [See Ex. XXXV, Nos. 22, 24.] When the ratio
is negative and numerically less than unity the sum of n
terms also constantly approaches nearer to the limiting sum
as n increases but swings alternately above and below it.
EXERCISE XXX VI 203
Each of the terms between the first and last term of a G.P.
is called a geometric mean.
6. State the common ratio of each of the following
sequences. Continue each for three terms hoth ways :
(i) .
, . . 4, 6. o, ITS
*t **> yi * o j>
. . . - 8, + 6, -
4'5, - - -
(iii) .
iii
' ' ' 6' 18' 54' ' '
(iv) .
P J P
q 1 ' q"'
i
q
(v) .
i
I
\ /
' a - b' a(a
- b/ J ' ' ' ' '
7. In No. G (i) give expressions for the 13th term to the
right of G and the 15th to the left of 13'fi in terms of positive
and negative powers of the common ratio.
8. In No. 6 (ii) give similar expressions for the 8th term
to the right and the nth term to the left of -H G. [The symbol
( - )" is used to denote the sign of a product of n negative
factors. ]
9. In No. G (iv) give similar expressions for the 9th term
1 v
to the right of -f - and the n term to the left of - -.
b <1 ^
10. In No. G (v) give similar expressions for the n\\\ terms
before and after a.
11. Eind the limiting sum (the " sum to infinity") of each
of the following series :
(i) S = i + J + | + . . . .
(ii) S= + x-4 + i-.-..
(iii) S = 24 + 20 + 16^ + . . . .
(iv) I7'5 + S'25 + i '575 +
(v) i - + ~ ' ' ' tla/b|<x].
Illustrate the summation of (i) and (ii) by two parallel
strips marked off as in No. 5.
Note. The symbol I a\ means the numerical value of a, i.e.
its value when its sign is removed.
12. Write formulae giving the difference between the sum
of n terms of No. 11 (i), (iv), and (v) and the limiting sum.
204 ALGEBRA
13. Let S H = the sum of n terms of a G.P. and S~ the limit-
ing sum. It is often convenient to write
S - S w = fcS.
Find the value of k in No. 11 (i), (in), and (vi).
14. Write a formula for the sum of n terms of each of the
following series :
(i) i + 2 + 4 + 8 + . . . .
(ii) 100 + 102 + 104*04 + . . . .
(iii) - 8 4- 10 - 12*5 -f . . . .
(v) q + pM- + . . . [p 2 >|q|].
15. In a o.p. let a=^ the first term, r = the common ratio,
?t=the number of terms. Write formulae for (i) the nth
term ; (ii) the sum of n terms, r being greater than unity ;
(iii) the same, r being less than unity ; (iv) the limiting sum
in the last case.
16. Write out full proofs of the formulae of No. 15 (ii) and
(iv). Prove by means of your formula} the properties de-
scribed in the third paragraph of the note to No. 6.
C.
17. A sum of P pounds is invested at compound interest
for n years, the rate of interest being 3 per cent per annum.
Show that its amount, A, is given by the formula
A = P x (1-03) M .
Change the subject of this formula to P, using a negative index
for the sake of conciseness. How will you describe P as
used in the second formula ?
18. Compound interest is given at the rate of i per pound
per annum. Write formulae for A, the amount to which a
sum P would accumulate in n years, and for P, the sum which
would, by accumulation, produce A in n years. (Note that
P is called the present value of A, A the amount of P.)
19. On 1 January, 1905, a man, X, determined to save 20
every year for the next five years and to invest it at the end
of each year in a business that promised him 3 per cent per
annum compound interest. Accordingly he invested 20 on
1 January, 1906, 1907, 1908, 1909, 1910. Show that the
total sum standing to his credit immediately after he paid
in the last 20 was
EXERCISE XXXVI 205
A - 20 { (1-03) 4 + (1-03) 3 + (1'03)' 2 + (1-03) 4- 1 }
- ~!(1W' - i}.
According to the Compound Interest Tables 1 will, at 3 per
cent compound interest, accumulate in live years to 1*1593.
Use this number to calculate the value of A.
20. Another man, Y, invested in the same business on 1
January, 1905, a single sum which by 1 January, 1910, had
accumulated to exactly the same amount as X's successive
investments in No. 19. Show that this single sum was
P _ 4f (T03)' -J.
'
Calculate its amount to the nearest tenth of a pound.
Note. Suppose that Y, instead of investing 91'6 in the
business on 1 January, 1905, had, on that date, lent this sum
to X. Suppose, further, that X undertook to return the loan
with interest by paying Y 20 on 1 January, 1906, 1907,
1908, 1909, 1910. Finally, suppose that Y on the days when
he received each instalment invested it as X did in No. 19.
Then the answers to Nos. 19 and 20 show that on 1 January,
1910, Y would have been in exactly the same position as he
would have been if he had invested his money as in No. 20.
In other words, to lend 91*6 and to receive in return five
annual payments of 20 is financially equivalent to investing
91-6 for five years at 3 per cent.
A number of equal payments made at regular intervals in
consideration of a lump sum previously received is called an
annuity. The lump sum is called the present value or the
cost of the annuity. The sum paid periodically is called the
rent of the annuity. The relation between the rent and the
cost is fixed not only by the number of payments (the term
of the annuity) but also by the rate of interest expected by
the person who buys the annuity or makes the loan which
is to be repaid by the annuity.
21. Let P = the present value, A the amount, a=the
rent, n the term of an annuity, and let i = the interest to
be earned by 1 in the interval between two payments.
Show that
206 ALGEBRA
i
and that P = A(l + /) - "
-a (lt!)!j
'
a
= CL . . .
1
22. Convert the last two formula) into formulae for finding
the rent of an annuity, given the present value, etc.
23. A man borrows 500 from a building society in order
to buy a house. The loan is to be returned with interest at
4 per cent by ten equal annual payments. The first payment
is to be made on the first anniversary of the loan. Calculate
the annual payment. [The amount of 1 in ten years at
4 per cent compound interest is 1*4802.]
24. Another man borrows 500 upon the same terms, ex-
cept that the re payment is to be spread over fifteen years.
Calculate the amount to be paid annually. [In fifteen years
1 becomes 1-8009 at 4 per cent compound interest.]
25. The Urban District Council of a seaside town invite
subscriptions to a loan of 21,000 for the construction of a
new pier. The loan is to be discharged by seven equal annual
amounts paid out of the rates, interest at 3^ per cent being
allowed. What annual charge upon the rates will be re-
quired ? [The amount of 1 for seven years at 3^ per cent
compound interest is 1*2723 ]
26. A man buys for 800 a house of which the lease has
forty-three years to run. The property is subject to a ground
rent of 10 per annum. (This means that for the next forty-
three years the man or his successors will receive the rent of
the house but must pay out of it 10 to the owner of the
land.) By a special arrangement the rent and ground rent
are paid once a year on the anniversary of the purchase.
Calculate the amount of the rent if the investment is to yield
5 per cent. [Amount of 1 for forty-three years at 5 per
cant compound interest = 8'1497.]
27. Show by means of the last formula of No. 21 that no
matter how long an annuity runs its cost cannot exceed a
limiting value given by the formula
P - a/f.
EXERCISE XXXVI 207
28. After the battle of Trafalgar, Parliament made a grant
of 6000 per annum to Nelson's heir and his descendants for
ever. What single sum would have been equivalent to this
grant, interest being reckoned at 3 per cent per annum ?
29. In ancient times a settled proportion of the produce
on agricultural land was set aside for the maintenance of the
clergy. It was called the tithe of produce. Since 1836 the
tithe has been paid in money. When land is taken for
building purposes, etc., the tithe is often redeemed, that is,
the tithe-owner surrenders his right to receive annual tithes
lor ever in consideration of an equivalent lump sum paid to
him by the landowner. The interest assumed in calculating
the redemption must not be more than 4 per cent
The tithe upon a certain piece of property is 3 10s. per
annum. What sum is needed to redeem it, interest being
reckoned at 3^ per cent ?
Note. The formula of No. 27 shows that a perpetual
annuity can be purchased by 1/z times the annual rent.
For this reason 1/i is called the number of years' pur-
chase. Thus if a tithe were redeemed or a freehold property
bought for twenty-five years' purchase the interest would
be 4 per cent.
30. A man takes a piece of land on a lease for 999 years
at an annual rent of 24. After a short time the landowner
agrees to let him have the freehold of the property for thirty
years' purchase. How much must the purchaser pay for the
land and at what rate is interest reckoned ?
EXERCISE XXXVIL
THE COMPLETE NUMBEi* SCALE.
Note. The " scale " required in some of the examples is a
long straight line graduated uniformly with plus or minus
numbers from an origin in the middle of its length. The
graduation should be carried to 50.
A.
1. Show on a scale the various positions of a point P which
represents in turn the values of a, a *f 6, a - b, ab, a/b,
ah 2 first when a = -2, b = + 5, and secondly when
a = + 2, b = ~ 5. (The successive positions should be
marked P , P t , P L >, etc.)
2. Show on a scale the movements of a point P which
marks the successive values of aft" (i) when a = - 1, b + 2 ;
and (ii) when a 4- 20, b ~ -, while n assumes in suc-
cession the values 0, 4- 1, 4- 2, . . . + 5 in each case.
3. Repeat the two investigations of No. 2, giving n in suc-
cession the values 0, - 1, - 2, ... 5 in each case.
4. Mark on a scale (where possible) the positions of a point
which registers the values of a(b + pc n ) when a = - 5, b =
+ 4, c= ^, p 9 and n assumes in succession the
values - 4, - 3, - 2, - 1, 0, + 1, + 2, + 3, + 4. What
will happen as the value of n continually rises ? What would
happen if the value of n were continually lowered ?
5. A point P occupies in succession the points 0, + 10,
-f 20, . . . + 50. Label these points P , Pj, etc., upon your
scale and label with the letters Q 0> Q I( Q 2 , etc., the approximate
positions of all the square roots of these numbers.
6. Indicate with suitable labels the numbers - 50, - 40,
- 30, ... 0, + 10 . . . and the appioximate position of
their cube roots.
Note. The symbol co is used to denote a very large num-
ber that is one whose reciprocal is very nearly zero.
7. Indicate on a scale the range of values assumed by the
expression (15 + 30#) / (1 -f 3#) as x moves ( rom - oo
208
EXERCISE XXXVII 20&
through /oro fco + oo . Move your pencil point along the
range in the way in which the representative point would
move as x passes through its various values. (Before cal-
culating values transform the expression as in Ex. XXXV,
No. 7.)
8. How would the results of No. 7 have been different if
the expression had been (15 4- 30#)/(1 - 3#) ?
9. Kepeat the investigation of No. 7 upon the values of
(15 + 30# 2 )/(1 + 3# 2 ), as x moves from - oo to + oo .
10. The point P marks the values of the expression - 4#.
Show that it will traverse the whole scale of numbers twice
in the negative direction while x traverses it once in the
positive direction.
B.
11. Find the value of the abstract variable x given that
I (2o? + 17) - 7 = 10 - (1 - 3a?).
Let two points P and Q record on parallel scales the values
of the left and right-hand sides of this relation. Indicate (by
the letters P , P 1? etc., and Q , Q,, etc.) the positions of P and
Q at the different stages of the solution and so justify it.
12. Solve the equation = -5 = 0. Justify the
AX> *\* i
various stages of the solution by the method of No. 11.
13. Solve the following equations :
(i) 4(2 - x) - HSx + 21) - (x + 3) - o.
(ii) 3 - ^ + (07 - x) - o.
(ill) (2X - I) (2X + 3) - (4X + 7) (X - 2) = 0.
(iv) 8 - 2 { sx - 7 (4 + 3x)} = o.
(v) v /(2x - 7) + 6 = o.
(VI) v/(2X - 7) - 6 - O.
(vii) #(sx + 8) + 3 = o.
(viii) 3 - t/(5x - 8) = o.
(ix) 2 X + /v /{( 4 x + i) (x - 4)} - 3 = o.
(x) (3x - i) 2 - (x - 3 ) 2 - o.
(xi) (x - 2>' - x(x - 4) (x - 5) + 4 (2x - x) = o.
(xii) (2x + i) (4x' J - 2X + i) -f (x + 2) (x 2 - 2x + 4) = o.
14. Express the sum of the two fractions P/(# - 1) and
Q/(# - 2) as a single fraction.
15. What is the numerator of the fraction obtained in No.
14 if P = +3 and Q - - 1 ?
U
210 ALGEBRA
16. What must be the values of P and Q in order that the
numerator may he (i) H- 1, (ii) x, (iii) 2,7; - 3 ?
17. Use the results of No. 16 to calculate the value of
2# - 3 _ x
~E 5 i T\ and o 'n
3a? + 2
(i) when x = 0, (ii) a? = 4* 4, (iii) # - 3.
Note. The fractions l/(x - 1), l/(x - 2) are called the
partial fractions of the fraction
(2a - 3)/(z 2 - 3<r + 2).
Similarly - !/(# - 1) and 2/(a? - 2) are the partial fractions
of xl(a? - 3x + 2).
18. Use the method of Nos. 14 and 16 to find the partial
fractions of the following :
v "' 2x 2 + x - 6> Viv ' I2x 2 + I 3 x - 14*
19. Use your results to find the value of (i) when x = - 4,
of (ii) when x = 4- 8, of (iii) when x = -f 7, and of (iv)
when # = - 13.
20. Find values of P and Q to satisfy the relation.
2a + 1 _ P Q
(x - I) 2 ~ x - 1 + (of-"!)" 2 '
Verify your result by putting x = + 4 and x = - 9.
21. Analyse the following into partial fractions after the
pattern of No. 20.
(i\ 2x + T . rii^ 2X ~ 9
U (x + 3 ) 2 ' W (2x r 3)* '
Verify each answer by substituting a positive or negative
value for x.
22. Express each of the following fractions as the sum of
an integral expression and a series of partial fractions.
6x 2 - IPX + 16 .... 9x <2 - 29* + 38 t
'*' 2x a + 3x - 2 ' ^ n ^ 3x a + x - 10 '
( 1U ) 4x 2 - I2x + 9 ; ^ 1V ^ (x + i) 8
Verify any two of your answers by substitution.
23. Reduce to single fractions (i) ^ ~ and
X JL\J X " f
EXKRCISK XXXVII 211
w x "- 9 # - 6*
Use your results to find the value of x which satisfies the
condition :
1 I ^ 1 1_
x _ 10 x - 1 " x - 9 a; -6
Also use them to find a value of x such that
x 9 # _ 5 # 8 x 5
a? - 10 ~ aT^7 ^ x - 9 ~ a; - 6'
[Keplace each fraction by its equivalent integral expression
and complementary fraction. The relation then becomes
identical with the former one.]
24. Find the values of the variable which comply with the
following conditions :
x - 3 2x-_3 3
c.
25. If (7 - 3#)/4 < what can be said about the value
of x ?
Justify your argument (as in Nos. 11 and 12) by means of
a pair of number-scales.
26. Show that if x is positive Sx/^x* - 9) > only if
x > + f , and that if x is negative 3x/(4x 2 - 9) > only if
x < ^
27. Show that 3a/(4o; 2 - 9) < only if + | > x > - f.
Mark on a scale the ranges of values of x that make
3x/(4:X 2 - 9) > and < respectively.
28. Mark on a scale the range of values of x within which
(i) (7 - 3*)/(* - 6)>o ; (ii) (22 + 7 x)/(8x - 5 o)<o ;
(iii) (x - 7) (x + 2)>o ; (iv) (x - 4) (x + 8)<o ;
(v) x 2 - 5x - 8< - 14 ; (vi) 6x 2 + I7x + 5> -f 19.
29. There are two directed numbers m and n. When
the first is multiplied by + 4 and the second by - 5 the
algebraic sum of the products is + 23. When the first is
multiplied by - 7 and the second by + 4 the algebraic sum
of the products is 7. Calculate the values of m and n by
the composition method, and justify your procedure.
14*
212 ALGKBRA
30. Find the values of directed numbers which satisfy
simultaneously the following sets of conditions :
(i) 1501 + ipn = 18, igm H- 150 = 50.
(ii) 3m - 4n + 2 5m - 6n - 2 = 7m + 2n + 4.
(iii) 4 - - -= - 7 ; - + -5 = + I0 .
v ' m n ' ' m n
,. v 6 2
(iv) + - = - i ; 2mn - 3m + sn = o.
(v) (m + i) (n + 5) = (m + 5) (n + i), mn -f m + n =
(m -i- 2) (n + 2).
(vi) m + n = i ; (2m - 3) (8n + i) = (4m - 5) (411 + 5).
(vii) m-fn + p= - i, 2m + 311 + p = +2,
4m -f 911 i p = + 14.
(viii) m - 2n = - 3, m 2 - 411^ = + 12.
(ix) m 4- n = - 4, mn = - 21.
(x) m - n = - 25, 4m - 4n ~ mn.-
EXERCISE XXXVIII.
FURTHER EXAMPLES ON DIRECTED NUMBERS.
A.
1. Given that i = w W (l + ~) . -
W -f w \ v/ v
calculate t when W = 190, w=2, V=-4, =+ 100
a - 0-1.
2. Calculate the value of
(i) whenoj = 0, (ii) when # = - 10, (iii) when x = -f 10,000
(iv) wheno; = - oo ; given thatjp = - ^, g = + J,r = - J.
3. Find the value of
1 4- b sin a
C COS a
(i) when x = 0, a = 12, (ii) when x = - 4, a = 55 ; given
that a = + 10, 6 = - 5, c = +4.
4. Evaluate
x
(i) when x = 0, p = + 2, g = - 3, (ii) when a? = + 10,
p = _ 3, g = + 2, (iii) when x = - 10, ^ = - 3, = + 2 ;
given that a = + 3, 6 = - 2.
5. In the formula
2'* _ 2"**
tan (a - /?) = 2M , + 2 . pj
y3 = 20 and p = -f ^. Find the value of a (i) when # = 0,
(ii) when x = +10. Prove also (iii) that when x is positive
and large a is practically constant with a value of 65.
B.
r 1 1 - 3a + a 3
6. Express a - _ + - rf -
214 ALGEBRA
as a single algebraic fraction. Show without further calcula-
tion that the expression
1 1 + 3a - a 3
has the same value. Verify your conclusion by substituting
(i) a = 0, (ii) a = -f 5, (iii) a - 7, in each of the original
expressions.
7. Simplify the product
x ~
x -
What new identity can be deduced from your result by
substituting x + 1 for # ? Verify both identities by putting
(i) x = + 2, (ii) a = - f
8. Keduce each of the following expressions to a single
algebraic fraction in its lowest terms :
(a + 2b) 2 (a - 2b) 2 '
i i 8b 2
a + 2b a - 2b a 3 -
+ i
v ' x 2 - i x 2 +' i x 4 + x 2 + I x 4 - x 2 + i*
(iv\ T + ** ~ 3 , * - i , 4?
(iv) i + x _ i + (x + i)2 + (x + i)2 ^ (x _ i} .
9. Show that the addition of the term (p + I) 2 to the ex-
pression
vp(p + 1) (2p + 1)
produces the same result as the substitution of p + 1 for p.
Verify by putting (i) p = + 9, (ii) p = - 6.
10. Show that the addition of the term (p + I) 3 to the ex-
pression
produces the same result as the substitution of p + 1 for p.
Verify by putting (i) p = + 2, (it) p = - 11.
11. Show that
a r ) rc r
EXERCISE XXXVIII
12. Use the identity of No. 11 to prove fchat
[Start from the expansion of 1/(1 - a) proved on p. 199.]
13. Deduce from this equivalence foimulue for calculating
1/(1 - x}* (i) to a first approximation, (ii) to a second ap-
proximation, (iii) to a third appioximation.
What are the values of 1/(0*999) 2 given by each of these
three approximations ?
14. Use Nos. 11 and 12 to derive formulas for calculating
(i) 1/(1 - x)*, (ii) 1/(1 - xY approximately, the approximation
being cairied in each case as far as the term involving x*.
15. What do the formula) of Nos. 12 and 14 become
when - x is substituted for x ?
16. Calculate to a third approximation the values of (i)
1/(0999) 3 , (ii) 1/(1-001) 3 , (iii) 1/(1-002) 4 - 1/(0 998) 4 .
C.
17. Solve the equations :
, I ,
__ __ _
n n - i n - 2 n(n - i) (n - 2) ~
t 2 i n
5n 4- 9 sn + ii
18. Find values of m and n which satisfy the following
pairs of relations simultaneously :
(i) sm - 2n = 12, 9m' 2 - 4n 2 = 576.
(ii) (m + n) + ^(m - n) = 2, 401 + n - i^
di\\ 5__ _ ___7 _ 3L"_ 2 . 6 + m
^ lll) m - 2n ~ 2m~- n ' 7 ~ 5 '
(iv) m
m n
- --
J '
n m m
216 ALGEBBA
19. Find values of /, m y and n which satisfy the following
relations simultaneously :
(i) 4! - 5m + n + 6 - o ;
7! - iim -f 2n + 12 = o ;
1 + m + $n - 9 = o.
(ii) 1 + m = 6;
m + n = 28 ;
n + 1 = 12.
(iii) 3! - 7111 + 411 = i ;
5! - 901 + n = - 22 ;
1 - 2m + n o.
(iv)
51
4- 2m
+ 3n = 18 ;
3l
+ 7m
- n = 5;
1 -
- 2m -
t- n = 6.
a
b
c
(v)
f 1-
- = 3
1
m
n
a
b
c
1 "
m
5" i;
2a
b
C
= O.
1
m
n
20. Express each of the following as a sum of partial
fractions with the simplest possible numerators and de-
nominators :
Sketch the graph of (ii) from n ~3tow= +3.
EXERCISE XXXIX.
LINEAR FUNCTIONS.
Note. A line when moved is supposed always to remain
para'lel to its original direction.
A.
1. Describe in words the positions of the straight lines
which correspond to the following relations :
(i) y = i'6x. (ii) y = i'6x + 47.
(iii) y = i'6x - 8 '2. (iv) y = - 2'5x + 14*3.
(v) y = - 7-8 - 47x.
2. Write down the relations which correspond to lines in
the following positions :
(i) Inclined at 31 to the x axis and raised 47 units.
(ii) Inclined at 116 to the x-axis and raised 22 units,
(iii) Inclined at 42 to the x-axis and lowered 12*3 units,
(iv) Inclined at 158 to the x-axis and lowered 11 units.
3. Throw the following equations into the form y = ax + b
and give the positions of the corresponding straight lines :
(i) 7x - icy - 26 = o. (ii) 2'8x - 4y+ 10 = o.
(iii) 3x - 4y + 12 = o. (iv) 4x + 3y - 20 = o.
(v) 5'4x + 3y - 24 = o. (vi) sx - 5'4y + 16-2 = o.
4. Draw on the same sheet the six lines of No. 3. How
could you have foreseen (i) that the first pair of lines would
he parallel ; (ii) that the lines in the second and third pairs
would be mutually perpendicular ?
5. Show that the lines corresponding to two equations of the
form ax + by 4- c x = and bx - ay 4- c 2 = must always
be at right angles to one another.
6. Find the linear relations that are satisfied by the follow-
ing pairs of values of the variables. Express them in the
standard equational form :
(i) (- 4, + 7) (+ 2, - 3). (ii) (+ 2, + 3) (- 4, + i).
(iii) (- 3, -f 87) (+ s ~ 14 '5>- (iv) (+ 7, - 32)(+ 7, + 4' 8 )-
(v) (- 147, - 2-3) (+ 6-4, - 2-3).
217
21 S ALGEBRA
7. Calculate (i) the crossing-point of the lines corresponding
to the relations in No. 6 (i) and (ii) ; (ii) the inclination of
each of these linos to the o>axis ; (iii) the angle between them.
Verify by drawing the lines and making the necessary measure-
ments. Decide without drawing a figure what are the posi-
tions of the lines corresponding to the relations in No. 6 (iv)
and (v) Verify by drawing.
8. Find the value of x for which the functions
Ix - 3 and - 5x + 6
have the same value. What is that value? Illustrate your
answer by means of the graphs of the functions.
9. Is it possible to find a value of x for which the three
functions - %x - 4, f# -f 17 and - 4# - 16 have the same
value ? If so, what is that value ? Illustrate by means of the
graphs of the functions.
10. For what value of a do the three functions
- 3x -f 2, -f %x - 3 and ax + 4
have a common value ? What is that value and what value
of x produces it ? Illustrate by graphs.
B.
11. Draw the graph of tan a from a = to a = 180.
Note. Take a line through the origin for example y = 2#.
Then y = %x -f 8 describes the same line raised through 8
units. A figure will show that the line could have been
transferred to the same position by moving it 4 units to the
left. Thus y = 2 (x + 4) implies that the line y %x has
been shifted 4 units to the left. Similarly y = 2 (x 4) is
equivalent to y = 2# - 8, and implies that the line has been
moved to the right. Since y = 2# 4- 8 can be written as
y - 8 = %x and y = 2# - 8 as y + 8 = %x we have the fol-
lowing rules :
Substitute y - 8 for y and the line is moved up 8 units.
Substitute y + 8 for y and it is moved down 8 units.
Substitute x - 4 for x and the line is moved 4 units to the
right.
Substitute x + 4 for x and it is moved 4 units to the left.
12. What horizontal movements would produce the same
results as the following vertical movements? Write each
relation in the forms corresponding to both kinds of
rnenti :
EXERCISE XXXIX 219
(i) An upward movement of the line?/ 3x through 12 units,
(ii) A downward movement of the Hue y = 2vLc through 7 '2
units,
(iii) An upward movement of the line y = - %x through 5 units,
(iv) A downward movement of the line y = - 3'1 through 24 '8
units.
13. Give the relations wh'ch correspond to the following
lines after they have been moved in the manner specified :
(i) The line y = 4x - 7 moved 12 units upwards,
(ii) The same lino moved 3 units to the left,
(iii) The line y = - 1*4# + 6'2 moved 5 places to the left,
(iv) The same line moved 7 units downwards,
(v) The line 3x - 2y + = moved 4 places to the right,
(vi) The same line moved G places downwards.
14. What movements (a) vertically and (b) horizontally will
effect the changes of position implied by changing the first of
each of the following pairs of relations into the second?
(i) y = 2x + ii into y = 2x - 3.
(ii) y = - sx + 8 into y = - SK - 7.
(iii) 3x - 5y + ii = o into 3 X ~ 57 ~ 34 ^ -
(iv) 2x + 77 - 18 = o into 2x + 77 + 10 = o.
15. The line y = %x can be made to pass through the point
( - 4, + 5) by moving it first 4 units to the left and then
5 units upwards. What will now be the corresponding re-
lation ?
16. Write down the relations corresponding to the follow-
ing lines :
(i) A line through the point ( + 3, - 7) and parallel to
y = l'5x.
(ii) A line through the point ( - 5, - 8) and parallel to
y = - 2-3x.
(iii) A line through the point ( - 2, +6) inclined at 41 to the
x-axis.
(iv) A line through the point ( + 8 - 4 '5) inclined at 122 to the
x-axis.
(v) A line through the point ( - 4, + 3) perpendicular to the
line y = |x. (See No. 5.)
(vi) A line through the point ( -f 3'2, + 1*8) perpendicular to
2x + 5y -7 = 0.
17. The line y w.c when moved so as to pass through the
point (p, q) is described by the form y mx + c. Show that
c = - mp + q.
18. Two lines y = m L x and y = m. 2 x when moved so as to
pass through (p, q) have as their corresponding relations
y = m^x -i- Cj and y = m,x + c.,.
220 ALGEBRA
Show that (m 2 - mj p = c l - c 2 .
19. A third lino y = m^x becomes y = m 3 x -f c 3 when
moved so that it also passes through the point (p t q). Use
the result of No. 18 to prove that when any three lines
y = m^ + c v y = m 2 x + c 2 , y =
pass through the same point
(m. 2 - mJKms - ?n. 2 ) - (c l -
20. Apply the test of No. 19 to determine which of the
following sets of lines is concurrent :
(i) y = i'2x + 8'6, y = - o-2x + 4-4, y = - 3x 4.
(ii) x - 2y - 13 = o, 3x -f 2y - 15 = o, 2x - 3y - 23 = o.
(iii) y = 3 x 7, y = - 2x + 3, y = - Sx + 9.
Confirm your conclusions by drawing the lines.
EXERCISE XL.
DIRECTED TRIGONOMETRICAL RATIOS.
A.
1. Draw the graph of sin a from a = to a = 180.
2. Directly below the former graph and with the same
scales draw the graph of cos a for the same values of a.
3. By comparing the graphs show that sin (a + 90 ') cos a,
a being less than 90. Complete the identity
cos (a + 90) . . . ,
a being less than 90.
4. Find equivalents for sin (a - 90) and cos (a - 90)
when 180 > a > 90.
5. From the results of Nos. 3 and 4 find equivalents for
tan (a + 90), (a < 90), and tan (a - 90), (180 > a > 90).
Do the results agree with the graph of Ex. XXXIX, No. 11 ?
Note. The reciprocal of the tangent of an angle is called
the cotangent of that angle. In symbols,
cot a I/tan a cos u/sin a.
6. Show by a figure that tan (90 - a) = cot a and that
cot (90 - a) = tan a, when a < 90.
7. Find similar equivalences for tan (a - 90) and cot
(a - 90) when 90 < a < 180.
8. Sketch roughly, for comparison, the graphs of tan a and
cot a from a = to a = 180. Find equivalences for tan
(a 4- 90), cot (a + 90). Do the results agree with those of
No. 5?
9. An officer in a battery on an island determines by his
range-finder that two ships are respectively 1200 and 1800
yards distant. He also observes that the angla between them
is 53. How far are they from one another?
10. Half an hour later the officer observes that the ships
are at the same distances as before from his battery, but thac
221
222 ALGEBRA
the angle between them is 113. What is now the distance
between the ships ?
11. The road from a certain village, X, runs practically
straight for 2 miles. It then changes its direction by 32.
The village of Y lies 1/3 miles from the turning. What is
the direct distance from X to Y? What angle does the line
joining the villages make with the road out of X ?
12. The difference in direction between two vectors is 8 and
their lengths are respectively a and b. Show that the length
of their resultant is J(a* -f b 2 -f %ab cos 8). If the resultant
makes an angle of J3 with the vector a, show that
sin fi = b sin BJ ^/(a 2 4- W + %ab cos 8).
B.
13. The sides of a triangle are proportional to the numbers
9, 12, 20 Calculate the angles.
14. Three church spires mark a triangle whose sides are
2 miles, 3 miles, and 4 miles long. Calculate the angle sub-
tended by each pair of spires as seen from the remaining one.
15. The base of a triangle is 120 yards long; and the
angles at the base are respectively 27 and 122. Calculate
the third angle and the lengths of the other two sides.
16. Calculate the area of the triangle of No. 15.
17. The length of the base of a triangle is c and the base
angles a-re respectively a and /3. Show that the area is given
by the formula
c 2 sin a sin (3
A = 2lm~(7+"0)'
IB. Calculate the area of a triangle in which c = 14,
tt = 7r, ft = 43.
19. Show that the altitude of, an isosceles triangle is
\c tan a, and that its area is \G~ tan a, a being the base
angle.
20. The magnitude of the angle-, A in a triangle is a. Show
that
Hence show that
sin <* = ' *^ a + b * c )( a + * ~~ C + c "
EXERCISE XL 22:]
21. Use the foregoing formula to calculate the angles of a
triangle whose sides are respectively 3, 4, and 5 inches long.
22. Show that if 25 is substituted for (a -\- b .+ c) the
equivalence of No. 20 becomes
2
sin a = j- *J{s(s - a)(s - b)(s - c)}.
Hence show that the area of the triangle is given by the
formula
A = J{s(s - a)(s - b)(s - c)}.
23. The sides of a triangle are respectively 60, 40, and 80
yards long. Calculate its area.
24. The sides AB, BC, CD, DA, of an irregular four-sided
field are, in order, 400 feet, 350 feet, 270 feet, and 320 feet
long. The diagonal AC is 520 feet long. Calculate the area
of the field and the size of its angles.
EXEKCISE XL1.
SURVEYING PROBLEMS.
A.
Note. Surveyors in making a map of a district begin by
fixing the relative positions of prominent points (e.g. a flag
on a church towor, a solitary tree on a hill) by means of a
series of triangles. Fig. 40 illustrates such a triangulation.
Fm. 40.
The length of a base AB is determined (on level ground) with
extreme care, and the angles 1, 2, 3, ... 14 are measured
with a theodolite. By the theodolite the angle of elevation
or depression of each station as viewed from the preceding
station is also determined. From these measurements the
lengths of the sides of the triangles and the heights or depths
of each station above or below A can be calculated. Finally
the co-ordinates of the stations B, C, D, etc., are calculated
with reference to the north and south and east and west lines
through A.
224
EXERCISE XLI 225
In No. 1 the calculation is to be divided among the class.
Group I are to calculate AC and BC ; Group II are to assume
AC = b and to calculate CD and AD ; Group III are to as-
sume CD = c and to calculate DE and DF ; and so on with
the other groups. When Group I have calculated AC, Group
II are to substitute its value in their expression for CD ; Group
III are to substitute Group II's result for CD in their expres-
sion for DE, and so on. While a group are waiting for the
result of the previous group they may solve Nos. 2, 3, and 4.
1. The length of the base AB is exactly 1 mile. The
angles are as follows :
Number: 12345678
Angle : 72 83 63 74 48 52 67 51
Number: 9 10 11 12 13 14
Angle: 56 49 76 72 60 55
Calculate the -lengths necessary for a careful drawing of the
triangles by graduated ruler and compasses.
2. The stations C and D are on the crest of a ridge. At
A the angle of elevation of C is 4. Calculate the height of
C above A in feet.
3. The angle of elevation of D from C is 2. Calculate
the height of D above A.
4. The angle of elevation of D from A is 4 42'. Calculate
the height of D above A. Does the calculation agree with
the result of No. 3 ?
5. E is on the eastern side of the ridge CD. The angle
of depression of E from C is 2 36' and from D 4 56'. Calcu-
late from both observations the height of E above A.
6. The angle BAS is 32. Calculate the bearings of (i)
C from A, (ii) D from C, (iii) E from D, . . . (vii) K from H.
Estimate all the bearings in degrees from the north round by
the east, (Thus a line bearing 30 E. of S. is to be given as
bearing 150 from the north.)
Note. In No. 7 the work is to be divided as in No. 1.
Group I are to find the distances of C from the NS and EW
lines through A; Group II the distances of D from the NS
and EW lines through C, etc. ; Group VII the distances of K
from the NS and EW lines through H. The results are to
be expressed in directed numbers.
7. Calculate the co-ordinates of each of the points C, D,
E, G, H, K, with respect to NS and EW lines through A, C,
P, E, G, H respectively. From the results determine the
15
226 ALGEBRA
co-ordinates of all the points with respect to NS and EW lines
through A.
8. Make a map in which the points A, B, . . . K are in-
serted in their correct positions.
B.
Note. An alternative, less elaborate method of surveying
is by making a traverse. The details of the country within
a triangulation are often fixed in this way. The method con-
sists in determining the lengths and the bearings of the lines
leading from each of the stations A, B, C, D, etc. (fig. 41) to
FIG. 41.
the next. The lengths are measured with a chain or a steel
tape, the bearings with a prismatic compass for rapid work
or a theodolite in careful surveying. In the former case the
angle between each of the directions AB, BO, CD, . . . GA
may be measured, and also the bearing of AB. From these
measurements the bearings of the other lines are calculated.
In the second case the angles 1, 2, 3, ... are accurately
determined. The map is made, either by drawing vectors,
or, in more accurate work, by calculating the co-ordinates of
the stations with respect to axes through A. If the lengths
and angles have been coi rectly measured the traverse ought
to close that is, the last line, GA, should make with the
others a closed polygon.
Bearings are always taken continuously from the north
round by the east. Thus a point 10 B. of S. is taken to
have a bearing of 170, a point 10 W. of S. one of 190, a point
40 W. of N. one of 320. In order to calculate easily the
co-ordinates of the points in the traverse every angle from
to 360 is supposed to have its own sine and cosine.
9. Draw a line NOS to represent the meridian. From
EXERCISE XLT
227
draw a, vector of length r with a bearing 220. Draw the
component western or southern vectors or resolved parts OQ,
QP. By consideration of these determine what sine, cosine,
and tangent you will assign to 220.
10. Answer the same questions with reference to a vector
whose bearing is 310.
11. State the equivalences by which you will obtain sin a,
cos a, and tan a from the tables on pp. 107 and 111 (i) when
180 > a > 90, (ii) 270 > a > 180, (iii) 360 > a > 270.
1 ^ The lengths and bearings of the lines in the traverse
shown in fig. 42 are given below. Each member of the class
FIG. 42.
is to calculate the components of two of the vectors. From
the combined results each one is to calculate the co-ordinates
of B, C, . . . G and A with reference to the NS and EW
lines through A.
AB BO CD DE EF FG GA
Distances 250 510 384 608 691 422 710 yards.
Bearings 326 64 156 52 165 254 297
13. How far is the traverse from being perfectly closed?
[Calculate the co-ordinates of A from GA.]
14. In a traverse connecting seven stations, A to G, angles
corresponding to the angles 1, 2, 3, ... 7 of fig. 41 are
measured with a theodolite and are as follows :
1
47
2
68
3
289
4
89
15*
5
242
6
233
7
302
228 ALGEBRA
Calculate the difference of bearing between AG and each of
the lines AB, BC, . . . FG. (That is, if G were due north
of A, what would be the bearings of the lines AB, BG, etc. ?)
15. The lengths of the lines in the traverse of No. 14 are
as follows. Calculate the co-ordinates of the points B, C, D,
E, F, G, with reference to axes through A respectively parallel
and perpendicular to AG.
AB BC CD DE KP FG
362 389 470 409 212 180 yards.
16. Sketch roughly the graphs of sin a, cos a, and tan a from
a = to a = 360. Do the identities of Ex. XL, Nos. 3
and 8 hold good for all angles between and 270 ? Do
those of Nos. 4 and 7 hold good for all angles between 90
and 360 ?
0.
Note. Except in surveying, an angle is reckoned positive
if measured in the anti-clockwise, and negative if measured in
the clockwise, direction.
17. Take a line of length OP making - 48 with the x-axis.
From P draw PM, PN perpendicular to OX and OY. What
values must be assigned to cos (- 48) and sin (- 48)
so that the rules PM = / cos a and PN = I sin a may be
observed ?
18. Determine by the same principle the sines and cosines
of : (i) - 100, (ii) - 200, (iii) - 300.
19. Find the tangents of : (i) - 23, (ii) - 135, (iii) - 250,
(iv) - 340.
20. Make a table showing the signs of (i) positive, and (ii)
negative angles in each of the four quadrants.
EXEKCISE XLIL
HYPEBBOLIC AND PARABOLIC FUNCTIONS.
A.
1. Draw on tracing paper, in accordance with instructions,
the rectangular hyperbola xy = k or y k/x, including its
asymptotes. Either choose your own value for k (selecting
some positive number) or work with the one assigned to you.
In Nos. 2 and 3 you are supposed to start with the asymptotes
of your curve coincident with axes of x and y drawn on a
sheet of squared paper lying beneath the tracing paper.
2. Carry out the following movements with your curve and
give in each case the formula which describes it in its new
position :
Move the curve y = k/x (i) 6 units upwards ; (ii) 14 units
to the left ; (iii) 17 units to the left ; (iv) 15 units downwards ;
(v) 23 units downwards and 10 units to the right ; (vi) 16
units to the left and 9 units upwards ; (vii) so that the cross-
ing-point of the asymptotes is at the point (- 8, + 12) ; (viii)
so that it is at the point ( + 18, - 15).
3. Move your curve successively into the positions corre-
sponding to the following relations. Describe the movements
in words :
(i) y = k/(x - 9). (ii) y - 8 = k/x.
(iii) y+ ii = k/(x - 13). (iv) y = k/(x -f 16) + 9.
4. Place your tracing paper upon the squared paper so
that the curve on it corresponds successively to the -following
relations :
(i) y = - k/x. (ii) y + 13 = - k/x.
(iii) y = 22 - k/(x + 7). (iv) y - 22 + k/(x + 7).
(v) y + ii = - k/(x - 23). (vi) y + n = k/(x - 23).
5. Throw each of the following equations into the form :
k
y + b =
y " x a
229
230 ALGEBRA
(i) xy + 3x - 12 = o. (ii) xy + 37 ~ 12 = o.
(iii) xy - 5x + 7 = o. (iv) 3xy + 4y + 15 = o.
(v) xy + 3x + 4y + 6 = O. (vi) xy + 3x + 5y -o.
(vii) 2xy + 7x -i- 3y - 8 = o. (viii) 3xy - 2x - 6y = o.
6. In the case of each of the curves of No. 5 state : (a)
what two rectangular movements would bring its asymptotes
into coincidence with the axes of x and y ; (b) what relation
would correspond to the curve in this position.
7. Find the positions of the two vertices of No. 5, (i), (iv),
(viii). [Find their positions when the asymptotes are co-
incident with the axes of x and y. Then suppose the curve
to be restored to the position in which it was given.]
8. Find the hyperbolic relations which are satisfied by the
following sets of values of x and y :
(i) (+ 8, + 9), (-& - 4) (+ I0 > + 6 )-
(ii) ( + 3, - 43), ( + 22, - 5), ( - 6, + 2).
(iii) (o, + 12), (- 5, + 14), ( - 20, + 8).
Illustrate your solution by graphs.
Note. Assume y + b = k/(x + a) and determine the values
of the constants. To draw the graphs easily, first fix the
position of the asymptotes ; then regard them as if they were
axes of x and y and plot the curve y = k/x.
B.
9. Draw on tracing paper as in No. 1 the parabola
y = kx 2
k being a positive number chosen by you or assigned to you.
Let your drawing include the axis and the tangent at the
vertex.
10. Move your curve successively from the position in
which its axis coincides with the ?/-axis and the tangent at the
vertex coincides with the #-axis into the positions which
correspond to the following relations. State in each case
whether the parabola is "head up" or "head down " and
give the co-ordinates of the vertex :
(i) y - kx 2 + 7. (ii) y = - kx 2 .
(iii) y = - kx 2 -h 7. (iv) y + 12 = kx 2 .
(v) y - 20 = - kx 2 . (vi) y = k(x - I4) 2 .
(vii) y = = k(x - I4) 2 . (viii) y = k(x + i6) 2 - 21.
(ix) y + 21 = - k(x - 13)*. (x) y - 15-5 = - k(x +7'8) 2 .
11. Give the relations corrresponding to your parabola when
it is held in the following positions :
EXERCISE XLII 231
(i) Head down with the vertex at'( 6, + 4).
(ii) Head up with the vertex at (- 9, - 11).
(iii) Head up with the vertex at ( + 14, - 7).
(iv) Head down with the vertex at (0, - 17).
12. In the case of each of the following functions of x state :
(a) whether it has an upper or a lower turning value ; (b) what
that value is ; (c) what value of x gives it the turning
value :
(i) y = 6(x - 7) 2 + 4- (ii) 7 = --3(* + 5) <J + 7-
(iii) y = - 3'6(x - 7'2; 3 - i'8. (iv) y - - 7x 2 - 9-3.
(v) y - 4(x + 2- 3 ) 2 . (vi) y = (x + 7'8) 2 - 21-3.
13. Express each of the following relations in the form
y =s + a(x b) 2 c.
Describe, as in No. 10, the position of each of the corre-
sponding parabolas :
(i) y = x 2 - 6x + 3. (ii) y = x' J + lox - 2.
(iii) y = - x 2 + I2x + 7. (iv) y = 3x 2 - I2X + 5.
(v) y = - 7x 2 + 28x - ii. (vi) y = x 2 + 5x - i.
(vii) y = - 2x 3 + I4x + 3. (viii) y = 6x 2 - i4x.
(ix) y - - 4x 2 + I3x. (x) y = 2'3x' 2 + ii'5x - 7-2.
14. Answer the questions of No. 12 with regard to each of
the functions of x in No. 13.
Note. Consider the relation y 4or &x 5. It can
be transformed as follows :
y = 4# 2 - 8# - 5
- 4(x 2 - 20 + 1) - 9
= 4 (a; - I) 2 - 3 a
= (Z(x - 1) +3} {2(x - 1) - 3}
- (2z + 1) (2.r - 5)
Similarly y - 3x 2 - 3Qx + 48
- 3(# J - Wx + 16)
= 3{(x 2 - lOx + 25) - 9}
- 3{(x - 5) 2 - 3 2 }
= 3(x - 8) (x - 2)
This process is described by saying that the parabolic (or
quadratic) function 4# 2 - Sx - 5 has been expressed as the
product of two linear functions of x, namely
2# + 1 and 2# - 5.
15. Where possible express each of the following parabolic
functions of x as a product of two linear functions. How
could you tell from the graphs of the functions in which
232 ALGEBRA
cases the transformation is possible and impossible respect-
ively ?
(i) y = x 2 + 6x ~ 7. (ii) y = x 2 + 6x + n.
(iii) y - 4x 2 - i6x - 34. (iv) y 4x 2 - i6x 4- 18.
(v) y = 3x 2 + 36x + 27. (vi) y = - 3x 2 + s6x - 27.
(vii) y = 5x 2 + I7x - 12. (viit) y = 2x 2 - 6"4x - 7*6.
16. How could you tell from the graphs of the functions
of No. 15 (a) which of them is capable of having the value
; (b) what values of x make the value of the function ?
Note. The easiest way to calculate where the parabola
y = 4:X 2 - &x - 5 crosses the #-axis is to express the quadra-
tic function 4#' 2 Qx 5 as a product of two linear functions.
We then have :
4z 2 - Sx - 5 - (2z 4- 1) O - 5) =
Therefore either 2x + I = or 2x - 5 = ;
that is, either x = - 1/2 or x = -f 5/2.
17. Find the values of x for which (where it is possible)
the various functions of No. 15 have zero value.
18. The vertex of a parabola is at tho point (+ 4, -f 7) and
passes through the point ( - 5, - 20). Find the formula.
19. Another parabola has its vertex at the same point and
passes through (+13, + 34). Find the corresponding re-
lation.
20. Indicate by a sketch the positions of the parabolas of
Nos. 18 and 19.
21. When x = - 6 a certain parabolic function of x has
a turning value of + 72. When x = its value is zero.
Express the function in the form px 2 + qx + r. What kind
of turning value has it ?
22. Find the parabola which has the line x = - 10 as its
axis, passes through the point ( + 4, -f 40 , and crosses the
#-axis at x = - 4. Where is its vertex?
23. The values of a parabolic function of x are the same
for all values of the variable which are at an equal distance
above and below + 6. Its value is zero when x = - 1 and
26 when x = + 12. Find the function and give its turn-
ing value.
C.
Note. The best way to determine whether a given curve
is a parabola is to measure a number of ordi nates, equidistant,
EXERCISE XLII
233
but otherwise taken at random. As you have seen, if the
curve is parabolic the second differences of the ordinates will
be constant.
When you know that the curve is a parabola the easiest
way to find its formula is to note the co-ordinates of its
+30
-5
+5
+ 75 +20 25 ' +30
+W
FIG. 43.
vertex and of some other point for example where it crosses
the z-axis or the */-axis. The method of Nos. 18, 19 should
then be applied. If the position of the vertex is not shown
find the co-ordinates of any three convenient points and
use the values of x and y to determine the constants in
y = px 2 + qx + r.
234 ALGEBRA
24. Determine whether the curve A in fig. 43 is a para-
bola. If it is find its formula.
25. Bepeat with curve B.
26. Repeat with curve C.
27. Determine whether curve D is a portion of a parabola.
If so find the formula and state the position of the vertex.
28. Eoll an oiled ball diagonally up a sloping drawing-
board as instructed. Show that the trace is a parabola.
Find its formula, taking a vertical and a horizontal edge of
your paper as axes. What is the connexion between the
vertical and horizontal distances of a projectile from the point
of projection ?
29. Arrange that a fine jet of water may be projected hori-
zontally with constant pressure and may fall into a sink some
distance away from, and below, the nozzle of the jet-tube.
Find whether the middle of the stream is a parabola.
30. Repeat, inclining the jet-tube upwards. Deal separately
with the inmost and outmost portions of the stream. If
they are parabolas calculate the positions of their turning-
points. Also determine by calculation whether they pass
through the mouth of the nozzle.
31. The following table gives the values of a certain
function for given values of x. Find whether the function
is parabolic. If so determine its precise form and its turning
value :
a : - 4 -3 -2 - 1 + 1 + 2 +3 +4
y: + 9-8 +7-9 + 6'2 +47 + 3'4 + 2-3 + 1-4 +07 + 0'2
32. Repeat the investigation upon the following data :
cc:-4 -3 -2 -1 +1 +2 +3 +4
y:+13 +0 +2 +0 -1 -2 -4 -8 -15
BXEECISE XLIII.
QUADRATIC EQUATIONS.
A.
1. Calculate to two decimal places the abscissae of the points
on the following parabolas where the ordinates have the
specified values:
(i) The parabola y = s(x - 2) 2 - 5 where y = + 7, y = - 2,
y = o.
(ii) The parabola y = - 4(x + 3)* + 20 where y = - 80,
y = -f 12, J = 20.
(iii) The parabola y = 2x 2 - 7x - 3 where y = + i^, y = - Sic-
(iv) The parabola y = 3 - 7x - 7x 2 where y = o, y = +3.
2. Find the values of x for which
(i) The function x 2 - 3x - 10 has the value + 375.
(ii) The function x 2 + o'px + 10 has the value - 37.
(iii) The function gx 2 + 3x - 8 has the value - 6.
(iv) The function 8 '2 - 5'ix - 3x 2 has the value 4- 10.
3. Find by direct factorization the roots of the following
quadratic equations :
(i) x 2 - 4x 4- 3 = o. (ii) x 2 + 4x - 5 = o.
(iii) x 2 4- I2x 4- 36 = o. (iv) x 2 - i3x + 40 = o.
(v) x 2 - 3x - 40 = o. (vi) x ? - i5x -f 36 = o.
(vii) x 2 - 7x = o. ( y iii) x2 + 5'3 X o.
(ix) 2x 2 - 7x + 3 - o. (x) 6x- - I3x +6=0.
(xi) 6x 2 - I3x +6 = 0. (xii) 8x 2 + 26x -7 = 0.
(xiii) i - 8x + iSx 2 = o. (xiv) 2x a + 7x 4- 6 = o.
(xv) 6x 2 - I3x - 5 = 0. (xvi) 4x 2 - 2ox -f 25 = o.
Note. How can you form a quadratic equation whose
roots shall be + 2 and + 3 ? - 2 and - 3 ? - 2 and + 3 ?
a and fi ? What is the relation between the roots and (i)
the coefficient of x, (ii) the constant term in the equation ?
4. Quadratic equations are to be formed having the follow-
ing pairs of roots. Set down in four parallel columns the
roots, the coefficients of x in the equations, the constant terms,
the completed equations :
235
236 ALGEBRA
(i) - 3, + 5- (") + 7 ~ 8.
(iii) + 4, + 3. (iv) - 10, - 10.
(v) - 10, + 10. (vi) + 5-2, - 2-5.
(vii) + 0-3, + 2-1. (viii) + 3 5-
(ix) + 2-3 1-2. (x) + 5 + v/3, + 5 - v>3.
(xi) - i x/2. (xii) + 1-3 v/5'4.
(xiii) a b. ( x iv) pa 2 qb 2 .
(xv) + v/2 v 3- (xvi) - v/5 N/7-
5. The roots of a quadratic equation are of the form
a *J h- Show that the coefficient of x and the constant
term are both rational. Is this the case if the roots are of
the form Ja b ?
6. The roots of a quadratic equation are of the form
^/a + ,Jb. Show that the constant term is rational but that
the coefficient of x is not. Is this the case if the roots are
of the form Ja b ?
7. State the sum of the roots and the pioduct of the roots
of each of the following quadratic equations :
(i) 3x 2 + sx - i = o. (ii) 2x 2 - 7^x + 47 = o.
(iii) 3'2x 2 - 8'4x - 5 = 0. (iv) px 2 + qx + v = o.
(v) (a + b)x 2 - (a - b)x - ab - o.
8. Write down, in a form clear of fractions, the quadratic
equation whose roots are :
(i) -*, + * (ii) + f , + f.
/ \ /\
(iv) - -, - B . (v) + p , + ? .
9. Use the results of No. 6 to solve the following
equations :
(i) x2 - 2N/3X + 2 = 0. (ii) x 2 + 2^sx - n = o.
(iii) x 2 - 2 v /i3x +5-0. (iv) x 2 + 2^15 -9 = 0.
(v) x 2 - 2>/ax + b = o. (vi) x 2 + 2>/(a + b)x + (a - b) = o.
(vii) x 2 - x/7x - j == o. (viii) x 2 - ^I4x +3 = 0.
10. If you are given one root of a quadratic equation how
can you calculate the other (i) from the coefficient of x, (ii)
from the constant term ? Use one of these methods alter-
nately to find the sacond roots in the following instances.
Use the other method to check the result in each case.
(i) x 2 +
(ii) x 2 -
77x + 14*4 = o ; given root, - 3*2.
- 1*52 = o ; given root, - 0*2.
(iii) x 2 + 3'4x - 46*11 = o ; given root, + 5-3.
(iv) x 2 - 77x + 14 62 = o ; given root, + 3*4.
(v) 33x a - yx - xo = o ; given root, + .
(vi) 63x 2 + 22x - 21 = o ; given root, + f.
EXERCISE XLIII 237
B.
11. Find which of the following parabolas crosses tho
ic-axis :
(i) y = x 2 - 4 x - 3. (ii) y = x 2 - 4 x + 5.
(iii) y = 3x 2 + 4x + 2. (iv) y = - 2x 2 + 3x + 7.
( v ) y = - 2x 2 - 3x - 5. (vi) y = 3x 2 - I2x + 12.
12. Describe in symbols the steps by which the problems
of No. 11 are solved. Use p, q, and r as symbols for the
coefficient of x 2 , the coefficient of x and the constant term re-
spectively. Hence show that (whether the parabola is " head
down " or " head up") it will cross the #-axis provided that
q 2 Apr is positive, and that it will touch the #-axis if
q 2 = 4pr.
13. Write down the condition (i) that the function
ax 2 + bx + c
may be capable of having zero value ; (ii) that it may be
represented as the product of two linear functions ; (iii) that
it may be a perfect square ; (iv) that the quadratic equation
ax 2 + bx + c =
may have roots ; (v) that the roots may be identical.
14. Apply the tests of No. 13 to classify the following
equations into those which have (a) identical roots, (b) unequal
roots, (c) no roots :
(i) 4x 2 - i2x + 9 = 0. (ii) 3x 2 - yx + 4 = o.
(iii) x 2 + px + p 2 = o. (iv) 5x 2 -f 6x - 8 = o.
(v) O'i6x a - o'8x +1 = 0. (vi) 3x 2 - 7x + 4 = o.
15. Determine which of the following equations have roots.
Determine the roots, where they exist, by expressing the left-
hand side of the equation as the difference between two squares
and then factorizing it. Leave the roots, if they are not
rational, in the form of a pair of conjugate surds.
(i) x 2 - nx + 12 = o. (ii) x 2 - nx - 12 = o.
(iii) x 2 + 4x - 7 = o. (iv) 2x a - 3 X + 5 = o.
(v) 4x 2 + 4x - 5 = o. (vi) 2x 2 - 5x_+ 32 = o.
(vii) 2x 2 + I3x -3 = 0. (viii) x 2 - 2\/nx + 20 = o.
(ix) x 2 - x/7x + x/15. (x) V3x 2 - x/i8x - I = o.
16. The roots of a quadratic equation are a and ft. Write
down the equation whose roots are ma and m/3
17. Give the equations whose roots are respectively :
238 ALGEBRA
(i) Four times those of the equation x 2 ~ 2x - 3 = 0.
(ii) Ten times those of the equation x 2 - i'3x - 7-14 -- o.
(iii) One-fifth of those of the equation x 2 + 5x - 150 = o.
Verify the transformation in each case.
18. Use the result of No. 16 to derive from each of the
following equations another in which (a) the coefficient of x 2
is unity, (b) the coefficient of x and the constant term are
both whole numbers. Find the roots of the transformed
equations and deduce from them the roots of the original
equations. Verify your results :
(i) 7x a + 4x - 3 = o. (ii) I4x 2 - IQX -f 6 = o.
(iii) I7x 2 - i ix - 6 = o. (iv) ipx 2 + px - 10 -= o.
19. Show how to calculate the values of a variable x denned
by relations of the following types :
(i) ax 2 - (a - b)x - b = o.
(ii) x 2 - p(x - q) + q 2 = 0.
(m) pqx 2 + (p - q)x = i.
(iv) abx 2 + (a 2 + b 2 )x + ab = o.
(v) 2(a - b)(x 2 + b) = (a + b) 2 x.
(vi) x a - 2ax + a 2 - b 2 .
(vii) (a - b) 2 x 2 - 2(a 2 - b' 2 )x + (a + b) 2 = b 2 .
(viii) (a - b) 2 x 2 + (a 3 - b :$ )x + ab(a 2 + b 2 ) = o.
20. Find, where possible, values of x for which the follow-
ing pairs of functions have the same value :
* (i) $x 2 - sx + 4*2 and x 2 + 5x - 5-8.
(ii) x 2 + 2x + 7 and 4x - 3.
(iii) 2x 2 - 3x + 10 and - 2x + 7.
(iv) x 2 + 2x + 7 and 2x 2 - 3x 4- 10.
(v) x 2 + 2x + 7 and ^x 2 + x + 3.
(vi) x 2 + 2x + 7 and - ^x 2 - x - 3,
(vii) 2x 2 - 3x + 10 and x 2 + x + 3.
(viii) 2x 2 - 3x + 10 and - x 2 - x - 3.
Explain the result in (i) by a sketch. Verify the results
in (ii)-(viii) by drawing graphs of the functions all upon the
same paper.
EXEECISE XLIV.
FURTHER EQUATIONS.
A.
1. Write the equation whose roots are 2o. times those of
ax 2 + bx + c = 0. Hence show that the roots of the latter
are { - b J(W - 4ac)}/2a. Does this result agree with
those of Ex. XLIII, No. 12 ?
Note. The roots of a quadratic equation may be calculated
directly by means of the formula of No. 1.
2. Which of the following equivalences is possible and
which impossible ? Determine in the former cases the values
of x which satisfy the relation.
(i) x + - = 2x + . (ii) x -\ 5- = 6.
(iii)x + ^ 1 = 4 . (iv) 2 (x- l) + - r L^=i.
' - 'n)I- '
x/ X-2 X+2 X/ X X-fl X + 2
(vii) 5 ? = _JL_. ( V iii) L __ = __3_.
V/ X X+I X+2 X ' X X + I X + 2
3. Explain the results of No. 2 (ii) and (iii), by drawing
the graph of y = x + 3/(# - 1) from x = to x = +6.
What turning value has the function x + S/(x - 1) in this
region ?
4. Eeplace the improper fractions in the following rela-
tions by their equivalent integral expressions and comple-
mentary fractions. Simplify the reduced relations and find
the values of x (if there are any) which satisfy them.
x.v 2X - I 4X
(i) 5 = o.
V ' X 2X + I
(ii) 2X ~ 3 _ L_7 37
X X + I 20
239
240 ALGEBRA
(iii) *-^-L3 - 6 = 2X * ~ 3* 1
X ' X 2X + I
(iv) I?LJi - 2X _ 2X + 5
4x -h 3 2x - i x + 2 '
5. Explain the result of No. 4 (i) by plotting the graph
of y = 2/(2# + 1) - 1/rc from #= - 4 to a? 4-4.
B.
6. Form equations of the lowest possible degree with the
following sets of roots :
(i) - i, + i, + 2. (ii) + i, + 2, + 3.
(iii) - 2, o, + 2. (iv) - 4, - 2, + 3.
(V) - 2, - I, + I, + 2. (vi) - 3, - 2, O, + I.
(vii) - 3, - i, + 2, + 4. (viii) - 3, o, o.
(ix) - 6, o, o, + 5. (x) - 2, - i, o, o, + 3.
Verify your result in (ii) and (v).
7. Find the roots of the following equations by factoriza-
tion :
(i) x 4 - i3x 2 + 36 = o.
(ii) x - 4 x/x + 3 = o.
(iii) x 2 - I7x + 16 = o.
(iv) x + 12 Jx + 35 = o.
(v) x ;J - 3x a - x + 3 = o.
(vi) x 3 + 2x 2 - 4x - 8 = o.
(vii) x 3 - 7x a + I2x = o.
(viii) x 4 4- 3x :J - 4ox 2 = o.
(ix) x 4 - 6x 3 + I3x 2 - I2x + 4 = 0.
(x) x 3 - 6x 2 + I2x - 8 = o.
(xi) x 4 - (sx 4- 6) 2 - o.
(xii) x 4 4- x j - x - 1=0.
(xiii) (x 2 - 9) 2 - 4x 2 + 36 = o.
(xiv) (x 2 - 3x - 8) 2 + 2(x a - 3x - 8) - 8 = o.
(xv) 2(x 2 ~ x - 2) 2 + s(x a - x - 2)(x + i) ~ 3(x + i) 2 = o.
8. One root of each of the following equations is given.
Find the others :
(i) x 3 - 8x 2 + i ix + 20 =o; given root, + 4.
(ii) x 8 + 7x 2 - 4x - 28 = o ; given root, + 2.
(iii) 4x** - 31x4- 15 = o; given root, + |.
(iv) 9x 8 - 73x + 24 = o ; given root, + |.
9. The roots of a cubic equation are (i) -f 8, - 5, - 3 ;
(ii) - 6, + 2, + 4 ; (iii) 0, - 3, -f- 3. Determine by inspec-
tion the values of the coefficient of x 1 and the constant term
in each case.
EXERCISE XLIV 241
10. A cubic equation has the form # 8 - px + q = 0. Show
that p = a 2 + aft + 13" and q = a/3(a + f3) where a and ft
are any two of its three roots. Test the conclusion by apply-
ing it to either No. 9 (i) or No. 9 (ii) and to either No. 8
(iii) or No 8 (iv).
11. The graph of the function
y = x* + Sx 2 - 4x - 12
is moved one unit to the right. Show that the formula cor-
responding to the new graph contains no term involving x 2 .
12. State the relation between the roots of the equations
a 3 + 3x 2 - 4x - 12 =
and x* - Ix - 6 - 0.
13. Draw on the same sheet of squared paper the graphs
of y x 3 and of y = Ix + 6. Deduce the values of the
roots of the equation
#3 + 32.2 _ 4^ __ J2 - 0.
14. Find how the graph of the function
y = x* - l-5a? 2 - 2-5# + 3
must be moved in order that it may correspond to a formula
of the type y = # 3 - px + q.
Use your result to find the roots of the cubic equation.
x*> - l'6x 2 - 2-5^ + 3 =
by the graphic method of No. 33. [Use the same graph of
y = *".]
15. Find by means of the graph of y x 3 the roots of the
following cubic equations :
(i) x' 5 + 3'sx 2 - yx - 20 = o.
(ii) x 3 - x 2 - ( j'72x + 0-576 = o.
16
EXERCISE XLV.
INVERSE PARABOLIC FUNCTIONS (I).
A.
1. A marble is rolled several times up a sloping board 2
metres long. The slope of the board is altered for each ex-
periment, and in the several experiments the marble starts
with a different velocity and from a different point of the
board. From the following data obtain formulae for the dis-
tance (S) of the marble from the upper end of the board t
seconds after projection. Determine in each case the highest
point reached by the marble and the moment when it reaches
that point. The velocity is measured in centimetres per
second.
(i) Starting-point, 40 cms. from lower end ; v = + 20 - 2t.
(ii) Starting-point, the middle ; v == 21 - Jtt.
(Hi) Starting-point, 8 cms. from lower end ; v = + 4.8 - 6.
(iv) Starting-point, 50 cms. from lower end ; v 40 - 5t.
What is the interpretation of the last result ?
How could the moments of turning be foretold from the
formulae for the velocity ?
2. Change the subject to t in each of the formulae for S
in No. 1. Use the formulae to determine in each case the
moments (i) when the marble crosses the middle line of the
board; (ii) when the marble falls over the lower or upper
end of the board.
How can the highest points reached by the marble be
deduced from these formulae?
3. The length of a cricket pitch is 22 yards. The path
followed by the centre of a ball from a certain bowler's hand
is described by the formula :
fc= 71-68 + 0-lQd - 0'02d 2
h being the height of the ball above the ground in inches,
242
EXERCISE XLV 243
d the horizontal distance in foot from the " howling crease "
(the line in which the wickets at the bowler's end are in-
serted). The vertical plane which contains the path of the ball
also passes through the middle wicket at each end. Calcu-
late the greatest height of the centre of the ball, its height
when it leaves the bowler's hand immediately above the
bowling crease, and its height when it reaches the plane of
the opposite wickets. Explain the meaning of the first and
third results. Sketch the path of the ball.
4. Obtain a formula for the distance of the ball from the
bowling crease when its centre is at a given height. Use it to
find where the ball (which has a diameter of 2 f 9 inches) would
hit the ground if allowed to do so. The batsman hits it when
its centre is 1 foot from the ground ; how far is it then from
his wicket?
5. When struck by a batsman a cricket ball follows the
path indicated by the formula :
h = 0-597 + O594d - O-OOStf 2
h and d being both measured in feet. Find the greatest
height reached by the ball.
6. Change the subject of the last formula to d. The bats-
man is caught out by a fielder who catches the ball when it is
6 feet from the ground. Whore may he have been ? What
would have been the horizontal range of the ball if it had not
been caugb t ?
7. The first term of an A. P. is 4- 29 and the common
difference - 2. Write a formula for S, the sum of n terms.
For what value of n is the sum highest? Why does it de-
crease when more terms are added?
8. Write a formula for the number of terms of the foregoing
series required to yield a given sum. For what numbers of
terms is the sum (i) 200, (ii) - 64 ? Explain the double re-
sult in each case.
9. Write a formula for the number of terms of the series
+ 16 + 14 + 12 + . . .
required to yield a given sum. Apply it to determine n when
S = + 72. Explain the result.
10. Show by arguments based upon the formulae both for
S and for n that the sum of the series in No. 9 cannot be
higher than 72 \. What is actually its highest value? Why
is it not actually 72 J?
11. Find in two ways a number above which the sum of
16*
244 ALGEBRA
the following series cannot rise. What is actually the highest
value of the sum and what is the number of terms which
gives it? Explain the result.
+ 19 + 16 + 13 + . . .
12. Find how many terms of the foregoing series will
yield (i) + 30, (ii) - 22. Are all your results valid? If riot,
which must be rejected and for what reasons ?
13. Find in the easiest way you know the number of terms
for which the sum of the series
- 81 - 77 - 73 - ...
reaches its lowest value. What is that value ?
14. Write a formula for the number of terms of the pre-
ceding series which must be taken in order to yield a given
sum. Find whether the following numbers are possible
values of the sum, and, where they are possible, find the cor-
responding number of terms. Explain why some of the re-
sults are impossible.
(i) 8= - 231, (ii) S= - 690, (iii) S= + 42, (iv) S - + 174.
15. Change the subject of the formula
S = %n {2a + (n - l)d} to n.
Show that S > - (2a - d) 2 /Sd.
B.
16. Write down relations expressing functions of x which
are respectively inverse to the functions given in the following
relations :
(i) y = 3X - 2. (ii) y = 2-8 - 07*.
(iii) 7x - 47 + 8 = 0. (iv) 3x + I2y -5 = 0.
(v) y = ~f-- (vi) y = ^ + 7.
(vii) (2x - s)(ay + 4) = i. (viii) 8xy + 2x - 207 - 5 = o.
(ix) y = 3x 3 -f 7x - i. (x) y = - 2x 2 + x ~ 8.
17. Find the inverse of the linear function ax + b. Show
that the graphs of a linear function and its inverse will always
intersect on the line through the origin which bisects the
angle between the axes. Verify by drawing on one sheet the
graphs of No. 16 (i), (ii), and (iii) and their inverse functions.
18. Show that the property described in No. 17 holds good
also between any hyperbolic function a/(x -f b) + c and its
inverse. Verify by drawing on the same sheet the graphs of
No. 16, (v) and (vi) and their inverse functions.
EXERCISE XLV 245
19. What (if any) are tlte limits to the possible values of
(a) the variable, (b) the function in No. 16 (ix) and (x)?
Answer the same questions with reference to the correspond-
ing inverse functions.
20. Find the inverse of each of the following functions :
(i) 2x 2 - sx + 4. (ii) 7 + ax - sx 2 .
(iii) ax 2 + bx + c. (iv) 2x/(3x - 4).
(v) ax/(bx + c). (vi) 3* a /(2x - i).
(vii) (2x 2 - sx + 5) - l . (viii) x(x - i) - 2 .
(ix) V(2x + 3). W \/x + V(x - 2).
21. Show that the product 3x fix - 5) is positive if x is
above + 2-5 or below zero, but negative if + 2*5>aj>0.
Show also that the product (3x + 2)(2# - 5) is positive only
if x is above + 2'5 or below - f .
22. What is the range of possible values of the variable in
each of the following functions ? Are there any limits to the
value of the function ?
(i) N/x a - sx. (ii)
(iii) x /(2x - i)(x + 3). (iv) N /(2x 2 + nx -
(v) x /{(x + 7)/(x - 3)}- (vi) x/{(x' J - 4)/(8 -
23. Show that the function of No. 20 (vi) has no values
between and + 3. Show that the function assumes these
values when x = and x = + 1 respectively.
24. Determine the sign of the function 3x 2 /(%x - 1) when
x is a very little below and again when it is a very little above
zero. What peculiarity of the value when x == is implied
by your results? Examine in a similar way the sign of the
function for values of x a little below and above -f 1. What
do you conclude about the value when x = + 1 ?
25. Sketch the graph of the function 3# 2 /(2# - 1) from
x = -lto#= +3. (Pay careful attention to the values of
the function a little below and above + 0*5.)
26. Show that the function in No. 20 (viii) has a turning
value of - 0*25. Is this a higher or a lower turning value ?
What is the corresponding value of x ? What statements, cor-
responding to these, may be made about the inverse function ?
Sketch the graph of the function from x = - 1 to a? = + 2
and the corresponding part of the inverse function.
27. What are the turning values of the functions inverse to
No. 22 (iii) and (iv) ? Sketch the two functions and the cor-
responding inverse functions.
246 ALGEBRA
28. Find the function inverse'to No. 22 (v). Sketch the
graph of the direct and inverse functions. Has either of them
turning values?
29. Find the inverse of the function in No. 22 (vi). Sketch
the graph of the direct and inverse functions.
30. Find the turning value of the function in No. 20 (vii)
Sketch the graph of the function and of its inverse.
EXEECISE XLV1.
INVERSE PARABOLIC FUNCTIONS (II).
A.
1. Show algebraically that the function x 2 - 2cx cos a + c 2
can be expressed in the form (x - c cos a) 2 + c 2 sin 2 a. Hence
show (i) that it is always positive and (ii) has a lower turning
value. (It is to be assumed that a is constant as well as c.)
2. Illustrate the results of No. 1 from the geometry of
the triangle, distinguishing between cases in which a is (i) < 90,
(ii) > 90.
3. Use the identity of No. 1 to change the subject of the
formula a 2 = b 2 + c 2 - %bc cos a to b. Is there any limit to
the values of a which will yield values of b ? How many
values of b correspond to a given value of a? In what case
is there only one value of b (or two equal values) ?
4. Two straight roads diverge at an angle of 42. A
treasure is hidden in the hedge of one road and is known to
be exactly 100 yards from a tree which grows in the hedge of
the other. The tree is 120 yards from the corner where the
two hedges meet. Calculate the distances from this corner at
which the treasure may be found and show the positions in a
diagram.
5. One dark night a ship foundered in the English Channel.
A coastguard on a cliff, noting the interval between seeing the
Hash of a gun fired on board and hearing the report, esti-
mated that it was 1 mile from him. At the coastguard
station 1^ miles east of this cliff the flash of the gun was ob-
served to bear 53 W. of S. but the report was inaudible.
Draw a diagram showing the two places in which the ship
may have sunk and calculate their distances from the coast-
guard station.
Note. Nos. 1-5 show that if two sides of a triangle are
247
248 ALGEBRA
given and the angle opposite to one of them, the length of the
third side cannot always be calculated with certainty. The
case is ambiguous.
6. Show from a consideration of the result of No. 3 that
if A is obtuse b has only one possible value.
7. Given a = 92, c = 120, A = 36 calculate the possible
values of b and of the angles B and C.
B.
8. In the function *J(b 2 + c 2 %bc cos a), b and c are
constant and non-directed but a varies from to 360. Show
that the value of the function ranges between b - c and b + c.
Illustrate by the following figure : Draw b (assumed to be
greater than c) from O along the o>axis. With O as centre
draw a circle of radius c. Let a be the angle between b and
c in its various positions. What lines give the various values
of the function ?
9. Draw a circle of radius r with centre C. Take any
point O outside the circle and let OC = d. Draw from a
straight line OP cutting the circle at P and making an angle
a with OC. Show that Z, the length of OP, is given by the
quadratic equation
V - 2ld cos a + d' 2 - r 2 = 0.
What is the product of the two values of OP which correspond
to a given value of a ? How does the equation show that
their product is equal (for all possible values of a) to the
square of the tangent to the circle from O ? flow does the
equation show that sin a cannot be greater than r/d ?
10. Take the point within the circle. Show that the
product of the two values of OP (corresponding to a single
value cf a) is now negative. What is the meaning of this
result ? What geometrical truth does the equation demon-
strate ?
11. In a straight line of length a a point is taken x from
one end. What is the product of the two segments of the
line? Show that it has its greatest value when the line is
bisected.
12. Find the sum of the squares of the two segments of the
line in No. 11. Show that it can be expressed in the form
EXERCISE XLVT 240
Hence show that the sum of the squares is least when the
line is bisected.
13. The sine of an angle is -f s . Show algebraically that
two cosines and two tangents correspond to this value. Ex-
hibit in a figure the possible values of the angle.
14. The tangent of an angle is 16/63. Find the corre-
sponding values of the sine and cosine and exhibit in a figure
the possible values of the angle.
15. The numbers - 12, + 16, and - 15 are said to be pro-
portional respectively to the sine, cosine, and tangent of a
certain angle. Is this the case, and, if so, what is the value
of the angle ?
EXERCISE XLVII.
AREA. FUNCTIONS.
A.
1. Write down the area functions which correspond to
the following ordinate functions :
(i) y = 2x 2 . (ii) y = r8x 2 .
(iii) y = - 0'9x 2 . (iv) y 6 */x.
(v) y = i'S \ /x - ( vi ) 7 = 5' 1 \/( - x).
2. Calculate the area between the curve, the #-axis and the
ordinate specified in the following cases taken from No. 1 :
The ordinate x = + 6in(i); x = + 10 in (iii); x -f 25
in (iv) ; x = - 100 in (vi) ; x = - 8 in (ii).
3. In No. 1 (i) calculate the area between the curve, the
ic-axis, and the ordi nates x 4- 2 and x -f 9.
4. In No. 1 (ii) calculate the area between the curve, the
#-axis, and the ordinates x 6 and x = -f 10.
5. In No. 1 (v) calculate the area between the curve, the
re-axis, and the oi'dinates x = + 4 and x = +49.
6. Fig. 43, A and C (p. 233), are parabolic. Calculate the area
above the &-axis in each figure, taking a small square of the
chequered background as the unit.
7. In fig. 43, C, calculate the area of the band between
two horizontal straight lines respectively 9 and 16 scale-units
below the vertex.
8. In fig. 43, B, calculate the area included between two
horizontal straight lines drawn respectively 5 and 10 scale-
units above the #-axis.
9. In fig. 44 measure a number of vertical lengths, such as
pq, intercepted between the curve AB and the line AC. Show
that they follow the law pq = kx 2 (where x = Am) and find
the value of k. What is the area of the space included
between AB, AC, and BC? (Compare Ex. XXX, Nos. 31,
32.)
250
EXERCISE XLVIT
261
10. In fig. 44 what is the area of ABD and of ABC ?
11. Give an expression' for the total height of an ordinate
such as pm at distance x from A. Write down the corre-
sponding area function, and show that the answer to No. 10
can be obtained from it.
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44.
12. Each of the following is the ordinate function of a
curve. Give the corresponding area functions :
(i) y = sx + 2-ix 2 .
(ii) y = i '6x - O'px 2 .
(iii) y = 12 + 4x + x 2
(iv) y = io'8 - 2'4x + i'5x a .
(v) y = 2x 2 - ax + J.
(vi) y = ^ + 5 x_- 8.
(vii) y i + 6\/x.
(viii) y = 5'2x - 4-5 Vx^
(ix) y = i'5x 2 - r2js/x.
(x) y = 0*6 - i'2 /s/x + 2'4x - 4'8x 9 .
H5H ALGEBRA
13. Calculate the area under the curve in No. 12 (i) from
x = to x = + 10 ; in No. 12 (iii) from x = - 5 to
x = + 5 ; in No. 12 (viii) from x = + 4 to x = + 36 ; in
No. 12 (ix) from x = + 9 to a? = +16.
Draw graphs to illustrate the second and the last of these
problems.
14. Draw the smooth curve passing through the points
whose co-ordinates are given in the following table. Find by
the method of differences the formula of the curve. Calculate
the area under it from x = to x = +20 :
x : 4 8 12 16 20
y : 20 32 36 32 20
B.
15. A solid figure is made by taking n circular discs 1 inch
thick and, respectively, 1, 2, 3 . . . n inches in radius. The
largest disc is placed on the table, the next largest on top of
it, and so on in order. A pin is driven through the centres
of the discs to keep them together, and projects 1 inch above
the highest. Another figure is made by piling n + 1 discs,
each n inches in radius and 1 inch thick, so that they form a
cylinder n + 1 inches high. What fraction is the volume of
the former solid of the volume of the cylinder ? Calculate the
actual volume when n = 10.
16. What formula in mensuration is obtained by supposing
the discs in No. 15 to be increased in number without limit ?
17. The angle between the hypothenuse and the perpen-
dicular of a right-angled triangle is 35. The triangle is made
to generate a cone by revolving about the perpendicular.
What is the area of the circular section situated at distance h
from the vertex ? What is the volume of the cone above this
section ?
18. I have a wooden model in the form of a truncated
pyramid upon a square base. It is 10 inches high, the length
of the side of the base is 23 inches, the length of the side of
the top 3 inches. Write formulae (i) for the length of the
side of a square section at a given distance from the top ; (ii)
for the area of that section. From the latter formula deduce
a formula for the volume of the solid above a given section.
By means of this formula calculate the volume of the model.
EXERCISE XLVTI 253
19. The base of a village cross consists of m + 1 square
slabs of stone each 1 foot thick. The side of the top slab
measures a feet, and the side of each of the others is b feet
longer than that of the one above it. Show that the total
volume is given by the formula :
V = (m + l){a 2 + mab + J-ra(2w + l)fc 2 }.
20. Draw the positive branch l of the curve
y = 12 - 3 Jx
from where it crosses the ?/-axis to where it crosses the a;- axis.
Imagine it to generate a solid by revolution about the #-axis.
Write formulae (i) for the area of the section of the solid at
the distance* x from the origin ; (ii) for the volume of the solid
from the base up to this section. Deduce the total volume of
the solid.
C.
21. A number m of rectangles of equal breadth are set side
by side. Their areas are I 3 , 2 3 , 3 3 , . . . m' 3 . Each lies upon
a rectangle which is of the same height and width as the one
whose area is y/& 3 . An additional rectangle of this size is
placed next before the recta agio under the first of the increas-
ing series. Thus the (m + 1) underlying rectangles form
together a rectangle of area (m + l)m 3 . Find, by co-opera-
tion with the rest of the class, what fraction of this last rect-
angle is covered by the increasing series when m has the values
1, 2, 3, . . . 10. What laws are followed by the numerators
and denominators of these fractions ?
22. The increasing rectangles are made so thin and at the
same time so numerous that they become indistinguishable
from the area under the curve y kx 3 . Show that the
formula for this area, from the origin up to the ordinate x from
the origin is A = kx*. What assumption have you made in
this proof?
23. OP is a portion of the curve y = kx 3 in the first quad-
rant, O being the origin. From P perpendiculars are drawn,
PM to the z-axis, PA to the 7/-axis. Show that the area AOP
is three-fourths of the rectangle AM.
1 That is, the curve obtained by taking only the positive square
roots of the successive values of x.
254 ALGEBRA
24. From the result of No. 23 deduce the area function of
the curve whose ordinate function is y = k IJx.
25. Bow can the result of No. 24 be brought under Wallis's
Law?
26. Write down the area functions corresponding to the
following ordinate functions :
(i) y = 2-4x-'.
(ii) y = - 3'2x s .
(iii) y = 1*2 Vx.
(iv) y = - 8 */x.
( v ) y = I - 2X ! .
(vi) y = i + 2x -f 3x 2 + 4x 3 .
(vii) y = 3-2 - i'4x + 2'7x 2 - 3*2x 3 .
(viii) y - 4 - 12 $x.
(ix) y - (x + x)/(i + x).
(x) y = (i - x*)/(i - x).
(xi) y - (i - x<)/(i + x).
(xii) y = (16 - 8xx*)/(2 ~ 3*).
27. Calculate the areas under the following cuives in No.
26:
(i) from x = o to x = +6.
(ii) from x = o to x = - 10.
(iv) from x = - 10 to x = + 10.
(vi) from x= -4tox= +4.
(x) from x=-5tox=+5-
(xi) from x=-5tox=+5.
Draw on the same sheet the curves of No. 26 (vi), (x), and
(xi) from #= - 5 to a? = 4-5.
28. Write out the proof by recurrence that the fraction
Q >2 + I 2 + 2* + . -_^+_^ _ 1 1_
(m+ l)m 2 "" 3 + 6m
for all values of m.
Note S> 2 sO a + 1 + 2 3 + 3* + . . . + ml
29. Find by the method of differences the formula for Jw 2 ,
30. Prove by the recurrence method that
S> 8 /( + IK = i + ^
for all values of m (See Ex. XXXVIII, No. 10.)
EXERCISE XLVIII.
DIFFERENTIAL FOKMUL^E.
A.
1. Find in each of the following cases the first difference
of y corresponding to any increment h in the value of x.
Deduce from your results the corresponding differential
formulae of the first order :
(i) y - 3x - ii. (ii) y - (a - b)x + (a + b).
(Hi] y = 4'ix' J . (iv) y = 4'ix 2 - 3'7x + 1*8.
(v) y = 7 x :J . (vi) y - i/2x.
2. Use Wallis's Law to derive differential formulae of the
first order from the following primitives :
(i) y = ix - f. (ii) y - 4*8 - 3'5x.
(hi) y = 2'7x 2 . (iv) y = 4*3x 2 - i'2x + 5-8.
(v) y = 8 + I2x - Sx 2 . (vi) y - 2x :{ .
(vii) y = x 3 - 4x + 3. (viii) y = 4x <{ + x 2 - yx + i.
(ix) y = 3x + i + i/3x. (x) y - (5x 2 - 7)/x.
3. Use Wallis's Law to obtain the primitive of each of the
following differential formulae :
(i) 8y/Sx = 2. (ii)
(iii) dy/5x = 8x. (iv) Sy/fix = 2x - 3.
(v) Sy/Sx = 5 - 3x. (vi) Sy/fix = 6x 2 - 4x + I.
(vii) 8y/Sx = 3/x 2 . (viii) Sy/5x = i - x/x a .
4. The primitives of No. 3 are satisfied respectively by the
following pairs of values of x and y. Use this information to
determine the values of the unknown constants in them :
(i) (- 7. + 5)- (u) (+ 4 o).
(iii) (o, o). (iv) ( - 2, + 5).
(v) (+ 4, - 3). (vi) (+ i, + i).
(vii) (- 3, + 3). (viii) (+ 2, + 2-5).
5. Derive a differential formula of the second order from
each of the following :
255
256 ALGEBRA
(i) y - ix 2 - 2x + 3. (ii) y = x 3 - 3x 2 + I4x - n.
(iii) y = 2(x - i) :{ .
6. Find the primitive of each of the following differential
formulae :
(i) fi' J y/fix 2 = - 3, given that it is satisfied by ( - 2, +5) and
,(0, +1).
(ii) d' 2 y/x 2 = 6x - 2, given that it describes a curve which
passes through the points (+ i, o) and ( - i, - 4).
(iii) 5 J y/5x <J = 12, given that its graph passes through the points
(o, o), (+ i, - 9), (- i, - 13).
7. Prove that, when y = 7^ 4 , By/8x can be calculated in
accordance with Wallis's Law.
8. Write down the primitive formulae which correspond
to:
(i) Sy/Sx = ax 3 . (ii) S 2 y/Sx 2 - 2x 2 - 3x + I.
(iii) #'y/5x 3 - Sx. (iv) ^y/Sx 4 = - i.
B.
9. Given that y is a function of x of the nth degree (where
n may be either 1, 2, 3, or 4), show that the differential
formula of the nih order expresses an exact equality for all
values of Sx.
10. A variable y is connected with x by the relation
y = 5x' 2 . Show that if the true value of Sy/Sx is to differ by
less than 0-001 per cent from the value expressed by the
differential formula fix must represent a number less than
| 20 | x 10~ 5 .
11. The range of values over which the relation of No. 10
holds good is (i) from x = + 1 upwards ; (ii) from x == 4-
10,000 upwards ; (iii) from x = + O'Ol upwards and from x
= -- O'Ol downwards ; (iv) from x = + 10~ 5 upwards and x
= - 10" 5 downwards. Find in each of these cases the largest
numerical value of the increment of x which can be called
" small".
12. "The differential formula Sy/8x = 10#, derived from
y = 5x 2 , can (by taking | Sx \ small enough) be regarded as
true to any required degree of accuracy for all values of x
except x = 0." Justify this statement.
Note. As x approaches zero (either from the positive or
the negative side) Sy/Sx = 10# also approaches zero. By
taking x near enough to zero, By/fix may thus be made numeri-
EXERCISE XLVITI 257
cally smaller than any number which can bo named. For
this reason, although (as was shown in No. 12) the differential
formula cannot really be applied when x 0, it is usual to
say that " %/&c - when x = ".
13. A variable y is connected with x by the relation
y = 5# 2 + 3# - 7 from x = upwards. In a certain calcu-
lation it is necessary that the true value of By/Sx should not
differ by more than one-thousandth part from the value given
by the differential formula. What is the greatest value of the
increment of x which can be symbolized by Sx ?
14. Show that when y = 5x 2 + 3x - 7 the differential
formula &y/$x = l(Lc + 3 may be made to hold good to
any required degree of accuracy for all values of x except
x = - 0'3, Explain carefully what is meant when it is
said that 8///&C = for this value of x.
Note. The investigation covered by Nos. 11-14 was neces-
sary to make clear what is meant by the statement that
?//& = for a certain value of x. When this statement is
understood the value of x in question is most easily deter-
mined by substituting zero for Syftx in the differential formula.
Thus, if y px 2 + qx + r, Sy/8x = %px + q. Hence 8y/ftx
= when %px + q = 0, that is, when x = - q/p. That is
to say, %/&c may be brought as near as we please to zero by
bringing x sufficiently near to - q/p, although the differential
formula, strictly speaking, fails when x = - q/p exactly.
Nos. 12 and 14 show that it cannot be applied in this case
because the condition that the formula shall be even approxi-
mately true is that the increment of x shall be zero an obvious
contradiction of the meaning of the word " increment".
15. Find the values of x (if there are any) for which 8y/$x
=* in the following cases :
(i) y = sx 2 - 4x - i. (ii) y = 6 - 2'sx.
(iii) y = 12-3 + 4'5x - i'5x 2 . (iv) y = 2x 3 + 3x 2 - 36x + 7.
(v) y = fx 3 - Vx 2 + I 5 x - 2.
16. Find the values of x (where there are any) for which
& 2 y/8x 2 = in the case of the relations specified in No. 15.
17
EXEKCISE XLIX.
GRADIENTS.
A.
Note. On the curve PT (fig. 45) take any two points
PP. Through PP' draw the secant PS. It is evident that
P' can always be taken so near to P that the curve between
the two points nowhere departs from the line PS by more
than a certain distance (say d) which may be chosen as small
y |yP Hi"
* n N
FIG. 45.
as we please. Another way of stating this same fact is to
say that if p is any point on the curve between P and P' the
ratio pq/Pq can be made to differ from the ratio P'Q'/PQ by
less than a number (say c) which may be taken as small as
we please. This being the case, PQ' or any part of it,
such as P#, may be called $x and the corresponding first
difference of the ordinates, i.e. P'Q' or pq, may be called $y.
258
EXERCISE XLTX 259
Thus the ratio P'Q'/PQ (or any of the ratios pqjYq) is the
value of 8y/&x at the point P. But P'Q'/PQ = tan a, a being
the inclination of PS to the #-axis. Now if PS were the
tangent to the curve at P, tan a would be called the gradient
of the curve at the point P. It follows that, in the circum-
stances described, the value of &y/Sx for the point P will give,
to as close an approximation as anyone chooses to name, the
gradient at that point.
If P were taken close to and on the left of the upper
turning point T, tan a, and therefore %/&r, would be positive
but would approach zero as P approached T. Again, if P
were close to T on the right, tan a, and therefore &y/Sx, would
be negative and would approach zero as P moved towards T.
At the point T itself 8y/&x may be said, then, to be zero
according to the convention explained in the notes to Nos.
13 and 15. The value of %/S.r would obviously exhibit
similar changes if P moved from left to right through the
lower turning-point, T', except that the change of sign would
be from minus to phis instead of from plus to minus.
An upper turning-point in a curve is usually called a
maximum and a lower turning-point a minimum. These
names are also given to the turning values of functions.
They will sometimes be used in future examples.
1. Calculate the gradients of the curve y = x' 3 - 3x + 2
at the points where x has the values (i) - 2 ; (ii) - 1 ;
(iii) ; (iv) 4- 1 ; (v) 4- 2. Determine whether the turning-
points are maxima or minima by considering the signs of the
gradients on each side of them.
2. Verify the results of No. 11 by drawing the graph of
the curve from x = - 2 '2 to x = +2*2. The part near the
2/-axis must be drawn with special care. At each of the
points specified in No. 11 draw a straight line having the
calculated gradient. Determine whether these lines appear
to be tangents.
3. Calculate the gradients of the curve
y - 2# 3 + 3x 2 - 36x + 7
at the points where x has the values (J)- 3'5 ; (ii) - 3;
(iii) - 1 ; (iv) ; (v) + 1 ; (vi) + 2 ; (vii) + 2-5. Deter-
mine whether the turning-points are maxima or minima.
4. Verify the results of No. 2 by drawing the graph of
the curve from x = - 4 to x = + 3. It will be convenient
to make the vertical ten times smaller than the horizontal
17 *
260
ALGEBRA
scale. The portion between x = - 1 and x = must be
drawn with special care. At each of the points specified in
No. 2 draw a straight line having the calculated gradient,
making allowance for the fact that the horizontal and vertical
scales are different. Determine whether the lines so drawn
appear to be tangents to the curve.
5. Show that a function of the form 2>x* + qx + r cannot
have turning values unless p and q are of opposite signs.
6. Find the condition that a function of the form
px* + q.c 2 + rx -f s
may have turning values.
7. Which of the following curves have turning-points?
Where they exist find their co-ordinates :
(i) y - sx 2 - 4x + 8 ; (n) y --= 7-1 - 2'yx. - 4'6x' 2 ;
(111) y = x :j 4- 3x 3 + 9x -h 5 ; (iv) y ^ 4x ; - i5x- - i8x + n ;
(v) y -= x 1 - x ! + x 2 - i.
13.
Note. In fig. 45 the curve in the neighbourhood of P is
below the tangent or concave towards the #-axis. In this
case it is evident that the differences of y, P'Q', P"Q", etc.,
* must decrease towards the
- right of P and that the cor-
responding differences to-
wards the left of P must
also decrease. In other
words, &n must be negative
for the value of x at P
whether the small differ-
ences in x are measured to
the right or left. Now
suppose that $ 2 y/8x 2 = R,
where B may be either a
constant number or an ex-
pression (such as 5x - 2)
whose value depends upon
x. Since it can be written
&y B(&r) 2 the sign of &y must be the same as the sign of
B whether x is positive or negative i.e. whether the equi-
distant ordinates are drawn to the right of P or to the left.
If, then, the curve is below the tangent, B, and therefore
FIG. 46.
EXERCISE XLIX 261
SPy/Sx 2 , must bo negative. Similarly, if the curve is above
the tangent, as at T', $ 2 y/Sx' 2 will be positive.
Sometimes it will happen that the curve lies above the
tangent on one side of P and below it on the other as in fig.
46. In this case 8 2 ?//&c 2 will be positive at any point close
enough to P on the one side and negative at any point close
enough to P on the other side. At P itself, therefore, tfy/Sx' 2
may be considered as being zero in the sense explained in the
note on p. 257. Such a point as P in fig. 46 is called a
point of inflexion.
Thus to find the situation of the curve with respect to the
tangent at any point it is best to determine the value of
S 2 7//&c 2 . If this is positive the curve is above the tangent, if
negative, below. If, as given by the formula, % 2 y/Sx 2 = we
must examine its value for points immediately to the right
and left of the given point. If the signs of these values are
different the point is a point of inflexion.
8. Examine the value of 8 2 7//&c a at each of the points
mentioned in No. 1, and so determine which are turning-
points or points of inflexion. Compare the results of your
calculation with the graph.
9. Eepeat this investigation upon the data of No. 3.
10. Show that the curve?/ = 3# 4 - 20# 3 + 48# 2 - 36o: +11
has two points of inflexion.
11. Examine the curves of No. 7 for points of inflexion.
12. Show that no curve of the type y = px^ -\-qx-\-r can
have a point of inflexion.
13. Show that all cubic curves of the type y = px* + qx 2
+ rx + s have one point of inflexion. Show also that in
those of the type y = px z + qx + r the point of inflexion
always lies on the ?/-axis. (See the graph of No. 1.)
14. Show that the curve y = px* + qx* + rx 2 + sx -f t has
either two points of inflexion or none. Show that if q is zero
the points of inflexion (if they exist) are equidistant from the
2/-axis.
C.
15. A rectangular concert hall AC (fig. 9, p. 34) is to be fitted
with a platform FH. AD must be 50 feet, but the length
AB is not fixed. FG is to be double of EF and EF is to be
262 ALOKBUA
one-tenth of the length (/) of the hall. Show that the area not
covered hy the platform is given by the formula
A = 50/ - \l".
Hence show that this area will be greatest if the hall is 125
feet long. Calculate the area in this case and verify that
/ = 120 and I = 130 both give smaller values. [Apply the
methods explained in the notes before Nos. 1 and 8 to find
the turning value of the function 50Z - ^ 2 and to show that
it is a maximum. The %/&r and SPy/Sx 2 of the notes will be
replaced hero by 8A/SZ and 8 2 A/SZ 2 .]
16. Fig. 17 (p. 35) is the surface of a plate whose thickness
is c. Supposing that b is always twice c show that the volume
of the plate is given by the formula
V = (3a - c)c-'.
Show that if a is a fixed length, but c may be varied, the
volume is greatest when c = a/2. Verify (as in No. 15) by
means of numbers chosen by yourself.
17. Fig. 18 (p. 35) is the surface of a plate whose thickness
is b. Show that the weight of the plate is given by the formula
W = (3a - 2b)b*w
where w is the weight of a unit cube of the material. Show
that if the length a is fixed but b may be varied the plate will
get constantly lighter as b increases from zero to a.
18. Fig. 28 (p. 94) represents a rectangular plate from
which two semi-circular pieces have been removed. Show
that if a is fixed but b may be varied, the area is greatest when
b = 3a/27r.
19. Fig. 28 may be taken as a central section of a block of
wood out of which two hemispherical pieces have been cut
of radius b. The height of the block is 2(6 + c). Thus two
of the sides perpendicular to the plane of the paper are
rectangles measuring a by 2(6 + c) but containing circular
holes of radius b. Show that the volume of the block is given
by
V = Qacb + 6a6 2 - fab*.
Show that if a and c are fixed but b is variable the maximum
and minimum values of the volume correspond to the values
of b which satisfy the equation
4ir . & 2 - 12a .b - &ac = 0.
Calculate the maximum and minimum volumes when a = 2?r
inches and c = 9 inches.
EXERCISE XLTX 263
20. Fig. 29 (p. 94) represents a star-shaped plate pierced
by a circular aperture. Show that if b is fixed and a varies
the area of the metal surface constantly increases as a increases
and never has a minimum or maximum.
21. A seaside camp offers to receive from a certain school
a party of not less than 5 at a cost per head of (2 + 200/n-)
shillings a week, n being the number in the party. Find the
number which it would be cheapest for the school to send.
22. The size of parcels which the Post Office will accept
for transmission is limited by the regulation that the length
and girth of the parcel when added together must not exceed
G feet. Find the dimensions of the largest box with square
en Is which it is possible to send through the post.
23. Take a square sheet of paper, PQBS, and draw in the
middle of it a square ABCD. Produce the sides of the inner
square to meet those of the outer square. Kemove, by
cutting, the squares AP, BQ, CE, DS. Fold the four wings
about the sides of the inner square so as to make a hollow
box of which the square is the base. Show that the volume
of this box is greatest when AB is two- thirds of PQ.
24. Draw a circle of radius r and inscribe in it a rectangle
of length 2a and breadth 26. Revolve the whole figure about
the diameter parallel to the sides of length 26 and so generate
a sphere with a cylinder inscribed in it. Show that the volume
of the cylinder is given by the formula
V - 27r6(r 2 - 6 2 ).
Hence show that the volume of the cylinder is greatest when
1) = r/^3, Calculate the volume in terms of r.
EXERCISE L.
THE CALCULATION OF TT AND THE SINE-TABLE.
A.
1. Let C = the length of any chord, AB, of a circle of unit
radius (fig. 47) and C x the length of AB', the chord which
bisects the arc AB. Write out a proof that
2. Take AB as a diameter of the circle and use the formula
to calculate the length of side of an inscribed square. Prove
independently that your result is correct.
3. Let E = the length of PQ (fig. 48) and G = the length
of AB. Write out a proof that
B ._. 2 ._
N/(* - C ' 2 )'
4. Use the formula of No. 3 to obtain an expression for the
B' -.A P
FIG. 47.
Fia. 48.
length of side of a square circumscribed about a circle (i) of
unit radius, (ii) of radius r.
5. What is the meaning of the result obtained from the
formula of No. 3 by taking AB as the diameter of the circle ?
2G4
EXERCISE L 265
6. Calculate to three places of decimals the perimeter of a
regular hexagon circumscribed about a circle of unit radius.
7. Within what limits is the value of ?r fixed by considera-
tion of the inscribed and circumscribed hexagons ?
8. Show that the perimeters of the regular 12-gons which
are inscribed within and circumscribed without a circle of unit
radius are respectively of length
12 7(2 - ^3) and
9. Given that ^3 = 1*732051, calculate the limits within
which the value of TT is fixed by consideration of the inscribed
and circumscribed 12-gons.
10. Give expressions for calculating the lengths of the
perimeters of the regular octagons inscribed within and cir-
cumscribed about a circle of unit radius.
11. Use these expressions to determine major and minor
limits for TT, given that </2 = 1-414213.
12. Change the subject of the formula of No. 1 to C.
13. Use the new formula to calculate the length of side of
an equilateral triangle inscribed in a circle of unit radius.
14. Calculate the length of side of an equilateral triangle
circumscribed about a circle of unit radius. What limits are
determined for TT by this result and that of No. 13?
15. Above each of the points 3, 4, 6, 8, 12 on the #-axis
plot a pair of points whose ordinates are respectively the
upper and lower limits of the value of TT determined by con-
sidering the perimeters of regular polygons of 3, 4, 6, 8, 12
sides. Join the two series of points by smooth curves.
Determine by extrapolation the probable subsequent course
of the curves and the ultimate value of TT so far as it can be
read from your graph. [Take as wide a vertical scale as
possible.]
B.
16. From the results used in No. 15 write down a table
giving to three decimal places the values of sin a and tan a when
a = ?i, 22f f 30, 45, 60. Compare the results with the
table on p. 111.
17. Change the formula of No. 1 into a formula for finding
sin ~ where sin a is known.
266 ALGEBRA
18. Use this formula to calculate the value of sin 3^ n .
[Take the value of sin 2 a from No.
]
19. Show that your value for sin
7-J is approximately twice that
obtained from su\ 3f . That being
the case you may, from the other
two sines, calculate sin 5 to three
places by proportion. Do so, and
deduce from it the value of cos 5
also to three places. Compare the
results with the table on p. 111.
FlG - 49> 20. Use fig. 49 to prove that
sin (a + ft) = sin a cos /3 + cos a sin fi.
[This proof can be used only if a + ft < 90. Why ?]
21. Prove by the same method that
sin (a - ft) = sin a cos ft - cos a sin /?.
[In fig. 49 let AOC - 2a and COB = 2/3, then AOB = 2(a - ft).
Produce AB and drop upon it from C a perpendicular CD.
Then AB = AD - BD.]
22. From the known values of sin 30, sin 45, sin 60, sin
5 and cos 5, calculate the values (to three places) of sin 25,
sin 35, sin 40, sin 50, sin 55, sin 65. The work is to be
divided among the class, each member calculating one or two
sines only. Collect the results into a table giving the sines
of every five degrees from 25 to 65. Compare this table
with the one on p. 111. Of what cosines may this also be
considered to be the table?
23. In fig. 47 let Z.AOB' = 2a so that 2_AOB = 4a. Prove
by the method of No. 20 that
sin 2a = 2 sin a . cos a.
24. Use the formula to calculate sin 70 and sin 80, taking
your data from the table constructed in No. 22. What cosines
have you simultaneously calculated ?
25. Give a careful summary of the method by which a
complete table of sines and cosines can be calculated for in-
tervals of 5.
SECTION III.
LOGABITHMS.
EXERCISE LI,
GBOWTH FACTORS.
TABLE OF AVERAGE HEIGHTS OF BOYS AND GIKLS.
Ago
61
6J
74
81
1
101
Hi
yearn
Boys
417
43-9
40-0
48-8
50-0
51-9
53-6
inches
Girls
41-3
4:3-3
457
477
497
517
53-8
Age
12J
131
144
154
16J
174
184
years
Boys
55-4
57-5
oo-o
62-9
04-9
66-5
67-4
inches
Girls
561
58*5
00-4
61-6
62-2
627
M
A.
Note. Imagine a boy to grow at such a rate that his height
at each age is exactly equal to the average height of all boys of
that age. He may be called, in respect of height, the average
boy.
1. How much taller does the average boy grow between 6 J
and 7 ? Between 12^ and 13-J- ? How much does the average
girl grow between 10| and 11?
Find the ratio of the second of each pair of heights to the
first. Which number best measures the growth of the child
the difference of the annual heights or their ratio ? Why ?
Arrange the three annual rates of growth in order of magni-
tude.
Note. The ratio of the boy's height on his llth birthday
269
270 ALGEBRA
to his height on his lOfch birthday may be called the growth-
factor of his height at ip years of age. How would you
find the growth-factor of his height at 12 years of age?
2. Calculate the growth-factor of the average boy's height
at 5 . If the growth-factor had the same value at (i) 12 and
at (ii) 14, how tall would the boy be at 13 J and 15^ respec-
tively ? Do these results agree with those given in the
table ?
3. If the growth-factor of the height of the average girl of
12 J- were the same as that of the average boy of the same age,
how tall would you expect her to be at 13 J?
4. Calculate the growth-factors for the average boy and the
average girl at 5 J, 9, 11^, and 14 years of age. What con-
clusions do you draw from the results ?
5. Suppose the growth -factor of the average girl of 9^ to
remain constant for the next few years. Calculate how tall
she would be at 10 , 11-J, . . . 15r}. Compare your results
with the table. What conclusion do you draw ?
6. Assuming the same growth-factor as in No. 5, calculate
what the average height would bo at 6^ and at 5.}. Compare
the results with the numbers given in the table,
7. The population of a town has been increasing for some
years at the rate of 20 per thousand per annum. What is the
annual growth-factor of the population ?
8. The present population of the town of No. 7 is 48,750.
What will it be two years hence? What was it two years
ago?
9. The population of a country parish is falling off to the
extent of 10 per thousand per annum. What is its annual
growth-factor ?
10. The population of the parish of No. 9 is 4520. What
will it be in three years ? What was it last year ?
11. A quantity is changing in such a way that its magni-
tudes at equal intervals of time form a geometrical progression
with a common ratio r. (In other words, the quantity has for
these intervals a constant growth-factor of r.) Its present
magnitude is Q . Show that its magnitude at the end of any
integral number of time-intervals, past or future, is given by
the formula
Q - Qr".
12. Write formulae for the population (P) of the places
EXERCISE LI 271
mentioned in Nos. 7 and 9 at any exact number of years before
or after the present moment.
13. The values of a number, Q, at regular intervals are
given by the formula Q Q x (l'l) n . Draw up a table by
which the values could be calculated to within 1 per cent for
values of n from - 5 to + 5.
14. Draw up similar tables to deal with the formulae
Q = Qo x (1-3)" and Q ~ Q x (0-8)".
B.
15. A man's business is increasing so rapidly that his income
is doubled regularly in the course of two years. What is the
annual growth-factor of his income ?
16. Find the annual growth-factor if the income were
doubled regularly in the course of four years.
17. What assumption did you make in solving Nos. 15 and
16?
Note. Nos. 18-20 can be solved by means of the tables of
Nos. 13 and 14.
18. Between 1 January, 1906, and 1 January, 1911, the
population of a London suburb increased from 5437 to 8758.
Assuming the increase to be due to a constant growth-factor
find (i) the value of the factor ; (ii) the population on 1
January, 1909; (iii) the population on 1 January, 1901.
19. A sapling grew in seven years from a height of 14
inches to a length of 87'8 inches. What constant annual
growth-factor would account for this increase ?
20. During an epidemic of scarlet fever in London it was
noticed that after a certain date the number of cases decreased
by the same fraction each week. In the course of eight weeks
they fell from 1540 per week to 258 per week. What was
the weekly growth-factor? How many cases occurred during
the fourth week after the decrease began ?
21. If the average boy's height increased by a constant
growth-factor between the ages of 9 and 14|, find how tall
he would be at 10 , 11-J-, 12 J, and 13. Compare the results
with the table.
[You would find by multiplication that (1/037) 5 =1-2
approximately.]
EXERCISE LTI.
GROWTH PROBLEMS.
A.
1. Plot graphs for the solution of problems in which the
growth-factor is (i) 1'3, (ii) 1-25, (Hi) 1'JL.
Five future and five past magnitudes should be plotted in
each case. The numbers needed can be taken from the re-
sults in Ex. LI, Nos. 13, 14. The thiee graphs may be plotted
on the same sheet of paper, the same vertical and horizontal
scale being used for each. These three graphs are to be
employed in solving the following problems ; they must
therefore be executed with great care and accuracy. They
are shown in fig. 50.
2. The value of the orders received daily by a certain firm
is doubled in 3'1 years. What annual percentage rate of
increase would lead to this result ?
3. The daily output of a gold mine increased from 55 oz.
to 60.} oz. in a month. If the rate of increase 'remained
constant, what would be the daily output (i) in two and a
half months, (ii) in three and a quarter months ? What was
the daily output four and a half months ago ?
4. Owing to the increase of motor vehicles, the number of
horses in a certain town, which was 3500 in 1905, had fallen
to 2173 in 1910. Assuming a constant annual rate of de-
crease, find the number in 1908.
5. The census of the population of the British Isles is taken
every ten years. The population of a certain district in
London was 8430 in 1861, and had risen to 18,510 in 189r,
What was probably the population in 1885 ? What would
you expect it to be in 1914 ?
6. Four tradesmen on comparing notes find that for every
pound they took a year ago they have taken to-day 1 2s,,
272
<JJ
cO
27 i ALGRBfcA
1 6s,, 16s., and 1 5s. respectively. Assuming these changes
to be due to constant growth -factors, find
(i) What they may each expect their takings to be in eighteen
months' time ;
(ii) what they were three months ago ;
(iii) what they were four and a quarter years ago ;
(iv) when the takings will be (or were) 2 for evory 1 taken
a year ago.
7. The soundings of an Admiralty surveying ship pursuing
a straight course show that the depth of the sea has been
and is increasing at the constant rate of 30 per cent per mile,
and that at a certain place it is 100 fathoms. What will the
depth be (i) 2-3 miles; (ii) 3-6 miles; (iii) 4-2 miles ahead?
What was the depth (iv) 0*7 miles; (v) 1-8 miles; (vi) 4*65
miles astern of the present position of the ship ?
8. What would be the depth of the sea (in the preceding
problem) at a place (i) 3 '3 miles ahead, (ii) 4-8 miles astern
of the place where it is 240 fathoms ?
9. It is found that the average score of a number of soldiers
who are practising rifle shooting falls off by 10 per cent for
every 100 yards that they recede from the target. This rule
is found to hold good for all distances from 50 to 1000 yards.
The average score at 300 yards is 24. What is it (i) at 400
yards ; (ii) at 720 yards ; (iii) at 80 yards ; (iv) at 210 yards?
10. The cost per yard of laying a railway through a moun-
tain range increases between two given points at the rate of
25 per cent for every 10 miles. At a certa ; n place it amounts
to 2 10s. per yard. How much is it (i) 7 miles ; (ii) 14
miles; (iii) 43 miles farther on? How much is it (iv) 17
miles ; (v) 33 miles farther back ?
11. The number of people per day who visit a certain
exhibition is increasing at the rate of 10 per cent per week
of six days. Yesterday (which was Monday) 3620 passed
the turnstiles. How many would you have expected to find
(i) on the previous Thursday ; (ii) on the previous Thursday
fortnight ? How many would you expect to find (iii) next
Tuesday ; (iv) next Fiiday week?
12. If the daily attendance at the exhibition began, from
the Friday mentioned in No. 11 (iv), to fall off at the rate of
25 per cent per week, how many visitors might be expected
(i) ten days later, (ii) fifteen days later ?
EXKtlCtSE LII 275
13. The population of a town has increased two and a half
times in the course of three and a half years, (i) What
annual growth-factor does this change represent ? (ii) What
increase of population per cent per annum would produce
this change ? (iii) Assuming a constant growth-factor, when
was the population (a) one-quarter, (b) one-fifth of what it is
to-day ?
14. Owing to the introduction of a new machine, the manu-
facturers in a certain industry are now employing only 64
per cent of the men whom they employed two years ago. If
this rate of decrease is maintained, how many per cent of the
men at present engaged will be displaced from the industry
(i) within the next eighteen months, (ii) within the next two
and a half years ?
15. A man bought in ditto rent parts of the same London
suburb two houses for 420 each. Owing to the opening of
a new railway the property rapidly rose in value, and he was
able after two and three-quarter years to sell one of the
houses for 860 and a year later to sell the other for 990.
Find which of the two houses had been increasing most
.rapidly in value, and state the annual rate of increase per cent
of each, assuming it to have been uniform.
B.
16. The number of persons employed by a certain firm in
one of their branches has increased at a uniform rate of 10
per cent annually, since the branch was opened fourteen years
ago. The number is now 195.
(i) What was the original number of tho employees ?
(ii) What was their number ten and a half years ago ?
(iii) What was it four and a quarter years ago ?
(iv) When was it exactly 100 ?
[Find on curve C tho point whose ordinate is 1*95 and take the
foot of that ordinate as the origin of the time-scale. Think out
the reason why it is permissible to do this.]
17. In another branch of the same firm (opened eleven years
ago) the employees number 155 and have also been increasing
annually at the rate of 10 per cent.
(i) How many were they when the branch opened ?
(ii) How many were they six and a half years ago ?
(iii) When did they number exactly 100 ?
276 ALGEBRA
18. At the headquarters of the same iirin (opened fifteen
years ago) there has been the same rate of increase in the
number of the staff. There are at present 636 engaged there.
(i) How many were there originally ?
(ii) How many five years ago 'i
(iii) How many eight and a half years ago ?
[Divide 636 by a number big enough to bring the quotient just
within the limits of the graph.]
19. The population of an Irish village in 1841 was 1020.
Every census since that date has shown a decrease of one fifth
of the population at the preceding census. What population
would you expect the census of 1911 to show?
20. All rateable property (i.e. houses, tramways, etc.) in
London is re -valued every five years for the purpose of deter-
mining the amount of rates to be paid. At the last quinquen-
nial valuation in 1910 two houses were both declared to have
a rateable value of 240 per annum. At the first valuation
(in 1870) one of them was rated at 84. The other was not
then in existence, but in 1880 was rated at 123. Assuming
that the values of both houses have risen uniformly, find
how much each has increased every five years.
EXERCISE LIII.
THE GUNTEB SCALE.
Note. The examples are to be solved by means of the
curves and Gunter scale of fig. 50, p. 273.
1. Find in each of the curves A, B, and C the abscissa of
the point where the ordinate is 1*4.
2. A quantity whose present magnitude is 1*0 is increasing
continuously with a growth-factor of i'4, Write formulas
for the abscissa, x, of the ordinate which indicates its magni-
tude at time t (i) in curve A, (ii) in curve B, (iii) in curve C.
3. Find the magnitude of this quantity when t = + 2-4,
by means of curve A.
4. Find the magnitude of the same quantity when
t =* - 0*75 using (i) curve B, (ii) curve C.
5. A quantity whose present magnitude is 8*4 is increasing
continuously with an hourly growth-factor of 1/2. Use
curve C to find :
(i) its magnitude after 2J hours ;
(ii) its magnitude 1J hours ago ;
(iii) the time when its magnitude was 4'2 ;
(iv) the time when its magnitude will be 12 6.
6. Find the time in which the magnitude of the quantity
mentioned in No. 5 will be trebled. Obtain your answer
from each of two curves.
7. A quantity whose present magnitude is 200 is decreasing
continuously with an annual growth-factor of 0*95. Find its
magnitude (i) in three years from now, (ii) four years ago,
using curve C.
8. The total length of a bean plant (fig. 2, p. 5) is to-day
(Monday) 3 inches and has been increasing for some time
277
278 ALGEBRA
with a constant weekly growth- factor of 1-5. Supposing the
rate of growth to remain constant, how long will it he at the
same hour (i) to-morrow, (ii) on Thursday? (iii) How long
was it at the same hour on Saturday last ?
9. Find (i) the seventh root of 3*62 ; (ii) the fifth root of
2'6; (iii) the cube root of O7.
10. Find (i) the cube root of the fifth power of 1*3 using
curve A ; (ii) the fifth power of the cube root of the same
number, using the same curve. Why must the results be the
same?
11. Find the seventh root of the cube of 1*4 using curve
B. How can you find the same by means of curve C ?
12. In a certain growth- curve the ordinate whose height is
a certain number, n, has an abscissa x. Write down the
abscissa of the ordinates whose heights are :
(i) the 8th root of the 5th power of n ;
(ii) the 5th power of the 8th root of n ;
(iii) the 6th root of the llth power of 1/n ;
(iv) the pth power of the qth root of n ;
(v) the qth root of the /?th power of 1/n 3 ;
(vi) the pth root of the #th power of n a .
B.
13. By means of the Gunter scale (GG, fig. 50, p. 273) and
a centimetre rule find the values of :
(i) (2-2)' ; (ii) (2-7)* ; (iii) (r 3 ) fl J (iv) s/6-8 ;
(v) ^8-x; (vi) #8-1 ; (vii) </ 4 -8 ; (viii) 4/9-57;
(ix) #(2'4) 2 ; 00 (J/2'4) 2 (xi) V(8'i) s ; (xii) (/8'i);
(xiii) (V 3'3) 5 J (xiv) /(4'8) 7 ; (xv) </(x-5).
14. By means of the Gunter scale and an inch rule find
the values of :
(i) (rs) 3 ; (ii) ^5-9; (iii) J/8'2 ;
(iv) */3'5 J W #(3'8) 4 ; (vi) #(i'9) l .
15. Take a narrow strip of paper rather more than three
times as long as GG. Mark its length into three sections
each equal to GG, leaving a margin at each end. Graduate
the middle section as a Gunter scale, transferring from GG
the graduations for the units only and leaving the positions
of fractional graduations unmarked. By means of GG con-
tinue the graduation of the scale to the right, marking the
positions of the tens (20, 30, ... 100) only. Also continue
EXERCISE LIII 279
the graduation to the left, marking the positions of the tenths
(0 9, 0'8, . . . O'l) only. All the graduations should be
marked with line lines in ink.
16. On the strip of No. 15 mark lightly in pencil the
graduation 1*85. Lay the strip across a sheet of squared
paper so that the graduations " 1 " and " 1*85 " are on
parallel lines of the sheet 1 cm. (or 1 half-inch) apart. Mark
on the strip the positions of the graduations which have the
following values :
(i) (1-85)'; (ii)V(l-85) 7 ;
(iii) the magnitude after 5 '7 units of time of a quantity whose
present magnitude is unity, the growth-factor being 1*85 ;
(iv) the magnitude of the same quantity 3*2 units of time before
the present moment.
Read off the numerical value of these magnitudes by means
of the Gunter scale on p. 273. Rub out the pencilled marks
when you have finished the example.
17. Find by similar methods the value of (i) (2 3) 5 ; (ii)
(2-3)-"; (iii) V(2'3) 5 ; (iv) V(2'3) 7 .
18. Lay the strip across the squared paper in such a way
that the graduations " 1 " and " 80 '' lie on parallel lines 12
cms. (or half-inches) apart. Mark the positions of the gradua-
tions which have the following values. Read the values off
by means of the Gunter scale on p. 273:
(i) 5/80; (ii) is/80; (iii) >#(80) 7 ; (iv) { l #(80)} 7 .
19. A quantity is increasing continuously with a constant
growth-factor. In 10 years its magnitude increases 60 times.
Find (i) the annual growth-factor ; (ii) the relative magnitude
at the end of 3*6 years ; (iii) the same at the end of 7*4 years ;
(iv) the same 2*5 years ago.
20. The magnitude of a quantity increases 45 times in
8*6 years. Assuming the increase to be due to a constant
growth-factor, find how much the magnitude increases in 5*8
years.
21. Cut a strip of paper rather more than twice as long as
the graduated base line in fig. 50, p. 273. Place the middle
of the strip at the origin and mark along the strip the positions
of the ordinates of curve A whose heights are 1, 2, 3, 4, and
0-9, 0-8, 0-7, . . . 04. Complete the graduation of the right-
hand section of the strip in units from 5 to 10 and of the left-
hand section in tenths from 0*3 to O'l,
280 ALGEBRA
22. Use the method of No. 18 to mark on the Gunter scale
of No. 21 the positions of the graduations whose values are
(i) yiO ; (ii) i^O'l. Find these values numerically by means
of the ordinates of curve A.
23. Write a full explanation of the method which you used
to complete the graduation of your Gunter scale in No. 21.
24. The graduation "r" is at the distance x from the
graduation " 1 " on a certain Gunter scale. Explain carefully
why you expect the graduation distant xt from the gradua-
tion " 1 " to give the magnitude of a unit which varies con-
tinuously with a constant growth-factor r. (Consider the case
when t is fractional as well as when it is integral.)
EXEKOISE LIV.
LOGABITHMS AND ANTILOGAKITHMS.
Note. The references are to the growth-curves and Gunter
scale of fig. 50, p. 273.
A.
1. Find by means of curve A the value of (i) 2'2 x 1-6 ;
(ii) 2-2 - 1-6; (iii) 1-6 ~ 2-2; (iv) 1-6 ~- (2-2)2.
2. Find by means of curve B the values of (i) 1'5 -f- 1*2 ;
(ii) 1-5-7- (i'2) 2 ; (iii) (V 1-5) + 1-2.
3. Answer No. 2 by means of curve C, comparing the
results with those obtained by using curve B.
4. Obtain by means of either the Gunter scale or a slide
rule the values of (i) 2-7 x 3-4; (ii) 3'4 4- 2*7 ; (iii) 9-2 4-
6-8; (iv) 3-05 x 2-4.
5 Obtain by the same means the value of (i) 27 x 340 ;
(ii) 0-36 4- 2800 ; (iii) 0-027 x 0-00035 ; (iv) 4800 ~ 37-5.
6. Obtain by means of the Gunter scale or (preferably) by
a slide rule the value of (i) 4'7 -i- 3'2 x 2-8; (ii) 9'2 -h 7'3
x 3-4 ; (iii) 9*2 x 3'4 ~ 7'3.
7. State in words the simplest rule for finding by the slide
rule the result of arithmetical operations like those of No. 6.
8. Find the value of
(i) 4'5 x ; (ii) 2'o x ^ ; (iii) 2-3 -f ;
3*4 i'7 4*7
(iv)os^; (v) 2 ^p; (vi) 3-.xJlJ.
9. Find the value of the following by a single operation
with the slide rule (or Gunter scale and paper strip). [Note
that the scale must be supposed to be continued to the left of
the first graduation.]
(i) il ^7 ; (ii) 3JL?2 ; ( iii) 8_ 2 JL7'J.
282 ALGEBRA
10. Find the value of :
370 x 0-028 . ,.,. 425 x 34P . C[[} 27'5_xj>'o82 .
W ^ > W ~~^o~ ' lu; "076
( 2joox3r5 ( V ) 7$45o_5.
v ' 0-58 ' v ' 89
11. Find the value of the following quotients : (i) 2'7 -f- 6'4;
(ii) 3-4 ^ 7.7 ; (Hi) 61 -f- 9-2 ; (iv) 370 ~ 52 ; (v) 0-078 ~
0-95.
12. Find the value of the following products : (i) 3*4 x 4'3 ;
(ii) 7-7 x 6-6 ; (iii) 230 x 72 ; (iv) 0'0058 x 0*063 ; (v)
7900 x 81.
B.
Note. The work in Nos. 13 and 14 may be divided among
the members of the class.
13. By means of curve B construct a table of logarithms
to base 1*25 for numbers at intervals of 0*1 from 0'5 to 2*0.
14. Construct a table of antilogarithms to base T25 for
logarithms at intervals of 0*2 from - 3 to + 3.
15. By means of the method of proportional parts calculate
from your table (i) log a 065; (ii) log, T23; (iii) log 1*72.
[a = 1-25.]
16. Calculate the value of (i) antilog a 2-63 ; (ii) antilog a 0'8 ;
(iii) antilog a (-l-54). [a =1-25.]
17. Use your tables of logarithms and antilogarithms to
calculate (i) 1-8 x 0'65 ; (ii) 1'23 -=- 1-4; (iii) 1-72 -f- 12-3.
18. Use your tables to calculate (i) 0'6 x 1*6 -~ 1-4;
(ii) (l-5) 2 ~l-8; (iii) */F7 x 155; (iv) */M -h VF9."
19. Complete the following identities and show how they
follow from the properties of the growth- curve :
(i) log rt P + loga Q = ; (ii) loga P - loga Q = J
(iii) loga P n - ; (iv) log* V p =
20. Complete the following identities and show how they
follow from the properties of the growth-curve :
(i) antiloga P x antiloga Q = ;
(ii) antiloga P -r antiloga Q --= ;
(iii) (antiloga P) = ;
(iv) y (antiloga P) -
EXEECISB LV.
THE BASE OF LOGARITHMS.
A.
Note. Nos. 1 to 6 are to be solved by means of a sheet of
squared paper and a strip of paper marked, as each case re-
quires, by reference to the Gunter scale on p. 273. [The
work is done much more easily and accurately if for the strip
of paper is substituted a Gunter scale cut from a sheet of
semi-logarithm paper.]
1. Find the value of the following logarithms, all to base
1-5: (i) log 1-8; (ii) log 3-2; (iii) log 4-7; (iv) log 6-3;
(v) log 7-9; (vi) log 10.
2. Find the value of the following antilogarithms, all to
base 1*5: (i) antilog 0*5; (ii) antilog 2'3 ; (iii) antilog 3'0;
(iv) antilog 4*7; (v) antilog 5 -2.
3. Find the value of: (i) logo 4-3 ; (ii) Iog 2 7'5; (iii) log,
10.
4. Find the value of : (i) antilog 2 0-8 ; (ii) antilog 2 1-4 ;
(iii) antilog 2 2 -8.
5. Given that log r 4*9 = 3, find (i) x ; (ii) log A 6-5 ; (iii)
antilog* 4.
G. Given that antilog* 1 -75 = 4, find (i) x\ (ii) antilog v
2-3; (iii) log, 10.
B.
Note. To multiply 2-357948 by 1-1 the work should be set
down thus :
2-357948
J2357948
2 7 593Y428
283
28 i ALGEBRA
7. A Gunter scale is to be constructed in which graduations
1 cm. apart are to have a ratio of 1*1. Calculate the gradua-
tions at the points which are 1, 2, 3, ... 10 cms. from the
beginning of the scale.
8. Enter these graduations in pencil upon a strip of centi-
metre squared paper. 1 Mark, first in pencil and afterwards in
ink, the positions of the graduations 1*1, 1*2, 1-3, . . . 2-6.
9. What would be the value of the graduation (i) 15 cms. ;
(ii) 16 cms. ; (iii) 20 cms. ; (iv) 21 cms. from the beginning
of the scale ? [Answers to three decimal places.]
10. A table of logarithms is to be constructed in which
log 1-1 = 0*2. What will be the base of these logarithms?
[Use the results of No. 7.]
11. What will be the value in this table of (i) log 1-949;
(ii) log 2-358 ; (iii) antilog 0*8 ; (iv) antilog 16; (v) antilog
3-2?
12. Using the same base find by the method of proportional
parts the value of (i) log 1'5 ; (ii) log 2 ; (iii) log 6 -5.
13. A table of logarithms is to be constructed in which
log 1-1 = 0-1. What is the base?
14. Find the value, with this number as base, of (i) log
1-4641; (ii) log 2-3579; (iii) log 2; (iv) antilog 0'3 ; (v)
antilog 2-1.
Note. Since 0'9 = 1 - / the easiest way to find
0-59049 x 0-9 is as follows :
0*59049
-059049
Q : 53144T
15. A table of logarithms is to be constructed in which,
while log 1 = 0, log 9 = 0-25. What is the base?
16. Calculate enough results to carry this table as far as
antilog 2-5.
17. Find the value in this table of (i) log 0'4783 ; (ii) log
0-729 ; (iii) log 0-4.
18. Find the value in this table of (i) antilog 1-25;
(ii) antilog 2 ; (iii) antilog 0-6.
19. A table of logarithms is constructed in which the base
1 If this is not available use inch squared paper and read " half
an inch " for " centimetre " in ]STos. 1 and 9,
EXERCISE LV 285
is 2-1436. Fincl (i) antilog 0'5 ; (ii) log 2-5937 ; (iii) antilog
0-625 ; (iv) log 1-1. [Use the results of No. 7.]
20. A table of logarithms is constructed in which the base
is 0-59049. Find (i) log 0*729 ; (ii) antilog 1-4 ; (iii) antilog 2.
[Use the results of Nos. 15 and 16.]
EXEECTSE LVI,
COMMON LOGARITHMS.
A.
Note. Nos, 1 and 3 are to be answered by means of a
Gunter scale and a sheet of squared paper. It is best to lay
the Gunter scale on the paper so that the divisions " 1 " and
" 10 " of the former are on two verticals of the squared paper
10 inches (or centimetres) apart. The first of these verticals
should be numbered " " and the second " 1 ". The inter-
mediate verticals divide this unit range into 100 equal parts.
All logarithms and antilogarithms mentioned in this exercise
are " common " logarithms and antilogarithms that is, the
base is 10.
1. Find to three places the value of (i) log 2 ; (ii) log 3*4 ;
(iii) 1-05; (iv)8-7; (v) 9-3.
2. From the results of No. 1 write down the logarithms of
(i) 2000 ; (ii) 3,400,000 ; (iii) 0-0034 ; (iv) 10,500 ; (v) 0-105 ;
(vi) 0-000087 ; (vii) 930.
3. Find to three significant figures (i.e. two decimal places)
the value of (i) antilog 0'4; (ii) antilog 0-75; (iii) antilog
0-368 ; (iv) antilog 0*945 ; (v) antilog 0-065.
4. From the results of No. 3 write down the antilogarithms
of (i) 2-4 ; (ii) 3-4 ; (iii) 1-75 ; (iv) 4-368 ; (v) 6*368 ; (vi)
1-945 ; (vii) 5-945 ; (viii) 2-065.
5. From the logarithms of No. 1 calculate the logarithms
of (i) /200; (ii) ^/(340) 2 ; (iii) /87000 ; (iv) (J/105) 4 ;
(v) 3-4x870; (vi) 105-93; (vii) (3400) 2 ~- ^9300;
(viii) /(930 - 87).
6. Find what numbers are obtained by the operations
indicated in No, 5 (i) to (viii).
286
EXERCISE LVT
28?
B.
Note. Since the mantissa of a common logarithm is
always positive while the characteristic may be either positive
or negative, it follows that in addition, subtraction, etc., of
logarithms the characteristics and the mantissae must often
be dealt with separately. Study the following examples :
0-0034 - 3-4 x 10 ~ 3
(a) log (0*3034 x 870) = log 01034 -flog 870
= 0-47
= log 2 -95
log 0-0034 -3 -531
log 870 -2- 939
0-470
The sum is (3 + 2) + (-531 + -939) = 1 + 1-470.
log 870 -2 -939
(6) log (870 -;- 0-0034) - log 870- log -0034
= 5-408
= log 256000
Jr408
2-56x10^256000
The difference is (2 - 3) + ('939 - -531) = 5 + -408.
log 0-0034 = 3-531
(o) log ^ - log 1 - log 930
= 3-032
log 0-00108
In this cue O'OOO - 2*968
log 1 = 0-000
log 930 = 2-968
3032
1-08x10 ~ 3 = 0-00108
1-000 - 3-968
0-093 = 9-3 x 10-'
3 log 0-093= 2-968 x 3
= 4 904
8 02 x 10 - 4 = 0-000802
(d) log (0-093) 3 = 3 log 0-093
= 4-904
_= log 0-000802
Hero 2-968 x 3 - 2 x 3 + -968 x 3 - 6 + 2'904
(0 lo g ^(0-093) - I of log 0-093 log 0'093 = S + 1 968
= 1-656 ^ of log 0-093 = 1-656
= log 0-453 4-53 x 10 - J = 0-453
The negative characteristic must first be made diviaible by 3.
7. Calculate the value of :
(i) 1050 x 0-00034 ;
(iii) 1/8700 ;
(ii) 0-87 ^ 930 ;
(iv) 1/0-087.
288 ALGEBRA
, 8. Calculate :
(i) (o-093> 4 ; (ii) ^(0-093) ; (iii) ^0-093 ;
(iv) (0-00087)' -r (o-oios) 4 ; (v) 1/^(0-93).
9. Calculate:
(i) 0-105 x 9300 - 87 ; (ii) ro-s
..... i ,. x
(111) ; (iv) 0*34 x -
34 x 87 \ ; M .
10. Calculate :
(i) V{<3-4)> x 93} ; (ii)
(iii) V^'OS x 93<> -f N/(o-o87)}.
EXERCISE LVII.
THE USE OF TABLES OF LOGARITHMS.
Note. These examples are to be solved by means of a
table of four-figure or five-figure logarithms.
A.
1. Write down the logarithms of the following numbers :
(i) 4-283 ; (ii) 42830 ; (iii) 0-0004283 ; (iv) 2035 ; (v) 20-35 ;
(vi) 0-7603 ; (vii) 530-2 ; (viii) 1007 ; (ix) 001007 ; (x) 8 ;
(xi) 10,000; (xii) 000001.
2. Write down the antilogarithms of the following numbers :
(i) 0-2430 ; Jii) 4-2430 ;_ (iii) 2*2430; (iv) 3*9124; (v)
0-0325 ; (vi) 3-0325 ; (vii) 1-0047 ; (viii) 0-0006 ; (ix) 3*0006 ;
(x) 5-2.
3. Use the logarithms of No. 1 to find the value of :
(i) log (4-283 - 20-35) ; (ii) log (20*35 x 0*001007) ; (iii)
log (530*2 + 0-0004283) ; (iv) log (10 ~ 0-001007) ; (v) log
(1 ~ 10007) ; (vi) log {1 - (530-2) 2 } ; (vii) log (0*7603) 3 ;
(viii) log (0-001007) 4 ; (ix) log ^42830 ; (x) log ^/0-0004283 ;
(xi) log -yO'7603 ; (xii) log yO-001007. [Leave a line blank
below each result.]
4. Complete the calculations of No. 3 by filling in the
antilogarithm of each result.
5. Find to four significant figures the value of :
3-28 x 724 . 0-4036 .
2308 ' W 2701 x 43'
? j8 ' 32 x ' i0g3N -
-
428-5 x 3*027 > (7*8)
____ ___
{6*237 x v/o-3856} 3 '
289 19
290 ALGEBRA
B.
Note. Assume IT = 31416, log TT = 0-49715.
6. Use the formula V = -J irr 3 to find (i) the volume of a
sphere of radius 247*6 cms. ; (ii) the radius of a sphere whose
volume is 48*2 cubic feet.
7. Calculate (i) the amount, (ii) the present value of 340
for 16 years, allowing 4 per cent per annum compound
interest.
8. How many years will it take for 1760 to amount to
3179 2s. at 3 per cent per annum compound interest ?
9. The present value of a sum of 1510 16s. due in twelve
years' time is stated to bo 890. What is the rate of
interest ?
10. Find (to four significant figures) (i) the value of (2-7)' 20 ;
(ii) the sum of 20 terms of a G.P. of which the first term
is 1'3 and the Common ratio 2 -7.
11. Find (i) the sum of the first 30 terms of the series
1 -f 0*8 + 0*64 + . . . ; (ii) the sum of the second 30 terms
of the same series ; (iii) the limiting value of the sum of all
the remaining terms of the series.
12. What percentage of the maximum sum of terms of the
series 20 + 17 + 14-45 + ... is included in (i) the first 25
terms ; (ii) the first 50 terms ?
13. A whole number is composed of n + 1 digits. What
is the characteristic of its logarithm ?
14. A certain number is a decimal fraction less than unity.
There are n - 1 noughts between the decimal point and the
first significant figure. What is the characteristic of its
logarithm ?
15. How many digits are there in (i) 2 100 ; (ii) (37) 500 ;
(iii) (132-73) 15 ?
16. How many noughts are there before the significant
figures in (i) (J) 1 ; (ii) GM 500 ?
17. Which is the first power of 2 whose value is above
1000 ?
18. Show that if P^Q then 1/Q^1/P'.
19. Which is the first power of j whose value is below
0-001 ?
20. How many terms of the series 1 + 3 + 9-1- 27 + . ,.
lie between 1Q 4 and 10 5 ?
EXERCISE LVII 2VU
How many terms of the series 1 -f- 4 + J. 4. .\ 4.
lie between 10' 4 and 10~ 5 ? a '
21. How many terms of the series 1 + + 4 , jj e
between 10~ 3 and 1CT 4 ? '
22. The first of a series of n terms in G.P. is a and the
common ratio is r. Show that r" = R(r - l)/a + 1.
23. What is the smallest number of terms of the series
1 + 4-2 + 17-64 + . . . whose sum exceeds 100,000? [Use
I>o. 22.]
24. What is the smallest number of terms of the series
I + 0-9 + 0'81 + . . . whose sum exceeds 8 ?
C.
25. I want to buy a certain house. Instead of paying for
it all at once 1 arrange to pay 110 for it at the end of each
year for seven years. Assuming that the seller expects to
receive interest at 5 per cent per annum, what is the price of
the house ?
26. I borrow a sum of 1250 from a Building Society on
the security of a house. It is to be paid back in twenty-five
half-yearly instalments, 2 per cent interest being paid per
half-year. Find the amount of the half-yearly instalment.
27. A man sets aside out of his salary 20 every quarter.
He deposits it in a financial institution which allows him
compound interest at the rate of 4 per cent per annum that
is | of 4 per cent per quarter paid quarterly. What is the
value of his investment at the end of seven years ?
28. Calculate (i) the cost of an annuity of 100 a year
paid half-yearly for six years, interest being reckoned at 3 per
cent ; (ii) the sum to which the annuity would amount by
the end of the period if the instalments, when paid, were
invested at 3 per cent compound interest paid half-yearly.
19
EXERCISE LVIII.
THE LOGARITHMIC AND ANTILOGARITHMIC FUNCTIONS.
A.
1. Draw two parallel and horizontal straight lines, and
graduate them both ways from zero to 10. Imagine a
point P to traverse the lower line from left to right, its dis-
tance at any moment from the zero point being called d. Let
another pointy move simultaneously along the upper line so
that its position always marks the value of antilog a d. Letter
successive positions of the former point P u P 2 , P 3 , etc., and
the corresponding positions of the latter p { , p 2 , p^ etc.
Choose the base yourself, but let it be greater than unity.
2. Below the lower scale in No. 1 draw a third graduated
line. Mark on it the positions P/, P 2 ', P 3 ', . . . which cor-
respond to the positions marked P lf P 2> etc., and p lt p< 2 , etc.,
in the case when the base is a number less than unity.
3. What peculiarities in (i) the logarithmic function, (ii)
the ant ilogarithrnic function, are illustrated by the movements
of the three points ?
4. Explain clearly why it is (i) permissible, (ii) useful, to
write
y s= a r instead of y = antilogy.
5. Given that y = (l'3) x find (by means of curve A, p. 273)
the values of y when (i) x = 4- 1*7, (ii) x = 4- 3*4, (iii)
x - - 2-35.
6. Write down the values of (i) (1-25) ~ 1 ' 3 , (ii) (0'8) 2 ' 7 ,
(iii) (1-1) ~ 3 * 15 , using the curves of fig. 50.
Note. Let there be a number h and two other numbers
a and b such that h p = a and h* = b, p and q being integers.
Also let k = l/p and k == l/#. Write down first the end-
less geometric sequence
292
EXERCISE LVITT 293
h " 3 , &- 2 , h ~\ 1, h, W,..., h p , .. , /*,... (A)
[a] [b]
and underneath it the two endless arithmetical sequences
...-9k, -2*. - fc, 0, fc, 2fc, ...,j*, ...,qk, ... (B)
[i]
... - 3k, - 2k, - k, 0, k, 2k, . . . , pk, . . . , qk, . . . (C)
[I]
Then, by the definition of logarithms, the terms of (B) are
the logarithms of the corresponding terms of (A) to the base
a, while those of (C) are logarithms to base b. Let x = h r be
one of the numbers in (A), then we have
Iog o; = rk and Iog 6 o? = rk.
Suppose, now, that given log a cc we want to find log b x or
vice versa. We have
rk
rk x v
ph
7. Prove, by a modification of this argument, that
log b x = Iog a ff/log a 6.
8. Find to base 10 the logarithms of the numbers whose
logarithms are (i) + 2-6 to base 2, (ii) 4*8 to base 5, (iii)
- 17'6 to base 0*46. Find also the numbers themselves.
9. Prove that if y = a* then log b y = x log^a.
10. Find by means of a table of common logarithms the
value of : (i) (10) 2 ' 36 , (ii) (10) - 3 - 72 , (iii) (100) - 2 - 75 , (iv) (27) 2 - 3 ,
(v) (82-1) - o- 46 , (vi) (0-034) ~ G ' 2 .
11. Let y' = logoff, y = log u x, and c = log^a. Then we
have proved that y' = cy.
Show that
6* = a? and that a = b c ,
and use the foregoing equivalence to prove that
(by - b cy .
Note. This result is important. The argument applies
equally to all kinds of values of c and y, integral and frac-
tional, positive and negative. It proves, therefore, not only
that fractional indices may be added and subtracted like
integral indices but also that the rule
294 ALGKBttA
(a" 1 )" = a mn
can be used when vi and n are both fractional.
12. Express each of the following as a single number with
a single power-index and find its value by the curves of
fig. 50: (i) {(1-3) - } 1 - 5 , (ii) {(1-3) - 2 ' 4 } - ' 5 , (iii) {(1-25) - 3 }i.
13. Find, by means of a table of common logarithms, the
value of : (i) {(10) - 3 - 4 } 1 -" 6 , (ii) {(2*3)-7} 3 , ^ {(Q-72) - 3 -*}a-.
14. Prove that x lln is a permissible way of expressing
JJ/aj, and x mln & permissible way of expressing both %/(x m ) and
! yx} m . Express in the index form : (i) ^(x 2 ), (ii) {/(x 3 ),
(iii){yx}\ (iv) {#*}*.
B.
15. Express the following relations in the form y ax m :
(i) y* = 9# 3 , (ii) ?y 3 - 27a? a , (iii) ?/ 4 = 16jJ 3 , (iv) x'tyb = 16.
Find, whenever possible, the value of y when x = 4- 10 and
when x =* - 10.
16. The relation between two variables x and y is of the
form
y 88 ax m
Putting Y = log y, X = log x and c = log a, show that the
same relation may be expressed iu the form
Y - mX + c
the logarithms being taken to any convenient base.
17. A number of corresponding values of two variables,
x and y, have been determined by measurement. When log y
( = Y) is plotted against log x ( = X) the resulting graph
is a straight line cutting the Y-axis at a point c above the
origin, and inclined to the X-axis at an angle whose tangent
is m. What is the relation between x and y ?
18. In a particular case the logarithms are taken to base
10, the line cuts the Y-axis where Y = 1-48 and is inclined
at 13 to the X-axis. Find the formula giving y in terms of x.
19. Find the relation between x and y in the following
cases : (i) the line cuts the Y-axis 0-8 above the origin and
cuts the X-axis where X = - 4 ; (ii) the line passes through
the points (0, + 1-85) and (+ 2-5, 0).
20. In an experiment the relation between the magnitudes
of two quantities d and H is believed to follow a law of the
form d = aH m . The following table gives a number of pairs
KXRRCISE LVTll 295
of values of the variables. Plot log d against log H and use
the graph to determine the constants a and m :
H 1, 2, 3, 4, 5, 6, 7, 8
d 2-2, 3-0, 3-6, 4-0, 4-5, 4 -9, 5-2, 5-6
21. In an experiment the following pairs of values were
determined for two quantities^ and u. Plot log p against log u
and use the resulting graph to determine the law which ex-
presses the dependence of p upon u :
u 1, 2, 3, 4, 5, 6, 7, 8
p 6-4, 3-7, 2-67, 2-15, 1-8, 1-54, 1-38, 1-23
C.
22. Show that any number x can be expressed in the form
a 10 ***, a being any positive number.
23. A table of logarithms is constructed in which the
logarithms increase by equal steps 1/n while the numbers
increase in a constant ratio (1 -f 1/n). What is the base of
the system ?
24. The base of the system described in No. 23 can be
calculated by arithmetic, but the calculation will be tedious
when n is large. On the other hand, it may be calculated by
means of a table of logarithms constructed upon any prin-
ciple. Use the table of common logarithms to calculate the
base in No. 23, (i) when n = 10, (ii) when n = 100, (iii)
when n 500, (iv) when n = 1000.
[Log 1-002 = 0-0008677; log 1-001 = 0-0004341.]
25. Obtain by direct calculation the square root of 10 cor-
rect to four places of decimals. From this obtain the fourth
root and then the eighth root. Taking these numbers as data,
obtain correctly to three decimal places the values of 10 I/S ,
10' 2 ' 8 , 10 3 ' 8 , . . . 10 7 ' 8 , 10 8/8 . That is, obtain the values of
antilog 10 0-125, antilog ]0 0-25, etc.
26. Plot the logarithms 0-125, 0-25, etc., against the anti-
logarithms, using as open a scale as possible. Read off from
your graph the logarithms of 1, 2, 3, ... 10. Compare
your readings with the value given in a table of common
logarithms.
27. Determine from your graph the values of (i) Iog 10 2*6,
(ii) Iog 10 84, (iii) Iog 10 0-38. Compare your readings with the
numbers in the table.
28. Simplify each of the following expressions :
296 ALGEBRA
(i) (aH-O^U-
' x (a'b -
i
(iv) (x' - y l) (x : ' + xV + y*)'
The student should observe that there are two typi-
cal ways of calculating logarithms. The first which is the
original method of Napier (c. 1594) is the one described in
No. 23 and illustrated in Ex. LV, Nos. 10, 13, 15. In this
method the value of the base cannot be foreseen but has to
be determined by calculation.
The second typical method is that of Henry Briggs who
suggested to Napier (1615) the advantage of taking 10 as the
base of the logarithms. In this method it is obvious that we
must start from the base. Briggs' procedure is in essence
that illustrated in Nos. 25, 26. That is to say, Briggs began
by repeated extraction of roots, starting from 10. Having
reached a number a very little above unity he built up his
table by a method analogous to that of No. 26. But the
number of root extractions, instead of being three was fifty-
four ! The logarithms corresponding to even distances in the
scale of numbers were determined by proportion and not by
graphic interpolation.
The tables of common logarithms in actual use are ulti-
mately derived from those of Briggs.
EXERCISE LTX.
NOMINAL AND EFFECTIVE GROWTH-FACTORS.
A.
1. Find the effective annual rate of interest' per pound and
per cent when the nominal rate is 5 per cent per annum, the
interest being added to the principal : (i) twice a year ; (ii)
every quarter ; (iii) twelve times a year.
2. Is it more profitable to invest in a concern which offers
interest at 2- per cent per annum and adds it to the principal
every half year, or in one which gives interest at 2 j per cent
per annum and adds it to the principal every quarter ?
3. A and B invest 1000 each, A in the former, B in the
latter of the concerns mentioned in No. 2. What is the dif-
ference between the values of their investments after three
years ?
4. Interest at the nominal rate of j per 1 per annum,
paid p times a year, is equivalent to an effective rate of i per
1 per annum. Show that
5. Use the formula of No. 4 to find the nominal rate equi-
valent to an effective rate of 21 per cent per annum, the
interest being converted into principal every half-year.
6. 1000 is invested in a concern in which the interest
earned is added to the principal every quarter. In five years
the principal has amounted to 1346 18s. Find the rate 'of
interest allowed per annum.
7. Write a formula for the present value (V) of a sum of
money (P) due in t years, interest being reckoned at j per 1
per annum convertible p times a year.
8. Show that the cost of an annuity of a per annum, pay-
297
29B ALGEBRA
able in equal instalments p times a year for n years, interest
being reckoned atj per 1 per annum, is given by the formula
B.
Note. Assume e = 2-718 and log e = 0-4343.
9. Interest at a nominal rate of (i) 3^ per cent per annum,
(ii) 5 per cent per annum is converted into principal every
moment. Find the equivalent effective rate in each case.
10. Interest at the nominal rate j per 1 per annum is
converted every moment into principal and is equivalent to an
effective annual rate of i per 1. Show that
j = 2-3 log (1 + i).
11. What nominal rate of interest is equivalent to an effec-
tive rate of 35*0 per cent per annum, when the interest is
added to the principal as fast as it is earned ?
12. Write a formula for the cost of an annuity payable
continuously, the total amount payable in a year being a.
Beckon interest at the nominal rate of j per 1 per annum.
[See No. 8.]
13. Interest at the nominal rate of 10 per cent per annum
is paid in four different cases : (i) half-yearly ; (ii) quarterly ;
(iii) monthly ; (iv) weekly. Calculate the effective annual
interest per cent in each case, using the following seven -
figured logarithms :
Number. Logarithm.
1-001923 0-0008346
1-008333 0-0036043
1-025000 0-0107239
1-050000 0-0211893
1-102500 0-0423786
1-103813 0-0428956
1-104718 0-0432516
1-105093 0-0433992
1-105170 00434295
[e =] 2-718282 0-4342945
14. Draw a rectangle about one centimetre wide and of a
convenient length (as great as the paper allows). Mark oft
areas to represent the respective values of the effective interest
calculated in No. 13.
15. A man invests a sum of money at compound interest, the
interest earned being added to the principal every week.
EXERCISE
Show that if the interest were converted continuously he
would possess at the end of a year only about 2d. more for
every 100 invested at 10 per cent per annum.
0.
16. State and prove a geometrical construction for de-
termining graphically the value of (1 + a/n) n for given values
of a and n, a and n being positive numbers.
17. Show by means of a figure that the construction also
applies when a is negative, so long as |a| <1.
18. Prove geometrically that (1 + a/n) 11 approaches a de-
finite value as n increases endlessly, a and n being both
positive.
19. Prove geometrically that (1 + a/n) n = {(1 + l/n) n }
when a is positive and n = oo .
20. State and prove a geometrical construction for finding
a, given the value of (1 + a/n) tl , a and n being both positive.
21. State and prove a geometrical construction for finding
a given the value of e a . Assume that a is positive.
22. Given that log 1-000001 ~ 0-000000434294, show how
to obtain an approximate value for 6.
LOGARITHMS.
(To be used in solving Nos. 1-6 above.)
Number. Logarithm.
1004167 0-0018060
1-005625 0-0024362
1-012500 0-0053950
1-015082 0-0064668
1-022692 0-0097448
1-025000 0-0107239
1*025156 00107900
1-050625 00214478
1-050945 0-0215800
1-051168 0-0216720
1-069632 0-0292344
1-077383 0-0323700
1-346900 0-1293354
SUPPLKMENTAU Y KX KKC1SES.
EXERCISE LX.
THE USE OF LOGARITHMS IN TRIGONOMETRY.
Note. When Napier invented logarithms (c. 1594) his
chief aim was to lighten the labour involved in trigonome-
trical calculations, and the first tables which he published
were tables of logarithms of sines. The present exercise is
intended to give practice in the use of such tables.
In some tables the logarithms of the trigonometrical
ratios are all increased by 10 in order to remove negative
characteristics. Such logarithms are called " tabular log-
arithms ". The symbol " L " is generally used (instead of
"log") to indicate a tabular logarithm. Thus log sin 40
= 1-8081, L sin 40 - 9-8081.
It is assumed in the following examples that ordinary, not
tabular logarithms are used. If this is not actually the case
" L " should be substituted for " log " in each question.
A.
1. Write down the logarithms of (i) sin 24 ; (ii) cos 36
30' ; (iii) tan 52 10' ; (iv) sin 72 20' ; (v) cos 48 40' ; (vi)
cot 15 ; (vii) sec 18 20' ; (viii) cosec 54 10'.
2. Find values of a such that (i) log sin a = 1-7622 ;
(ii) log cos a = 1-8365 ; (iii) log tan a = 0-0354 ; (iv) log cot a
= 2-6101 ; (v) log sec a = 0-0582 ; (vi) log cosec a = 00632.
3. Write down the logarithms of (i) sin 18 23' ; (ii) cos
18 23' ; (iii) tan 54 36' ; (iv) cot 54 36' ; (v) sec 37 45' ;
(vi) cosec 37 45'.
4. Find values of a such that (i) log sin a = T6630 ; (ii)
log cos a = f-6630 ; (iii) log tan a = 1*8140 ; (iv) log cot a
= 0-1836 ; (v) log sec a = 0-0340 ; (vi) log cosec a = 0-0960.
Note. Since no negative numbers have logarithms such
803
304 ALGEBRA
numbers must be replaced in logarithmic calculations by the
corresponding positive numbers. The correct sign of the
final result must bo determined separately.
5. Find the values of a sin a cos fi for each set of values
of the variables given in the following table :
0)
(yi)
(iii)
(iv)
(v)
a
14-8
-0'30
127
-3*281
1/38
a
27
131
76
248
307
ft
62
25
123
106
142
6.
Given
that
a - 22
and p =
29 find
the value
of
(i) log sec }(a + 0) ; (ii) log sin (2a -f 3/2) ; (iii) log (237 sin
a/sin /3) ; (iv) log {sin (2a - /3)/cos- (a + 2/3) }.
B.
7. Given the angles a and ft and the side a of a triangle,
show that the side b can be calculated by the logarithmic
formula
log b = log a + log sin ft - log sin a.
8. Calculate the angle y and the sides b and c in the
triangles in which the following values of a, /?, and a are
given :
a
ft
a
(i) 41
02
107
(ii) 07 30'
53 12'
19-6
(iii) 115 45'
20 15'
243-8
(iv) 12 17'
57 54'
82-28
9. Use logarithmic tables to calculate the remaining parts
of the triangles in which (i) A = 52, B = 73, and c = 427
yards ; (ii) A - 98, B - 36, and c - 1000 feet ; (iii) B -
108 20', G = 57 43', b - 1582 yards.
Note. A formula can be expressed in logarithmic form
only when it consists entirely of products, quotients, powers,
or roots. Thus the formula b = a sin /?/sin a can bo turned
into a logarithmic form ; the formula
cos a (6* + c 2 - a 2 )/26c
cannot. To obtain a logarithmic formula to replace the
latter we must turn to relations of the kind suggested in
Ex. XL, Nos. 20-22.
10. Show by means of the equivalence
EXERCISE LX 305
COS a = 2 COS 2 ^ - 1
fu f ' , a (6 + c) 2 - a 2
that cos j ^ = v rj
<*_+ C ^
and hence that cos? - J( a + 6 + c )
2 \ 4
11. Show that if s = (a + b + c) then the foregoing
formula becomes
a /
003 2 - V S
(s - a)
Express this formula in the logarithmic form.
12. The sides of a triangle are respectively 120, 240, and
180 feet long. Calculate the angles.
13. Demonstrate by means of the equivale ce
cos a = 1 - 2 sin 2 -~-
the formula
14. Deduce from the results of Nos. 11 and 13 formulae
suitable for calculating (i) tan ^ ( n ) sm by means of log-
A
arithms.
15. Show that the area of a triangle whose sides are given
can be calculated by the formula
A = J{s(s - a) (s - b) (s - c)}.
Note. For calculating the angles of a triangle, given its
sides, any one of the formulae for cos g, sin ^, and tan 5, will
suffice. The first is the easiest to remember.
16. Calculate the angles and the area of each of the tri-
angles whose sides are given in the following table :
a b c
(i) 47 53 82
(ii) 196 212 183
(iii) 23-62 74'8 62'04
(iv) 381-7 240-08 317'<36
20
306
ALGEBRA
C.
Note. It will be seen that the formula of No. 11 makes it
an easy matter to calculate a when a, b, and c are known,
but that the formula cannot be used with logarithms to calculate
a when b, c, and a are known. To obtain a logarithmic
formula suitable for dealing with this case we have to make
use of the rather complicated relation
tan
fi-v
- G a
co fc _
b -f c 2
Given b, c, and a this formula enables us to calculate ft - y.
But since ft + y = 180 - a it is easy to deduce the values
of ft and y. When these are known a can be calculated by
the logarithmic formula of No. 7.
Nos. 17 to 20 show how the formula for tan % (ft - y) can
be demonstrated. It is assumed that b>c.
B
FIG. 51.
17. In the triangle ABC (fig. 51) take AD = c on AC and
AE = c on CA produced. Join BD, BE. In the triangle
BAD show that /-ABD = ^ADB = 90 - a . In the tri-
angle ABE show that /-AEB = /-ABE = |a. Hence show
(i) that /-DBC = | (ft - y) and (ii) that /-EBC = 90 +
(/8 - y). [Kemember that 90 = | (a + ft + y),]
18. In the triangle BCD prove that
b - c a
sin -J (ft - y) cos
EXERCISE LX 307
19. In the triangle BCE prove that
b + c
_
cos (/3 - y)
20. From the last two results show that
fi - y b - c A a
____
ton
,___.
21. The following parts of a triangle are given. Find the
other parts in each case ;
(i) 6 - 24-8 c = 16-2 a = 42
(ii) 6 - 243 c - 328 a - 106 30'
(iii) a - 167 b -= 98 "6 y - 50 20'
(iv) c = 820-2 a = 004 '5 =-- 122
(v) a = 120 c - 38 j8 - 8 36'
22. In Ex. XLVI, Nos. 1-5, it was shown that when the data
are two sides of a triangle and the angle opposite to one of
them the position of this side is sometimes ambiguous. De-
termine (by the aid of figures) in which of the following cases
the ambiguity exists. Find, by the use of appropriate
logarithmic formulas, the remaining parts of each possible
triangle.
(i) a = 143 b - 63 ft ~ 38
(ii) a - 183 b = 206 ft - 38
(iii) b - 238 __ c = 307 y - 96
, (iv) b - H N /3 c - 33 ft - 30
EXERCISE LXI.
POLAR CO-OUDINATES.
Note. Let P be any point whose co-ordinates are x, y.
Let OP = r and the angle POX = a. Then r and a are
called the polar co-ordinates of P. When it is necessary
to distinguish x and y from the polar co-ordinates they are
called the Cartesian co-ordinates after the great French
philosopher and mathematician Descartes (1596-1650) who
invented their use.
1. If x, y are the Cartesian and r, a the polar co-ordinates
of a point P, show that x = r cos a, y = r sin a, and
x* + y 2 = v 2 .
2. When Halley's comet returned in 1909-10 it followed
a path which is most conveniently described by the follow-
ing polar formula, the origin being the sun. The. symbol
d = the comet's least distance from the sun (about 14 x 10
miles). Determine the values of r corresponding to values
of a for every 15 from to 360 and so plot the path of 'the
cornet :
r =
"0-97 cos a
3. Draw the graph corresponding to the polar formula
r = a(l - cos a)
assigning any convenient value to a. The curve is called
the cardioid.
4. Draw the HmaQOn whose polar formula is
r = b - a cos a.
[Different members of the class should choose different values
for a and b.] What does this curve become when a = b ?
5. Draw one of the two " three-leaved roses " whose
polar formulae are
r = a sin 3a and r = a cos 3a.
EXERCISE LXI 309
6. Draw one of the " four-leaved roses "
r a sin 2a and r a cos 2u.
7. State and account for the differences between the mem-
bers of each pair of " roses ".
8. Draw the lemniscate or figure-of-eight whose polai'
formula is
r 2 = a 2 cos 2a.
Explain its relation to the four-leaved rose.
9. Draw the conchoid whose polar formula is
r = a sec a 4- b
or r = a/cos a -f b
10. As is well known, a properly thrown boomerang, if it
does not hit its object, will return through the air towards the
thrower. Experiments have been made by Mr. G. T. Walker
to trace its path. The following table gives the results of
one such experiment. A, B, C, . . . W, are points on the
path of the boomerang ; r, a are the polar co-ordinates of the
points on the ground immediately below A, B, C, etc., r
being measured in feet ; h is the height of the boomerang in
feet, (i) Draw a plan showing a " bird's-eye view " of the
boomerang's flight, (ii) Join up the origin A with one of the
more distant points on the plan, by a straight line. Draw a
grciph showing the " elevation " of the boomerang's path as
viewed from a distant point along a perpendicular to that
straight line.
ABCDEFGHIJ
a 10 20 30 40 50 60 70 80 90
r 26-5 65*5 103 124 132 134 134 134 130
h 5 10 20 26 27 30 42 51 57 55
KLMNOPQRST
a 100 110 115 115 110 100 90 80 70 60
r 129 115 100 72 65 59 59 62 64'5 65
h 43 33 27 19 18 18 17 17 18 19
U V W
a 60 40
r 62 41
h 15 40
EXEECISE LXII.
SOME IMPORTANT TRIGONOMETRICAL IDENTITIES.
Note. Except in taking bearings in surveying it may always
be assumed that angles are measured by the anti-clockwise
rotation of a line about a fixed point.
A.
1. In fig. 52 OP and PQ are both vectors of length r. The
inclination of OP to the #-axis is /3, that of PQ, a. What is
Fia. 52.
their difference of direction APQ ? What is the angle AOQ ?
The magnitude of the resultant vector OQ ? Its inclination
to the #-axis ?
2. Find the co-ordinates of Q (i) by projecting OP and PQ
upon OX and OY ; (ii) by projecting OQ upon OX and OY.
Hence complete the identities
cos a -h cos ft . . ; sin a + sin ft = . . ,
310
EXERCISE LXTI
311
3. Answer the questions of No. 1 with reference to the
vectors in figs. 53 to 55, the length of each vector being r
and their inclinations to the #-axis a and /?. Use your
FIG. 53.
Y'
FIG, 54.
FIG. 55.
results to determine whether the identities of No. 2 hold good
in all the cases shown. Are there any other possible cases ?
If so, draw the appropriate figures and test the identities for
therq,
312
ALGEBRA
4. In fig. 56 OQ is a resultant of two vectors, OP and PQ.
The length of OQ and OP is in each case r, while their in-
clinations to the #-axis are respectively a and fi. Show that
the length of PQ is 2r sin - (a - /?), and that its inclination
to the tf-axis is 90 + % (a + (3).
X
FIG. 56.
5. Find the length of the projection of PQ upon the z-axis
(i) by taking the difference between the projections of OP and
OQ ; (ii) by projecting PQ directly, using the results of No.
4. Obtain similarly two equivalent expressions for the pro-
jection of PQ upon the 7/-axis. Hence complete the identities :
cos ft - cos a ; sin a - sin ft
6. What products involving angles less than 90 are equiva-
lent to :
(i) cos 23 -f cos 37.
(iii) cos 72 -f cos 48.
(v) cos 34 - cos 82.
Vii)
(vii) cos 342 - cos 128.
Verify any two results from the tables,
(ii) sin 23 -t- sin 37.
(iv) sin 123 + sin 76.
(vi) sin 164 - sin 56.
(viii) sin 342 - sin 128 ?
EXKltUlSU JLiXll 313
7. Express the following products as sums or differ-
ences :
(i) 2 cos 15 cos 32. (ii) 2 sin 87 sin 35.
(iii) 2 sin 43 cos 27. (iv) 2 cos 128 sin 53.
(v) cos 237 sin 92. (vi) 28 sin 17 sin 54.
8. Show that the following expressions are equivalent each
to the tangent or co-tangent of a certain angle :
C\ ^L43_t_^ n 6l ("\ s i fl 81 - sin 24
~~
(iii) ^3^J3in 5 / iv \ sin 5 - sin a
cos 30 4- cos 50' ^ cos 5a 4- cos a*
( v ) C S 2/3 - COS 2a . COSJ3 I - COS a
sin 2/3 4- 'Jii 2(1 ' ' sin a - sin ft '
9. Draw a figure like fig. 56 but let ft = 0. Use it to prove
that
cos a = 2 cos 2 x - 1
A
T ~ . a a
and sm a = 2 sin -^ cos -.
A A
Do the equivalences hold good for all values of a from to
360?
10. Draw a figure like fig. 56 but let ft 0. What
equivalences appear when PQ is projected upon the axes of x
and y ? Do they agree with those of No. 9 ?
11. Find the simplest expressions by which the following
may be replaced in a formula :
/ \ o <t / \
(i) cos- a - sin- a. (11)
x ' '
.....
(")
.
I + cos a
sin a ,. . cos 2a
I - cos a ' cos a - sin a
2 tan a , ., 2 tan a
(vi) -
(viii)
i - tan' 2 a"
sin a + sin 2a
I 4- COS a 4- COS 2a
12. Show that the equivalences of Nos. 9 and 10 can be
deduced from those of Nos. 2 and 5 by substituting for ft.
B.
13. In fig. 57 OP is a vector of length r making an angle
of ft with a straight line OA which is inclined to the ic-axis
at an angle a. OQ and QP are the vectors along and at
314:
ALGEBRA
right angles to OA by which OP may be replaced. What
are the lengths of OQ and QP ? Find two expressions for
the abscissa of P, first by projecting OP directly on to the
#-axis, secondly by projecting its component vectors OQ and
QP on to the #-axis. Hence prove that
cos (a + ft) = cos a cos ft - sin a sin /3.
14. Find in a similar manner two expressions for the
ordinate of P. Hence prove that
sin (a + ft) = sin a cos /3 + cos a sin /3.
15. Test the general validity of these equivalences by means
of figures in which OA and OP fall in various positions in the
four quadrants.
16. Draw a figure similar to fig. 57 but let OP lie within
the angle XOA, the angle AOP being /3 as before. Use this
figure to prove that
cos (a - ft) ~= cos a cos + sin a sin ft
and sin (a - ft) = sin a cos ft - cos a sin ft.
FIG. 57.
17. Use the equivalences of Nos. 13, 14, 16 to prove
tKat
EXERCISE LX1I 315
COS (a + /3) + COS (a - /?) = 2 COS a COS ft
cos (a - /?) - cos (a + ft) = 2 sin a sin ft
sin (a + ft) 4- sin (a - fi) = 2 sin a cos /?
sin (a + ft) - sin (a - fi) 2 cos a sin (3.
Do these identities agree with those of Nos. 2 and 5 ?
18. Deduce the equivalences of Nos. 13, 14, and 16 from
those of Nos. 2 and 5 by substituting A for -(a + ft) and B
for*(a-/3).
19. Use the identities of Nos. 13, 14, and 16 to find equival-
ences for
sin (90 + a), cos (90 + a),
sin (180 + a), cos (180 + a),
sin (270 a), cos (270 + a).
Do the results agree with those deduced in Ex. XLT, No. 16,
from the graphs?
20. Show that sin (45 a) = .- (cos a sin a)
and that cos (45 a) = ._ (cos a + sin a).
\/2
. C1U 1.1 L COS a + Sm a L SAKO \
21. Show that : = tan (45 + a)
cos a - sin a v '
_ . COS a - sin a /AKQ .
and that - = tan (45 - a).
COS a + Sin a x '
Verify the identities by means of the tables, choosing any
angle you please for a.
22. Show that cos a + 7. sm a = ^ +
^J3 COS a sin a
, ,, , x /3 COS a - sin a
and that ____ . _ tan (60 - a).
COS a + \/3 Sm a
23. Demonstrate the following equivalences :
... i + tan a _ , .
/.., i - tan a
V3 - tan a
310 ALGEBRA
21. Establish the following identities :
,. N tan a 4- tan ft , ,
-
(ii) .? * -_ t
v ' i + tan a tan /3
(Hi) _*H^- = tan 2a.
x ' i - tan 2 a
Show how the identities of No. 23 can be deduced from these.
EXERCISE LXIIt.
THE PARABOLIC FUNCTION.
A.
1. A ball is thrown into the air. Its vertical height in feet
above the point of projection is given by the formula
h = 72t - 16J 3 ,
the distance it has travelled horizontally from that point by
the formula d 12, I being the number of seconds since it
left the thrower's hand. Show by eliminating t that its path
through the air is a parabola. Assuming that the ball was
projected from a point 5 feet above the ground and that it
falls on the roof of a house 35 feet above the level of the
ground at the place where it was thrown, find the horizontal
range of the ball and its time of flight. Find the horizontal
range and time of flight if the ball falls into a hollow where
the ground is 14 feet below the level of the ground at the place
of projection. Find also the greatest height reached.
2. Suppose the ball in the previous question to be thrown
straight up a hill which has a uniform slope of $. Taking
as origin the point where the ball leaves tfie thrower's hand
(5 feet above the ground), write the formula which describes
the line of greatest slope. Find the co-ordinates of the point
where the ball strikes the ground and calculate the range
that is, the distance measured along the slope from the last
point to the foot of the vertical through the point of projection.
Note. Square roots may be calculated by the approxima-
tion method or taken from tables.
3. The ball in No. 2 is now thrown straight down the hill.
Calculate the range and the time of flight.
4. A ball is hit across a cricket field by a batsman. If
another ball had been thrown vertically with such speed as
317
318 ALGEBRA
to be throughout its flight on the same horizontal level as the
former its velocity would be given by the formula
v = 80 - 32*.
If a third ball had been rolled along the ground at a constant
velocity of 20 feet/sec, it would have kept directly underneath
the first ball throughout its flight. Give formulae for the
height (h) of the ball above the ground and its horizontal
distance (d) from the point of projection at the different mo-
ments of its flight. Find the formula for h in terms of d.
Find when and where the ball reaches its highest point and
calculate its horizontal range. (It may be assumed that the
ball was struck when on the ground and that the field was
level.)
5. If the ball had been hit as in No. 4 but straight up a hill
of slope 1/10, find (i) how far up the hill it would travel, and
(ii) its time of flight.
6. Answer the questions of No. 5 in the case when the ball
is hit down the hill.
7. Eliminate t from the relations y = at - bt 2 and x = ct.
Find the turning value of the resulting function and the values
of x and t to which it corresponds. Find the values common
to the resulting relation and the relation y = px + q. Show
that the values of t which correspond with these common
values of x and y can be obtained either from y = at - bt' 2
or from x = ct.
8. Two variables x and y are connected with a third variable
z by the relations x = 1 + 3z and y = 1 - 2z + 4^ 2 . Show
that y can be expressed as a parabolic function of x and find
its turning value.
9. Express y as a function of x, given that
4:Z 2
y = _- and that x = 2z - 3.
JL oZ
Find the values of x and z when y = + 1.
B.
10. Show that the parabola y = x 2 /p can also be described
by the relation r p tan a sec a.
11. Verify No. 10 by drawing the graph of r = 4 tan a sec a,
giving a values from to 360.
EXERCISE LXTTI 319
(Note. By r is meant the distance along OP from O to-
wards P; account is to be taken, therefore, only of positive
values of r.)
12. Move the parabola y = x 2 /p vertically through a dis-
tance p/4: downward if the parabola is "head down,"
upward if it is " head up ". Let the #-axis now cut the curve
in the points Q, Q'. Show that QQ' = p.
(Note.The line QQ' is called the latus rectum of the
parabola ; the point now at the origin is called its focus.)
13. Show that a parabola with its focus at the origin and
its latus rectum along the a;- axis can be described by the polar
equation
4r' 2 - (2r sin a + p)' 2 = 0.
Hence show that it can also be described by either of the
relations
r -- + ^-q ^-- - and r = - -.- P-~ -.
2(1 - sin a) 2(L + sin a)
In these formulae how will a parabola with head down be
distinguished from one with head up ?
14. Verify your conclusions in No. 13 by drawing the
graphs of both relations on the same sheet, one half of the
class putting p = + 6 in the former and p = - 6 in the latter,
the other half reversing theae substitutions.
EXEECISE LXIV.
IMPLICIT QUADRATIC FUNCTIONS (I).
A.
1. What is the relation of polar co-ordinates equivalent to
x* + y 2 ~ a 2
where a is a constant? What curve is described by this
relation ?
Note. A relation such as x 1 + y 2 = a 2 is said to express y
implicitly as a function of x. This means that substitution
of some value for x will not suffice to give the value of y.
To express the function explicitly we must write
y = V(a* - * 2 ).
Substitution of a value for x now suffices to give the value of
the function.
2. The centre of the circle x 2 + y 2 = 16 "is moved to the
point ( 3, + 4). To what implicit relation between x and?/
does it now correspond ?
3. The centre of the circle which corresponds to the re-
lation x 2 + y 2 = a 2 is moved to the point (g, /). Show that
it now corresponds to the relation x 2 + 2/ 2 -f %yx + 2/?/ + c =
where c = g l + f 2 - a 2 .
4. What are the graphs corresponding to the following
implicit functions ?
(i) x 2 + y a - 25 = o.
(ii) 9 (x* + y 2 ) - 16 = o.
(iii) x' 2 + y 2 - 4x + 6y - 12 = 0.
(iv) x 2 + y 2 -f I4x - 8y + i = o.
(v) x a -f y 2 + IDX -f 247 = o.
(vi) sx 2 -f sy 2 - 7x + 2 = o.
(vii) 2x 2 + 2y 2 - 5y = o.
(viii) x 2 + y 2 - 2ax + 2by - 2(a 2 + ab + b 2 ) = o.
(Note. When the graph of a function is a circle its radius
and position of its centre should be specified.)
320
EXERCISE LXTV 321
5. Show that No. 4 (iii) is equivalent to the pair of
explicit functions expressed by
y = - 3 V{25 - (x - 2) a }.
Hence prove that all possible values of y are included in the
range from 8 to + 2 and those of x in the range from - 3
to + 7. Show also that the values of the function are
symmetrical about - 3 that is, that for every value of the
function above - 3 there is a corresponding value an equal
distance below.
How does the graph of the function exhibit the truth of the
foregoing conclusions ?
6. Express No. 4 (i) and (vii) as explicit functions of x.
Use the results to determine the limits within which the
values of x and y must lie. Confirm your results by drawing
the graphs of the functions.
7. Show (i) algebraically, (ii) by considering what the
corresponding graph should be, that no values of x and y can
satisfy the relation x 2 + y' 2 + 9 = 0.
8. Apply alternately the two methods of No. 7 to find
whether any values of x and y can satisfy the following re-
lations :
(i) x 2 + ]r - 4x + 6y + 20 = o.
(ii) x 2 + y 2 - 7x - i8y + 100 = o.
(iii) x 2 + y 2 - icx + 47 + 29 = o.
(iv) x 2 -f y 2 -f 8x - I2y + 52 = o.
9. Find the values of x and y (if there are any) which
satisfy simultaneously the relation x 2 + y' 2 25 and the
following linear relations :
(i) y - 3(x + i). (ii) 4x - 6y - 13 = o.
(iii) sx - 47 + 25 - o. (iv) 9x -f 77 - 49 - o.
Illustrate your answers by the corresponding graphs, paying
particular attention to (iii).
10. Find whether the relation x 2 + y- - 4x + 6y - 12 =
has any values in common with the linear relations (i)
y = 3(x + 1) and (ii) 4# - 6y - 13 = 0. Illustrate by
graphs drawn on the same sheet as those of No. 9.
11. Show that the relations 3x - ty -f 25 = and
25# 2 + 25y* - lOOz + 15(ty - 1524 =
have only one pair of values of x and y in common (or two
identical pairs). Illustrate by a graph drawn on the same
sheet as those of No. 9.
21
322 ALGEBRA
12. How could you have foretold by considering the
functions No. 4 (i) and (iii) that the straight line
x - 6y - 13 = "
would pass through the intersections of their graphs? What
line will pass through the intersections of the graphs of No.
4 (iii) and (iv) ? Verify graphically.
13. Write down the linear relations which are satisfied by
the values of x and y (if there are any) which are common
to the following pairs of relations.
(i) x 2 4- y 2 = 4 and x 2 + y 2 + 4x - 2y - 4 = o.
(ii) x 2 + y 2 - 6x = o and x 2 + y 2 + 8y = o.
(iiij x 2 -f y 2 + 4x + 2y - 4 and x 2 + y 2 - lox + 2y + 10 = o.
14. Use the results of No. 13 to find the actual values of
x and y (where they exist) which are common to the various
pairs of relations.
B.
15. What connexion between x and y is obtained by
eliminating a from the relations x = a cos a, y = b sin a?
16. Use the results of No. 15 to plot the graph of the
x 2 ?/ 2
implicit function -^- + *j- - 1. (Here x = 3 cos a, y = 2
sin a. With O as centre draw a circle of radius 3. , Take
any point p on its circumference; join Op and draw the
ordinate pM.. Then if >OX = a, OM = 3 cos a and pM. = 3
sin a. Now if P is the point on the required graph which
corresponds to this value of a its abscissa should be 3 cos a,
that is equal to OM, but its ordinate should be 2 sin a, that
is | of >M. To obtain the graph, therefore, it is sufficient to
draw a number of ordinates of the circle, to mark points
two-thirds of the way up or down them from the #-axis, and
to draw a smooth curve through these points.)
Note. The graph of a relation of the form x 2 /a 2 + y' 2 /b' 2 = 1
is called an ellipse. It is obvious from No. 16 that its
greatest length is 2a and its greatest breadth 2&. These
lengths measure its major and minor axes. No. 16 also
shows that any chord POP which passes through O is
bisected at 0. For this reason O is called the centre and
the chords through O diameters of the ellipse. The circle
of radius a by the aid of which the graph is drawn is called
the auxiliary circle and a the eccentric angle.
EXERCISE LXTV 323
17. A piece of squared paper with axes drawn on it is fixed
so that the light from the sun falls on it perpendicularly. A
circular ring of radius a is held parallel to the paper and
a short distance from it, the centre of the ring being directly
between the centre of the sun aud the origin of co-ordinates.
The ring is now revolved through an angle /3 about the diameter
whose shadow would fall on the #-axis. Show by a figure
that the shadow of the ring (or the " projection " of the ring)
is the ellipse # 2 /a 2 + y' 2 /b' 2 = 1 where b a cos (3.
18. Use No. 17 to prove that the area of an ellipse is irab
where a and b are the semi-axes. (Imagine the circular ring
to be divided up into a vast number of strips peipendicular to
the diameter about wuich it is revolved. What happens to
the projected breadth, length, and area of each of these strips
when the ring is turned ?)
19. Hold the paper used in No. 16 so that the curve may
nrSi nil
correspond to the relation - + -= 1.
4 y
20. Indicate the shape and position of the graphs of the
following relations :
(i) x 2 + 9y 2 = 9. (ii) 9x 2 + 25y 2 = 225.
(iii) 4x 2 + 9y 2 = I. (iv) i6x 2 + y 2 = i.
(v) 2x 2 + sy 2 = 6. (vi) 7* 2 + 37 2 = 10.
21. Express No. 20 (ii), (iv) and (v) as explicit functions
of x. Find algebraically the ranges of the possible values of
x and of the function. Do they agree with the limits sug-
gested by geometrical considerations?
22. The centre of the ellipse described by No. 20 (i) is
moved to the point ( 2, + 1). To what relation between
x and y does it now correspond?
23. The ellipses described by No. 20 (v) and (vi) are moved
so that their centres coincide respectively with the points
(+ 3, - 2) and (- 1, - 1). To what relations do they cor-
respond in these positions ? Does either of the curves pass
through the origin ?
24. Show (by reversing the method of No3. 22 and 23) that
the graphs of the following functions are ellipses. State the
lengths and positions of their axes and the positions of their
centres.
(i) x 2 + 4y' J - 2x + 247 + 33 = o.
(ii) 9x 2 + 4y 2 + 36x - 24y + 36 = o.
(iii) 2x 3 + sy 2 - 28x + 6y -f 96 = o.
21 *
324 ALGKBRA
25. Determine whether the following r&awons are capable
of being satisfied by any values of x and y :
(i) 2x 2 + 3y 2 + 8x - 6y + 12 = o.
(ii) 5x 2 + 3y 2 + I ox - 247 + 60 = o.
26. Find the values of x and y (if there are any) which are
common to the following pairs of relations :
(i) 4x 2 + 9y' 2 = 36 and 2x - 3y = 6.
(ii) 4x 2 + 9y 2 = 36 and 2x - y + 5 = o.
(iii) No. 38 (i) and x + 2y + 3 = o.
Illustrate the results by graphs drawn on the same sheet.
EXEKCISE LXV.
IMPLICIT QUADRATIC FUNCTIONS (II).
A.
Note. Consider any point P at a distance r from the
origin O. Let OP revolve in the anti-clockwise direction
about O through an angle a and so carry P to a new position
P'. Let the angle P'OX be /?. Then the new co-ordinates
of the point are x = r cos fi and y = r sin ft while the old ones
were r cos (J3 - a) and r sin (p - a). But since
r cos (/3 a) = r cos /3 cos a + r sin ft sin a
and r sin (/? - a) = r sin ft cos a - r cos /? sin a
the old co-ordinates can be written
x cos a + y sin a and ?/ cos a - x sin a.
Let P be one of the points of a graph which corresponds to
a given relation between x and y. Let the whole figure be
rotated anti-clockwise about O through an angle a. Then the
relation between x and y which corresponds to the graph in
its new position may evidently be obtained by substituting
x cos a + y sin a for x and y cos a x sin a for y in the
original relation.
1. A graph is rotated through an angle a in the clockwise
direction about O. Show that the relation to which it
corresponds in its new position can be derived from the
relation to which it corresponded in the former position by
substituting x cos a - y sin a for x and y cos a 4- x sin a
for y.
2. If the circle x 1 + y' 2 = a 2 is rotated either way about
its centre O its appearance remains unchanged. The fore-
going substitutions ought, therefore, to make no difference
to the relation. Show by actual substitution that this is the
case.
3. The straight line y = a is obviously a horizontal tan-
825
326 ALGEBRA
gent to the circle # 2 + y^ = a 2 . Show, by rotating the figure
(anti-clockwise) through any angle a, that ail tangents to the
circle come under the general description
y = x tan a + a sec a
where a may be any angle from to 360.
4. Show by means of No. 3 that the line
y = p x + a Jl + y 2
is a tangent to the circle a? 2 + # 2 = & 2 for all values of p.
a 2 .
5. The rectangular hyperbola corresponding to xy = ~ is
A
rotated in the clockwise direction through 45. Show that
it now corresponds to the relation # 2 y' 2 a 2 . (Remember
that sin 45 = 1/^2"= cos 45.)
6. Show that the same hyperbola when rotated through
45 in the anti-clockwise direction corresponds to the relation
#2 _ y<2 = _ fit.
7. Express the implicit hyperbolic function x 2 - y 2 = 9
as an explicit function of x. Show that while the value of
the function has no limits x can have no values between - 3
and + 3. Compare these results with those obtained in the
case of the function # 2 + y 2 = 9.
8. Draw on one sheet the circle x 2 + y 2 = 9 and the
rectangular hyperbola x 2 - y^ = 9. (The easiest way to
obtain the latter is to draw the asymptotes making 45 with
the axes, to regard them as axes, and to draw xy = -| with
regard to them. See No. 5.)
9. Draw any line OPP' to cut the circle of No. 8 in P and
the hyperbola in P'. Let the angle P'OX = a. Prove that
the co-ordinates of F are 3 cos a/^cos 2 a - sin 2 a) and
3 sin a/^cos 2 a - sin 2 a). Compare these with the co-
ordinates of the corresponding point P upon the circle.
What limits are there to the value of a ?
10. Draw a series of double ordinates of the circle
x 2 -f 7/ 2 = 9 and of the rectangular hyperbola x 2 - y 2 = 9.
Take points on them (as in Ex. LXIV, No. 16) two-thirds up
or down each ordinate from the #-axis. Draw smooth curves
through the points. The curve derived from the circle is, of
course, the ellipse # 2 /9 -f 2/ 2 /4 = 1. Use the expressions in
No. 9 to show that the curve derived from the hyperbola is
the graph of the relation # 2 /9 - y 2 /4 =1.
11. Take points on the same ordinates (produced) four-
EXERCISE LXV 327
thirds of the way up or down them from the #-axis. Draw
smooth curves through the points. To what relations do
they correspond?
12. Points are taken on the ordinates of the rectangular
hyperbola x 2 - 7/ 2 = a 2 at distances from the #-axis which
are in each case b/a of the height of the ordinate. Show
that they lie on a curve which corresponds to the relation
/j.2 yl
__. _ -f. = 1. Show that the curve has asymptotes which
a 2 b*
make angles with the #-axis whoso tangents are respectively
+ h/a and - b/a.
Note. All these curves are called hyperbolas. They differ
from the rectangular Hyperbola in the fact that the angles
between their asymptotes is always greater or less than 90.
Remember that they are derived from the rectangular hyper-
bola exactly as ellipses are derived from their auxiliary circle.
If the line FO in No. 9 is produced to meet the other branch
of the curve in P" it is obvious that P'P" is bisected at 0.
Chords through *O are therefore diameters and O is the
centre of the hyperbola. The length 2& (which answers to
the length of the major axis of the ellipse) is called the axis
of the hyperbola. The length 26 (which answers to the
minor axis of the ellipse) is not really an axis of the hyper-
bola x' 2 /a 2 - y' 2 /b 2 = 1 ; but, as it is the axis of the conjugate
hyperbola x*/a? - y 2 /b 2 = - 1 it is called the conjugate
axis of the former hyperbola.
13. Show that the following are hyperbolic functions.
Indicate the positions of the corresponding curves ; give the
lengths of their axes and state the angle between their
asymptotes.
(i) x 2 - y 2 + 4x + 6y - 9 = o.
(ii) x 2 - 4y 2 + 8x + 247 - 24 = o.
(iii) 4x 2 - 9y 2 - 8x - 367 + 4 = o.
(iv) 3x 2 - 2y 2 + I2x + I2y - 12 = o.
14. Show that values of x and y can always be found
which will satisfy relations of the forms
(x + p) (2 (y + g? _ , (x + pY (y^qf _ ,
& P " a "a? "" "6" a ~~~ "" "
What limits (if any) are there to the values of x and y in
each case?
^ 15. Show that if a and b are positive a relation of the
328 ALGEBRA
form ax* - by 2 %gx 2/T/ c = can always be thrown
into one of the forms in No. 14 and hence will always be
satisfied by some values of x and y.
B.
16. The parabola y = j9 2 # 2 is revolved (i) clockwise through
90; (ii) anti-clockwise through 90; (iii) anti- clock wise through
180. Apply the substitution formulae to determine the rela-
tions describing it in each of these positions. Do they agree
with former results ?
17. The co-ordinates of a point P are p and q. The line
OP is rotated, anti- clock wise, through an angle a. Show that
the new co-ordinates of P are
p cos a - q sin a. and q cos a + p sin a.
Find the new co-ordinates of a point P, originally at (+ 5, - 10)
after OP has been rotated anti-clockwise through the angle
whose tangent is 3/4.
18. The line joining O to the centre of the circle
x 2 + y 2 - Wx + 20?/ + 109 =
is rotated anti-clockwise through the angle whose tangent is
3/4. Find (by the substitution of the Note to No. 1) the
relation corresponding to the circle in its new position. State
the old and new positions of the centre of the circle. Com-
pare the results with those of No. 17. Do they agree? Is
the radius of the circle unaltered (as, of course, it should
be) by the rotation ?
19. Find the relation corresponding to the parabola
y = 2#' 2 after it has been (i) rotated anti-clockwise through
30 and (ii) afterwards moved parallel to itself until its vertex
is at the point (- 3, + 10).
20. Give a formula descriptive of the ellipse 2# 2 -f 3y 2 = 4
after it has been (i) turned clockwise through 45, and (ii)
moved parallel to itself until its centre is at the point
(+ 4, - 2).
21. Find the implicit function of x whose graph is the
hyperbola 4# 2 - 3y* = 2 (i) rotated anti-clockwise through
the angle whose tangent is -J, and (ii) moved parallel to itself
until its centre is at the point ( - 1, - 1).
22. Find the relation between x and y the graph of which
is a rectangular hyperbola whose axis is 5 and is incline
the #-axis at an angle whose tangent is J.
KXKRCISK LXV 329
Note. Nos. 16-22, taken with earlier examples, bring out
the following facts about functions in which both x and y are
present in the second degree (and possibly in the first degree
also) :
(i) If the graph is a circle there can be no co-efficient of xy and
the co-efficients of x 2 and y 2 must be identical ;
(ii) If the graph is a parabola, an ellipse or a hyperbola, xy will
be present unless the axis is parallel either to the x-axis
or the y-axis ;
(iii) If the function is parabolic the three terms of the second
degree will form a perfect square ;
(iv) If the function is elliptic the co- efficients of x 2 and t/ 2 will
bo unequal (unless a = 45) but both of the same sign ;
(v) If the graph : > a rectangular hyperbola the co-efficients of
x 2 and y* v\ ill bo equal but opposite in s gii ;
(vi) If the graph is a non -rectangular hyperbola the co-efficients
of x' 2 and y' 2 will be opposite in sign and unequal ;
(vii) Values of x and y can always be found which will satisfy
a given function if it is parabolic or hyperbolic in form but
not always if it is circular or elliptic in form.
23. State what you can determine by inspection about the
character of the following functions :
(i) (3* ~ 4y) 2 - 7* + 8y - ii = o.
(ii) sx 2 + sy 2 - 4x + 6y + i = o.
(iii) sx 2 - 3y 2 - 4x + yy + i = o.
(iv) 9x' J + 4x7 + 7y 2 - 8x + 5y + 3 = o.
(v) 5x 2 + y 2 + 7* ~ 13 = o.
(vi) 2x' J - 5x7, - 3y 2 + 2x - 7y + 18 = O.
Note.' If the term involving xy has been brought into the
formula of a curve by rotating the curve anti-clockwise about
the origin, it can obviously be removed by rotating it back
through an equal angle.
24. Show that the relation 14# 2 - kxy + ll?/ 2 = 5 describes
an ellipse which has been rotated anti-clockwise through the
angle whose tangent is 2.
25. Show that the graph of the function
Ux 2 - 3Gxy - y 2 - 13 =
is a hyperbola which, if rotated clockwise through the angle
whose tangent is f , would have its axis coincident with the
26. The rectangular hyperbola 3x 2 + Sxy - 3y' 2 = 10 is
rotated backwards (i.e. clockwise) until its axis is coincident
with the rr-axis. Show that tne angle is given by the
equation
330 ALGEBRA
4(cos 2 a - sin 2 a) - 6 sin a cos a = 0.
Show that this equation can be written in the form
3 sin 2a = 4 cos 2a
Hence show that a = 26.i
27. Find the angle through which the ellipse
37x 2 - 18xy + I3y' 2 = 10
must be turned back so that its major axis may coincide with
the ic-axis.
28. Show that the graph of the function
ax 2 + 2hxy + by 1 + %# + 2/7/ + c =
will have its axis parallel to the #-axis if it is rotated clock-
wise through an angle a such that tan 2a = 2/&/(a b).
(Note that the terms %gx and 2/?/ affect only the position of
the centre, not the inclination of the axis.)
29. Show that the relation
5x 2 + 2xy + 5?/' 2 - 6, + ISy + 11 =
may be expressed as an explicit function of x in either of the
forms
y = - l(x + 9) i V{50 - 24(a; - I) 2 }
or y - - -J(# + 9)
1^(5 72 - 2 ^6 + 2 V6o;)(5 72 + 2 76 - 2 J6x).
How can you deduce from these expressions (i) that the line
y = _ ^(x + 9) bisects all vertical chords of the ellipse;
(ii) that it meets the curve at the points
(+ 1 + *V3, -1- JV3)
and (+ 1 - |V^ - 1 + i\/3) J ( m> ) fcnafc tne tangents at
these points are vertical ; (iv) that the tangents at the points
(+ 1, - 2 + ,/2) and (+ 1, - 2 - ^2) are parallel to the
line y = - ^(x -f 9) ? Use these results to draw with rough
accuracy the graph of the function.
30. As we have seen, the relation p 2 x - q 2 y 2 = r- repre-
sents a hyperbola with its centre at the origin. Show that
the graph of p 2 x 2 - q 2 y 2 = is a pair of straight lines through
the origin. What is % the connexion between this pair of
lines and the hyperbola ?
Find, by converting it into an explicit function of x, whether
the relation
% X 2 + xy - 6y' 2 + 7y - 2 =
corresponds to a hyperbola or a pair of straight lines.
EXEECISE LXVI.
MEAN POSITION.
A.
1. Fig. 58 represents a target 4 feet long at which a man
has been shooting with the object of hitting the vertical line
CO'. Find the mean distance of his hits from CO' by adding
the various distances (regarded as directed numbers) and divid-
ing the resultant by the total number of shots.
Note. The above directions for finding the mean distance
could be expressed by the formula x = (%x)/n where # = mean
distance from CC' and % = number of shots.
B
C
FIG. 58.
B
2. Let a line MM' be drawn across the target so that its
distance from CO' is the mean distance calculated in No. 1.
Show that the position of MM' can also be obtained by cal-
culating the mean distance of the hits from A A'.
3. What is the mean distance of the hits from MM'?
From a line 1'2 feet to the left of MM 7 ? From a point O6
feet to the right of CO' ? (These questions can be answered
without calculation.)
4. Fig. 59 shows the attempts of a marksman to hit the
centre of a target 4 feet square. Find the mean distance of
his hits from each of the lines YY', XX'.
Note. Mark in fig 59 a point whose co-ordinates are the
values of # and y calculated in No. 4. This point is the
381
332
ALGEBRA
mean position of the hits. It is called the centroid of the
points where the shots hit the target.
5. What is the distance of the centroid from the centre of
the target in No. 4 ?
Y
Y
FIG. 59.
6, When another marksman shot at the same target the
co-ordinates of the points at which he hit it were :
(- 1-2, + 0-6), (+ 21, + 0-4), (0, - 02), (- 0-7, - 0-5),
(+08, 0), (0, 0), (+ 01, - 0-3), (+ 0-9, + 0-6),
(- 1-3, + 0-4), (0, 0), (+ 06, - 0-4), (0, + 1). Calculate
the co-ordinates of the centroid and its dis^an^e* from the
centre of the target.
EXERCISE LXVI 333
B.
7. Take any triangle OAB and from O draw a perpendicu-
lar OP meeting AB in P. Let OP h and AB = Teh.
Imagine the surface of the triangle to consist of a very large
number of very small squares arranged in rows parallel to
AB. Let there be n squares in each unit of length. How
many squares will there be in the row AB ? How many in
a row distant x from O? How many in the whole triangle?
8. Assuming that the squares are so small that they may
be treated as points, show that the centroids of all the rows
lie on a certain straight line.
9. On the same assumption show that the mean distance
of the surface of the triangle from a line through O parallel
to AB is | ft.
10. What is (from Nos. 8 and 9) the exact position of the
centroid of a triangle ?
11. Find the centroid of a triangle by calculating the mean
distance of the elements of its surface from the base.
12. Apply the methods of Nos. 7-10 to find the centroid
of a hollow cone without a base.
13. Find the centroid of a solid cone. [The cone must be
thought of as built up of a vast number of very small cubes
or " elements of volume ".]
14. Show that the centroid of a pyramid on a square base
is three-quarters of the distance from the vertex to the cent-
roid of the base along the line joining them.
15. A cap is cut off from a thin india-rubber or metal ball.
Where is the centroid of the residue?
16. A conical shell (without a base) is fashioned from a
block of material in such a way that the thickness of the shell
at any point is proportional to its distance from the vertex.
Where is the centroid ?
17. Two sides AB, CD, of a quadrilateral are parallel and
are respectively 8 inches and 35 inches long. The perpendi-
cular distance between them is 9 inches. What is the length
of a line x inches from AB ? Where is the centroid?
18. On a sheet of paper outline any two figures of area A l
and A 2 . Mark two points, G l and C 2 , to represent their cent-
roids. Let the co-ordinates of G 1 and C 2 be respectively
($ v $j) and (d? 2 , f 2 ). Let C be the centroid of all the elements
of area included in the two figures, and let its co-ordinates be
334 ALGEBRA
X, Y. Show that X = (A fa + A^)/^ + A 2 ) and that
Y = (A^ + A 2 y 2 )/(A 1 + A 2 ).
19. Supposing that you know X, Y and x v y v by what for-
mulae would you calculate x 2 , $ 2 ?
20. Find the centroid of a figure which consists of an
isosceles triangle of height h standing on one side of a square
measuring a each way.
21. Instead of standing on the side of the square the
isosceles triangle has been cut out of the square (h being < a).
Where is the centroid of the remaining area ?
22. A figure is composed of two circles which touch one
another externally, their radii being 5 inches and 12 inches
respectively. Where is the centroid ?
23. In another figure the smaller circle has been drawn so
as to touch the larger internally and has then been cut out.
Where is the centroid of the remaining area?
24. A portion of a cone has* been removed by a section
parallel to the base. The radius of the top is 10 cms., that
of the base 24 cms., while the line joining their centres is 42
cms. long. Write a formula for the area of a section distant
x from the top. Hence calculate the position of the cent-
roid.
25. Find by measurement and calculation the position of
the centroid in figs. 11 and 12 (p. 34).
C.
Note. The Greek mathematician Pappus (c. 350 A.D.)
and, in modern times, the Swiss Guldinus (c. 1635) demon-
strated two important theorems of mensuration :
Imagine the curve P (fig. 60) to revolve in space about
the external axis YY', so as to mark out a solid ring-like
figure. L- 1 I be the perimeter of the curve and A its area.
Also let G! be the centroid of the perimeter of the curve and
G 2 the centroid of its surface. Then we have
surface of ring- solid = / x circumference of circle traced
out by Cj . . (I)
volume of ring- solid = A x circumference of circle traced
out by C ;2 (II)
Nos. 26-28 give the argument leading up to I, Nos. 29-31
that leading up to II.
EXERCISE LXV1
335
26. Suppose the perimeter of the curve to be divided up
into a large number (n) of equal
segments of length SZ. Mark
the mid-length of each segment
by points of which P is a speci-
men. Let the distance of P
from the axis be x, and let the
distance of G v the centroid, be
x. Prove that
FIG. 60.
27. Write down an expres-
sion for the area of the surface
traced out by the segment P on
the assumption that it is a straight
line of length SI. Note that by
taking SI small enough this as-
sumption may be made true to
within c per cent for every seg-
ment, c being chosen as small as
we please.
28. Hence show that
surface of ring- solid = 2nx . I
to any nameable degree of accuracy.
29. Next suppose the surface of the curve to be mapped
out into n squares of area ^A of which Q is a specimen. Let
the length of the side of the square be 2h and the distance
of its mid-point from YY' be x. Also let the distance from
YY of C 2 , the centroid of the area, be x. Show as before that
^lirX = %-nllX.
30. Find an expression for the volume of the solid of
square section traced out by the revolution of Q.
31. Hence show that
volume of ring-solid = 2?r# . A.
32. The cross-section of the iron of an anchor-ring is a
circle of radius r. The diameter of the whole ring is 2E.
Write down formulae for finding (i) its surface A and (ii) its
volume V.
33. Calculate the surface and volume of an anchor-ring in
which r = 3 inches and R = 8*2 inches. [Assume ?r 2 = 10.]
34. ABC is an isosceles triangle in which BC = 10 inches
and AB = AC = 13 inches. A solid figure is produced by
revolving the triangle about an axis 16 inches from, and
336 ALGKBRA
parallel to, its base. Calculate the area and volume of the
solid.
35. Fig. 27 (p. 94) is identical with the cross-section of
the rim of a wheel. The mean radius of the wheel is r.
Write a formula for calculating its area and its volume.
36. An anchor-ring has an elliptical cross-section, the major
axis being 4 inches and the minor 3 inches long. The mean
radius of the ring is 6 inches. Find its weight, given that
1 cubic inch of iron weighs 0'28 Ib. [See Ex. LXIV, No. 18.]
EXERCISE LXVII.
ROOT-MEAN-BQUARE DEVFATION.
A.
Note. If a measure is required of the inaccuracy of the
shooting in l^x. LXVT, No. 1, it must be noted (a) that a devia-
tion (any) of 1 foot to the right of the centre of the target and a
deviation of 1 foot to the left have exactly the same importance,
and (b) that a deviation of 1 foot, is more than 12 times as
serious as a deviation of 1 inch that is, the badness of a shot
must not bo measured simply by the distance of the point hit
from the point aimed at. For both these reasons it is usual,
in estimating the departure of the hits from perfect accuracy,
(i) to square their actual distances from the point aimed at,
(ii) to find the mean of the squares, and (iii) to take the square
root of the mean. For by squaring the distances a negative
distance is rendered of the same importance as a positive dis-
tance of the same magnitude, and larger deviations acquire
relatively more influence upon the estimate than smaller
deviations. The result is called the square root of the
mean square of the deviations or, more concisely, the
root-mean-square deviation.
1. Calculate the square root of the mean square of the
deviations from CC' in fig. 58.
2. In fig. 59 find the square root of the mean square of
the distances of the points hit (i) from XX', (ii) from YY'.
3. Show that the mean square of the distances of a number
of points from the origin is the sum of the mean squares of
feheir distances from the axes. Apply this principle to calculate
the square root of the mean square of the distances of the hits
from the centre of the target in fig. 59.
4. Calculate the square root of the mean square of the
distances from the centre of the hits recorded in Ex. LXVT,
No. 6.'
837 22
338 ALGEBRA
5. In an examination in arithmetic ten questions were set
and the answers were marked as either " Right " or " Wrong ".
The class was divided into two divisions, A and B, each con-
taining 15 pupils. The following table gives the number of
pupils in each division who obtained 10, 9, 8 ... correct
answers. Calculate (i) the mean number of correct answers
in each division, (ii) the square root of the mean square of
the deviations from perfect accuracy in each division. Which
division did best ? Why do you think so ?
No. correct: 10 9876543210
A: 10115402010
B: 30022310310
B.
Note. In studying the behaviour of rotating bodies and
in other problems it is often necessary to answer questions
like Nos. 6-20.
6. On a line I cms. long are strung m -f 1 equal spherical
beads touching one another. Their centres are d cms. apart
that is, I = md. Show that the root mean square of the
distances of the centres of the beads from the centre of the
end one is
I
7. What is the root-mean-square of the distances of the
points of a straight line of length I from one of its ends ?
8. Find the root-mean-square of the distances of the
points of a line of length I from its mid-point.
9. Calculate the root-mean-square of the distances of all
the points of the rectangle AB' (fig. 58) from the line CC'.
10. Calculate the root mean square of the distances of the
superficial elements of a soap bubble (i) from a tangent
plane, (ii) from a plane through its centre.
11. OAB is any triangle. The perpendicular distance
between O and AB is h. A number of equidistant lines are
drawn across the triangle parallel to AB. Including AB
and the line of zero length through there are m + 1 of them
altogether. Show that the root- mean- square of the distances
of all the points on these lines from a line through O parallel
to AB is
EXERCISE LXVII 339
12. What is the root- mean-square of the distances of the
points of the triangle OAB from the line through O parallel
to AB?
13. Find the root-mean-square of the distances of the
points of the figure in Ex. LXVI, No. 17, from the side AB.
14. Repeat the last investigation substituting CD for AB.
Does the answer agree with that of No. 13?
15. A figure is composed of a series of concentric circles
whose radii (if the centre be regarded as one of the circles)
are in direct proportion to the numbers 0, 1, 2, . . . m, the
radius of the largest being r. Find the root-mean-square of
the distances from the centre of all the points on the circum-
ferences of these circles.
16. What is the root-mean-square of the distances from
the centre of all points in the area of a circle of radius r ?
17. Deduce from the result of No. 16 by means of the
principle of No. 3 the root-mean-square of the distances of
the points within a circle from one of the diameters.
18. Find the root -mean-square of the distances of the
points on the circumference of a circle from one of the
diameters.
19. Show that the root-mean-square of the distances of the
points on the surface of a sphere from a plane through its
centre is rj ^3.
20. Calculate the root-mean-sqnare of the distances of the
points in fig. 14 (p. 35) from the line joining two opposite
vertices, assuming a 2, b = 1. Deduce the root-mean-
square of the distances from the centre of the figure.
EXERCISE LXVIII.
THE BINOMIAL THEOREM.
Note. If we want to know exactly how much a sum of
money P will amount to in n years at i per pound per annum,
compound interest, we must use the formula A = P (1 + i) n
and evaluate the second factor either by direct calculation or
by the aid of logarithms. But if an approximate value will
suffice it can be obtained by " expanding" (1 + i) n in ac-
cordance with Stifel's table (Ex. XXXI) and replacing (1 + i) n
by as many terms of the expansion as will suffice for the degree
of accuracy in view. Thus, suppose that we want to know
the amount of 100 for 6 years at 3 per cent, compound
interest. By Stifel's table we have
whence
(1-OS) 6 (1 + 0-03) = 1 + 0-03 x 6 + (0-03) 2 x 15 +
(0-03) 3 x 20 + ...
1 + 0-18 + 0-0135 + 0-00054 + . . .
= 1-19404 +
.-. A = 1-19404 x 100 +
= 119 8s. +
The last figure of the approximate factor is ignored because
the next term of the series might affect it. (This term is, in
fact, 0-00001215 a number which affects the ultimate re-
sult to the extent of rather more than a farthing.)
This method of approximation can always be used provided
(i) that i < 1 and (ii) that n is a positive integer. If i were
> 1 the approximation would be impossible because the terms
would, as a rule, increase successively instead of decreasing.
If n is not a positive integer Stifel's table will not enable us
to determine the coefficients of the terms.
EX K ROTS E LXVIIT 341
In dealing with a long term of years Stifel's method of
finding the coefficients required for the expansion of (1 + i) n
is inconvenient. Fortunately Sir Isaac Newton discovered
(about 1665) a formula by which the coefficients required for
a given value of n can be determined without reference to
those needed for other values. His rule is that the co-
efficients of the terms (including the first) are :
n n n-1 n(n - 1) n- % n(n - 1) (n - 2) n - 3
1> l x J> j x 2~ "T7T" X ~3~' 1.2.3 X T~'
Thus when n = 6, the coefficients are 1, 1 x |, 6 x [I,
15 x |-, 20 x |, 15 x |, 6 x . The law of the decreasing
numerators and increasing denominators of the successive
multipliers is obvious.
The products which constitute the denominators of these
coefficients are called factorials. Thus the product 1.2.3
... r is called factorial r. It is generally expressed by
the symbolism [_r or r !
In honour of their discoverer the above expressions will be
called the Newtonian coefficients. It is most convenient
to refer to the first (which is always unity) as the " zeroth,"
and the others as the first, second, . . . Newtonian co-
efficients.
A.
1. Expand (i) (1 + i)\ (ii) (1 + i)*, (Hi) (1 + i) l by
moans of Newton's rule and compare the results with those
obtained by Stifel's table.
2. Write down the formulae for (i) the r th Newtonian co-
efficient (c r ), (ii) the (r + l) th term of the expansion of
(1 + *).
3. Show that c r = n l/r I (n - r) I Which other coeffici-
ent is described by the same expression ?
4. Find by algebraic multiplication the product of
n
> _
T I
by (1 + i), arranging the result in order of the powers of i.
Note. No. 4 is very important for it shows that if Newton's
formula holds good for the expansion of (1 + i) n it also holds
342 ALGEBRA
good for the expansion of (1 + i) n+l . But we know by direct
multiplication that it holds good for (1 -j- t) 2 . Hence it holds
good for all positive integral values of n.
5. Write out from memory the argument of No. 4 and the
preceding note as a proof by recurrence (or " mathematical
induction ") that Newton's expansion of (1 -f i) n is true for
all positive integral values of n.
6. Use No. 3 to prove that, when n is a positive integer,
(i) the expansion of (1 + i) n contains n + 1 terms ; (ii) if n
is even there is a middle term and the Newtonian coeffici-
ents of the terms at equal distances on each side of it are the
same ; (iii) if n is odd there are two middle terms whose
Newtonian coefficients are the same.
7. Calculate to the nearest shilling how much 1000 will
amount to at compound interest (i) in ten years at 4 per cent
per annum paid annually ; (ii) in twenty years at 3 per
cent per annum paid annually ; (iii) in thirty years at 4 per
cent per annum when the interest is added to the principal at
half-yearly intervals.
8. Find to the nearest pound the compound interest on
100 for 100 years at 2 per cent per annum when the interest
is added to the principal every quarter.
9. Write down the first five terms of the expansion of (i)
(1 - 2#) 10 ; (ii) (2 + #) 8 ; (iii) (3a - 4a) 6 ; (iv) (1 - ^a?) 20 ;
(v) (2* + 3) 7 .
10. Write down (i) the fourth term in the expansion of
(2 - 3#) 10 ; (ii) the middle term in the expansion of
/ 1 \ 8
(a - bx 2 )*; the term independent of x in f 2# - x- ) \ (iv)
the term independent of x in (ax 1 -f bjx) 6 ; (v) the middle
terms of (2a - 3te) 10 .
B.
Note. Let V be the present value of a sum of P for n
years at compound interest i per pound, paid annually, then
we know that
V = P(l + i) ~ n .
To calculate present values approximately we need, there-
fore, a rule for expanding (1 + i) ~ n when n is a positive
whole number.
A suggestion towards such a rule will be found in Ex.
EXERCISE LXVIIT 343
XXXVIII, Nos. 11-14, where it was shown that the approxi-
mate values of certain negative powers of the number 1 - i
can be calculated by the formulae
(1 - i) - l - 1 + i + i* + i 3 + t* + . . .
- -2 = 2 3 4
The principle by which the coefficients are determined is
derived from Ex. XXXVIII, No. 11. Let the coefficients
in any one of the lines be, in order, 1, a 1} a 2 , a 3 , a 4 , . . . Then
the corresponding coefficients in the next line are
1, (1 + aj), (1 + aj + Og), (1 + ^ + a 2 + a s ),
(1 + ! + a 2 + a 3 + a 4 ), . . .
If we call these coefficients 1, A v A. 2 , A 3 , A 4 , . . . then it is
obvious that
A 1 = 1 + a A 2 = A! 4- a,, A 3 - A 2 + a 3 ,
A 4 = A 3 + a 4 , etc.
Thus the coefficients in one expansion are derived from
those of the preceding expansion by a law very similar to
that of Stifel's table. Any coefficient is simply the sum of
the coefficient directly before it in the same expansion plus
the coefficient directly above it in the preceding expansion.
For instance, 10, the coefficient of i 3 in the expansion of
(1 - i) ~ 3 is the sum of the 6 immediately before it and the
4 immediately above it.
11. Starting with the row
1, 1, 1, 1, 1, 1, 1,
which may be regarded as the coefficients of the first seven
terms of the expansion of (1 - i) ~ 1 , make a table of the co-
efficients of the first seven terms in the expansion of (1 - i) ~ n
where n = 2, 3, ... 7.
12. Now note that the expansion of (1 + i) ~ n can be ob-
tained from the expansion of (1 - i) ~ n by substituting - i
for i. Use this principle to obtain the first seven terms in
the expansions of (i) (1 -f i) ~ 4 , (ii) (1 + i) ~ 7 .
13. Verify that the coefficients in any of the rows of the
table in No. 11 obey the following law of derivation :
- ^ n-fl n(n+l) n+2 n(n+l) (n + 2) n+3 A
l,lxn,nx -- , x -^ - x ' -y 3 j - ~ x ~f "> etc -
In other words, verify that if n = 1, 2, 3, ... 7,
344 ALGEBRA
n(n+ 1) (w+_2) (n,+ 3)
+ .^- a + . . .
14. Assuming that the foregoing expansion holds good for
a given value of n show by applying the rules
AJ = 1 + a 1? A 2 = A! + a 2 , A 3 = A 2 + a 3 , etc.,
that it holds good also for n + 1.
15. Obtain an approximation -expansion for (1 + i)~ n by
substituting - i for i in the expansion of No. 13.
Note. There is an extremely important difference between
the expansions of Nos, 4 and 15. The expansion of (1 i) n
has a definite number of terms, namely n + 1. It can,
therefore, be used, not only for approximations, but also for
calculating the exact value of (1 i) n whatever be the value
of i. On the other hand, the expansion of (1 i) ~ n has no
definite number of terms but is endless. Thus it can be
used for approximations only. Moreover it can be used
for this purpose only on condition that a definite number of
terms is sufficient to give the result to the required degree of
accuracy and that all the subsequent terms may be ignored.
The best way to make sure that this condition is fulfilled
is to calculate the value of the complementary fraction (see
p. 197) after a given number of terms. Thus we know
(p. 199) in the case of the expansion of (1 - x)~ l that after
the term x r has been calculated there is a complementary
fraction a? 7 + 1 /(1 - x). Since the value of this complement
can easily be calculated, we can tell in any given case how
much the value of the approximation is affected by ignoring
it. Again we found (p. 215) that in the expansion of
(1 - x) ~ 2 this complementary fraction is
(r+l)af + l x r ^ 1
1 - x + (1 - x)*
which can again be calculated, but not so easily. Similarly,
the complementary fraction could be found for other values of
n. It is to be observed, however, that this method of finding
the degree of accuracy of an approximation-expansion is not
only troublesome in practice but also theoretically unsound.
For, as will be seen from the two examples just considered,
the complementary fr.iction contains the very expression
KXKRCISE LXVIII 345
which it is required to expand. It is better, therefore, lo
make use of a simpler, though indirect method.
16. Suppose in a given case that the expansion of (1 - i) " n
has heen calculated by the formula of No. 15 as far as the
term containing^" 1 . Show that the next term is derived
from this one by multiplying by the factor
and that this factor decreases as r increases.
17. The expression (1 - 0-05) ~ 4 has been expanded as
far as the fifth term (i.e. the term containing the fourth power
of 0'05). What is the factor for calculating the sixth
term?
18. Show that the sum of all the terms after the fifth is less
than the sum of the series
0-0000175 x {1 + 0-08 + (0-08) a + (008) 3 + . . .}
i.e. less than 0-0000191.
19. Show that the error involved in using the first five terms
of the expansion of (1 + 0*05) ~ 4 as an approximation-formula
is less than 0-0000163.
20. What sum of money invested at 5 per cent per annum
compound interest, paid annually, will amount in four years to
1000? Use enough terms of the expansion to give the
answer correctly to the nearest sixpence.
21. Find the upper limit of the error involved in neglecting
all terms after the third in the expansion of (i) (1 - 0'03) ~ 5 ,
(ii) (1 + 0-03) - 5 .
22. I want to know approximately the present value of
100 due in five years at 3 per cent compound interest, paid
annually. Calculate it by using three terms of the expansion
of (1 -f 0'03) ~ 5 and determine within what amount the
answer is correct.
23. Calculate the height of the ordinates where x = - 6
in each of the curves of fig. 50 (p. 273). [It is, of course,
useless to carry the approximation beyond the point at which
the terms cease to yield results measurable in the graph.]
24. Write down
(i) the 4th term in the expansion of (1 - 2.c) ~ 7 ;
(ii) the 8th term in tht> expansion of (a + b-jr) ~ * ;
(Hi) the 5th term in the expansion of (10 - 3.c) ~ i} .
346 ALGKBRA
C.
Note. The preceding discussion has shown that if n is a
whole number and i is numerically less than unity, we may
always make use of the approximation- formula)
-I). t 7i(w-l)(w-2).,
2! V + l - 3 Y V+ ... (I)
. . . (II)
It can easily be shown that these two formula) may be re-
garded as special cases of a single formula
, .., . m(m-i). 9 m(m- i)(m-2)... , IT .
(i-f i) m =n-mi+ 2 i f " ' ' ' ^
in which w may be either a positive integer (n) or a negative
integer ( - n).
25. Prove this statement by substituting - n for m in
formula III.
26. The formula ?/ = r e which describes the " exponential
curves " of fig. 50 (p. 273) can also be written
?/=(! + i) r
where r = 1 + i. If r is less than 2, i is less than 1. It is
possible, therefore, to calculate y approximately by means of
formula III for all integral values of x, positive and negative.
But if the expansion gives the ordinates of the curves when
x = + 1, + 2, + 3 . . . - 1, - 2, - 3 . . .it seems
probable that it will also give them when x has fractional
values such as + 1-3, + 4'2, - O6, etc. Let each student
select in one of the three curves of fig. 50 any two ordinates
whose abscissae are fractional numbers. Substitute the chosen
values of i and x in the formula
. x(x - 1) . x(x - 1) (x - 2) ,
(1 + i) = 1 + xi + - 2 y- } -# + - 37 - * +
carrying the calculation out until the terms cease to be of
measurable magnitude. Compare the values thus obtained
with the measured ordinates of the curves. They will be
found to agree perfectly.
Note. The investigation of No. 26 cannot, of course, be
called a proof. A strict proof is, in fact, too difficult to be
considered at this stage. 1 Nevertheless, the agreement be-
tween the lengths of measured ordinates selected at random
1 It is given towards the end of Part II of this work.
KXKROISE LXVIII 347
and the values calculated by the formula leaves no reasonable
doubt that the latter can be considered as an expansion of
(1 + i)" 1 for all values of m integral or fractional, positive or
negative. In this statement it is to be understood that the
value of i is such that the expansion is an approximation-
formula in which a given degree of accuracy can be obtained
by taking into account only a limited number of terms.
The statement that formula 111 can be used in this way for
all values of m is called the Binomial Theorem.
27. Verify the Binomial Theorem by using it to calculate
(i) yi'l, (ii) ''i/l'%, each to three places of decimals and
comparing the results with those obtained by means of
logarithms. [Note that 71-1 = (1 + 0-1)', */l*2 = (1 4- 0-2)'.]
28. Find the sum of the first four terms in the expansion of
(i) (ro4) aft ; (ji) (0-98) ~ 7 ;
(iii) (i + x)' a ; (iv) (i - x) - '*.
29. Obtain by tho Binomial Theorem a solution correct to
0-1 oz. of Ux. LII, No. 3.
30. Obtain by the Binomial Theorems answers to Ex. LIT,
No. 7, correct to the nearest fathom.
31. Tho pull in pounds weight needed to tow a certain
canal boat at a spead of v miles per hour was found to' follow
the law
P 32y ltlS .
Calculate to the nearest pound the pull required when the
speed is (i) 1*3 miles per hour; (ii) 0*8 miles per hour.
Check your results by means of the curves of p. 273.
32. When air is suddenly compressed (for example in using
a bicycle-pump) its temperature rises. The rule for the rise
of temperature is
T + 273' J = (T -
where T and p^ are the original temperature and pressure of
the air and T and p the temperature and pressure immedi-
ately after the compression, the temperature being measured
in degrees centigrade. On a day when the temperature is
7 C. the air in a bicycle-tyro is compressed ten times in
rapid succession, the pressure being increased by one-fifth at
each stroke. Find to the nearest degree the temperature of
the air in the tyre, assuming that there is no time for cooling
to take place.
348 ALGEBRA
D.
33. In Ex. LIX it was shown that as n increases the
value of (1 + !/?&)" approaches a certain number, " e," which
it never quite reaches. It was shown by a graphic method
that e 2-71. . . . By writing out the first few terms of the
expansions of the following expressions you should be able to
find a very simple expansion which can obviously be regarded
as an approximation -formula for e. Carry out the subse-
quent lines of your work upon the model shown in the first
line.
10.9.8.7 / 1 \ 4
+
/ 1 \ 4
-(id)
4!
_ l l 0-9 0-72 O-504
(1 + 0-01) 100 -
(1 + 0-001) 1000 =
(1 + 0-0001) 1000() =
(1 + 0-00001) 100000 -
34. Calculate e to three decimal places by means of the
expansion
_ 1 1 i !_ JL
35. In Ex. LIX it was proved that as n increases
(l + -} approaches e a .
Find by the method indicated in No. 34 an approximation -
formula for calculating the value of e fl .
36. Use the formula of No. 35 to calculate approximate
answers to Ex. LIX, No. 9.
EXEECISE LXIX.
THE GENERALIZATION OF WALLIS'S LAW.
A.
1. Prove by the Binomial Theorem that if y = ax n then
ty n-i
/- = nax n l
%x
n being any positive whole number.
2. Prove similarly that if y = ax ~ " then
So;
3. Show that the two preceding results are included in the
statement that, if y = ax m then ST//&C = ma#" 1 ~ \ m being
any whole number, positive or negative.
4. Write down the value of Syl&x in the following cases :
(i) y - x\ (ii) y = i/x 5 . (iii) y - 4 x 7 .
(iv) y = 5/x 8 . (v) y = 5x 4 - i2x 3 + 2x 2 - lox + i.
(vi) y = x (] - 2x ; + 3x' J - 4 + 3/x 2 - 2/x 1 + i/x t5 .
5. Write down the value of $ 2 yjSx' 2 in the case of No. 4
(v) and (vi).
6. Write down the value of S 3 7//&E 3 in the case of No. 4
(i) to (iv).
Note. The foregoing results are extremely important
because they show that Wallis's Law holds good for all in-
tegral values of m'my = ax m . This conclusion follows from
the fact that the binomial expansion has been proved in the
preceding exercise to hold good as an approximation-formula
whenever m is integral. We also saw good reason to think
that it holds good when m is fractional. If this were proved
to be the case Wallis's Law would be proved to be true for
all values of m. Fortunately, in order to derive a differ-
ential formula from y = ax m it is sufficient to know the first
349
350 ALGEBRA
term and the coefficient of x. We need not know the co-
efficients of the higher powers of x if, as usual, these
higher powers may be neglected in forming the differential
formula. Now it is easy to prove (with a certain assumption)
that the first two terms of tho expansion of (1 + x) 1 ' 1 ' 1 are
That is, that they follow the law of the Binomial Theorem.
Tho proof is as follows :
Let (1 + #)"''' == 1 + a?'
then (1 + xY = (1 + x)' 1 .
Since p and q are both integers, positive or negative, it
follows that
1 + px + R = 1 + qx' + R'
when R and R' symbolize the remaining terms of the two
expansions. Assuming, then, that (as on p. 348) when x is
small enough these remainders are negligible, we have
px = qx'
or x' = *^x
whence (1 + xY lg = 1 + x + ...
</
7. Prove that the formula S?//S# = max'* ~ 1 holds good
when m = plq, p and q being integers of either sign.
8. Write down the value of 8?//8# when
(i)y = xl; (ii)y=s/x 2 ; (iii) y - i/J/x 3 ;
(iv) y = iox 3 ' 4 ; (v) y = 4X- 1 ' 8 .
9. Show that
(a + x + h) m = (a + x)" + mh . (a + x) m - l + R
where R is a series of terms in each of which h occurs as
a factor in a power above the first. Hence prove that if
y = (a -f #) m , 8y/Sx = m(a + x)' n ~ l . Write down the cor-
responding result when y = (a - x)' n .
10. Write down a formula for 8yj8x when
= i/( 2 - x)< ;
(2x + 3) :i ;
y i/2/(ax - 4)-
(i) y = (n- x) 3 ; (ii) y = i/
(iii) y = 4 (x - 2'S) 1 ' 8 ; (iv) y = (2
(v) y = (ax + b) ; (vi) y i/
Note. In fig. 61 the whole space under the curve is sup-
posed to be divided into strips of equal width. Some of the
P:XERCISE LXIX
351
strips are shown as specimens. Let the abscissa of any
point P l be x and let the area under the curve, from the
y-axis up to P, be A. Then, just as NN' is called an incre-
ment of x, so the area of the strip PNN'P' may bo called the
corresponding increment of A. This area will, in general, be
different from that of the rectangle PNN'Q', but it is evident
that by making the strips sufficiently narrow the difference
can be reduced to less than c per cent of the area of the
rectangles in every case, however small c may be. Lob w bo
the greatest width for which the difference is less than c per
N rT
N N r
FIG. 61.
cent. Then for still smaller widths the increment of the
area is, proportional, within less than c per cent, to the in-
crement of x. That is, in order to calculate the area of the
strip to the specified degree of accuracy we need not take ac-
count of any powers of the width except the first.
In these circumstances we may, in accordance with previous
definitions, call NN' the differential of x, and the area of
the strip the differential of A. Moreover (since PN = y) y
we have the differential formula
8 A = ySx or SA/8& = y.
Now if the ordinate-function of the curve is known we can
substitute it for y in this formula and find the primitive. 2
1 Two points on the curve are marked P. The argument applies
indifferently to either of them.
2 For example, if y = 4x :< we have &A/5x = 4x :j , whence A = x* + p
where p is the usual undetermined constant.
452 ALGEBRA
Assuming that we have additional information sufficient to
determine the constants, we have now a formula expressing A
as a function of x. As was shown in the discussion preceding
Ex. XLVII, values of A obtained from this formula cannot
be more than 2c per cent in error. Moreover, since the
ordinates may be supposed as close together as we please,
2c may be considered smaller than any number that can be
named. In other words, the primitive deduced from 8A =
y$x may be regarded as the area-functaon of the curve.
Fig. 61 illustrates this conclusion. Across the "column-
graph " composed of the rectangles a broken curve is drawn
in such a way that the area under it between any two
bounding ordinates of the rectangles is exactly the same as
the total area of the rectangles between those same ordin-
ates. Let A' be the area-function of this curve. Then the
successive rectangles are the exact first differences of the
successive values of A' at intervals of &c. Now by hypo-
thesis the curve whose area-function is the primitive of
SA/ftx y may, in any part of the figure, include anything
up to c per cent more or c per cent less than the dotted curve
includes. That is, the curve calculated from the differential
formula may run like the firm curve or like the curve marked
out by alternate dots and dashes in fig. 61 ; but in no region
of the figure will the space between its extreme possible
positions be more than 2c per cent of the area of the corre-
sponding rectangles. But even in the most unfavourable
case by making &x small enough 2c may be made as small as
wo please. That is, the added curves may be made to close
up to the given curve so as to become indistinguishable from
it. This is, of course, simply another way of saying that the
primitive deduced from the formula SA/So; = y may be re-
garded as the area-function of the curve.
11. Find tho area-function when the ordiiiato-function is
(i) y = 3# 4 , (ii) y = x\ (iii) y = 3# 2 - 4 . [Find the primitive
of 8A = y$x by Wallis's Law, and determine the constant by
the consideration that when x is zero A is zero.]
12. Find the area between the jr-axis, the curve
y = 3# 4 - 2a; 3 + x - 7
and the ordinates where x = - 2 and x = -f 10.
13. Find the area-functions of the curves whose ordinate-
fuuctions are
EXERCISE LXIX 353
(i) y = (2 + x)i ; (H) y = * (2* - i)' ;
(iii) y = i/(i - 3x) 4 ; (iv) y - i/(3x + i)' 4 .
14. " Wallis's Law gives the primitive of ty//<$x = x m ' l
for all values of m except m = 0." Why does the law break
down for this value ?
C.
Note. Imagine a point to be moving with variable spend
along a straight line. At time t let its distance from a fixed
origin, O, be s, t being measured from the moment when it
passes through O. Suppose it to travel in time h from a
certain point P to another point P'. Then we have
average velocity between P and P' = PP'//.
Since the point is moving with variable speed the value of
PP'/k will depend upon the length of PP. Let P be fixed
but let P' be taken successively nearer and nearer to it.
Then the average velocity will be di Heron t for each of these
different positions of P. It is obvious, however, that when
PP', and therefore k, are small enough the subsequent
values of the average values will not differ by more than c
per cent where c is any number as small as we please. Tbat
is, equal distances within tho range PP' will be covered in
times which are equal, to the specified degree of exactness,
however high that degree may bo. If v be this final value of
the average velocity, wo can write 8t for h and 8s for PP' and
we have the differential formula
8,9 = vM.
In fig. 61 let t be measured along the base and let the
ordinates at equal distances St have for their heights the
corresponding values of v. Then the areas of the successive
rectangles, such as PNN'Q', will be the successive values of
v&t or Ss. It follows that the total area of the column-graph
up to the ordinate t will differ by less than c ger cent of its
value from the area which measures the actual distance
covered by the moving point in time t. That ia to say, this
distance will be measured by the area under the dotted curve
p the figure. But (as in the Note before No. 11) if 8 is
taken small enough this area will also differ by less than
c per cent from that under the curve corresponding to the
primitive of 8s/8t = v, when for v we substitute the function
by which it can be calculated from t. In the most unfavour-
.354 ALGEBRA
able case the cuive giving the true values of 5 and that which
corresponds to the primitive of &s/&t = v may lie on opposite
sides of the dotted curve, but they will include between them
an area which is less than 2c per cent of the area of the
column-graph. 1 Thus the primitive of 8s/8 = v enables us to
calculate the actual distance covered by the moving point to
a degree of accuracy limited only by the degree of exactness
adopted in measuring v.
The proper way to describe v is to call it the average
velocity of the point during a short time &t after it
passes a point P. This definition warns us that it is not
an absolutely fixed value but only one whose variation is
limited within a range of given minuteness. For brevity it
may be called the velocity of the moving point at P, but
the full meaning of this expression should be borne in mind.
15. A point moves along a straight line in such a way that
its distance s from a fixed point O is given by the relation
(i) s = 2/(t + i) ; (ii) s == + 3-4 + st 1 '" ;
(iii) s = - lot + 4t' J ' a ; (iv) s = - io/(2t + 3)*.
Write down a formula giving the velocity (v = &s/ftt) at time
t in each case. Find the velocity of the point in each case
when t = and when t = 1.
16. The velocity of a point moving in a straight line is
given at different times by one of the following formulae. In
each case its position is measured by its distance (s) from
the point which it occupies when t = 0. Find the position-
formulae :
(i) Ss/fit = + 2 - 4t* ; (ii) fis/St = i + 2t + 3t 2 ;
(iii) v = xooMi + st) ; (iv) v = - 20/^(2 - 3 t).
Calculate in each case the position of the point after it has
been moving for 1 unit of time.
Note. If a point is moving along a line the number Ws/W
measures what should, strictly speaking, be called the average
acceleration of its velocity during a short time St after
it passes any given point situated s from the origin. Less
correctly this number is sometimes called the acceleration
at P.
1 It should be noted that neither of these curves need pass
through the corners of the rectangles as in fig. til.
L.A1A. 355
17. Write down formula expressing the acceleration of the
point in No. 15, (i) and (ii) and in No. 16, (i) and (h).
18. 'The acceleration of a point is given by the formulae
(i) 8s/8t* = + 10 - Jt ; (ii) S 2 s/St a = - 5 + ^t ;
(iii) a - 3(1 + 4 t)'-' ; (iv) a = - 4 /(i + 2t) 3 .
The time is in each case measured from the moment when
the velocity is zero. Write down the velocity-formulae.
19. Write down also the position-formulae corresponding
to the acceleration -formulae of No. 18, given that the position
of the moving point is measured from the point which it
occupies when t = 0.
20. A point moves along a line with constant acceleration
p. Show that its distance from any fixed point in the line is
a parabolic function of the time.
D.
21. Write down the formula which gives the gradient at
any point of the curves specified in No. 11.
22. Find, in degrees, the inclination to the #-axis of the
tangent at the point on the curve where x = + 1 in each of
the cases of No. 11.
23. Calculate the ordinatcs at these points. Knowing the
slope of each tangent and the co-ordi nates of its point of
contact with its curve, write down the formula which de-
scribes it. [See Ex. XXXIX, No. 16.]
24. Find where each of the tangents of No. 23 cuts the
cc-axis. Gall this point T, the point of contact P, and the
foot of the ordinate through P N. Then TN is called the
subtangent. Calculate the length of the subtangent in
each of the cases of No. 23.
25. Let p be the abscissa of, any point P of the parabola
y = a 2 x 2 . What is the ordinate of P ? Show that the
formula of the tangent through P is
y = a?p(%x - p).
Hence prove that the subtangent is one half of the abscissa.
Illustrate by a diagram.
26. The parabola of No. 25 is turned into the position in
which its formula is
y-aj*
23*
356 ALGEBRA
Show that the subtangent at any point is bisected by the
vertex.
27. Prove that the lines y = mx and y = - x are at
7?i
right angles for all values of m. Show that it follows that
the linos y - b = m(x - a) and y - b' - (# - a') arc
y \ j j m . )
also at right angles, whatever be the values of a, a', b, b'.
28. The straight line drawn at right angles to the tangent
to a curve at its point of contact is called a normal of the
curve. Write down the formula} of the normals in the case
of each of the tangents of No. 23.
29. Let the normal at a point P meet the #-axis in the
point Q. Then NQ is called the subnormal. Calculate the
length of the subnormal in each case of No. 28.
30. Show that in the case of a parabola in the position
corresponding to y = a*x 2 the subnormal is proportional to
the cube of the abscissa. Show that in tho position corre-
sponding to y = - Jx it is constant.
ANSWEES TO THE EXAMPLES.
*#*x. The solidus notation for division is used throughout these
answers for convenience in printing. The pupil should be taught
both to read and to use it freely, but it should be remembered that
the fractional notation is often more appropriate.
2. The answers to trigonometrical problems in Sections I. and
II. are those which are obtained by using the three-figure tables on
pp. 107 and 111.
3. The answers to graphic problems are in some cases given to
a higher degree of approximation than the average pupil is likely
to reach. In other cases slightly different answers might be ob-
tained by equally competent draughtsmen.
4. The references above each set of answers are to the chapters
in The Teaching of Algebra.
24
SECTION I.
EXEKGISE I.
See ch. in. ; ch. vi., 3.
1. A = Ib. 2. b = A/1.
3. (i) = np ; (ii) p - 0/n.
4. - Nc/n. 5. p = 12s. 6. s = p/12.
7. (i) s - 20L ; (ii) p = 240L.
8. (i) L - s/20 ; (ii) L = p/240.
9. (i) A = 31b. ; (ii) A = nib.
10. (i) A = s 2 ; (ii) A = ns 2 .
11. (i) V - Ah ; (ii) V - Ibh.
12. (i) d = V/A ; (ii) d - V/lb.
13. C = nc/N. 14. C - nc/20N. 15. t = p + 1.
16. (i) P - 12s. + p ; (ii) S - 20L + a ; (hi) P - 240L + p.
17. C - 6 + in. 18. A = 2 + Jn. 19. t = Jw + J.
20. a = A - 27. 21. a = ^A - 3. 22. p = ^n - 30.
23. G = np - NP. L - NP - np.
24. I = Pnr/100. 25. A = P + Prir/100.
EXERCISE II.
See ch. iv., 2-5
A.
2. (i) CD ; (ii) equal ; (iir) AB.
8. 11.52a.m. (i) 7*52 ins. ; (ii) 10'46 ins.
B.
(i) 075 inch ; (ii; 1-65 inches ; (iii) 3^ Ib. ; (iv) 13} Ib.
i) 18 cms. ; (ii) 148 cms. ; (iii) 2'5 seconds ; (iv) 3'54 seconds.
35 2 cms. ; (ii) 65'6 cms.
15.s. ; (ii) 21s. 6d. ; (iii) 39s.
50; (ii) 12.2 p.m. ; (iii) 5 a.m., 7.6 p.m.
10-83 gins. ; (ii) 25 '86 cms.
3 feet 3 inches ; (ii) 0'97 second.
6.4 p.m. ; (ii) 8.22 p.m. ; (iii) April 19.
43 13s. ; (ii) 55 15s.
359 24*
360 ALGEBRA
0.
22. (i) 3*35 acres ; (ii) 16'1 acres.
25. (i) 8680 ; (ii) 7480 ; (iii) 1340 ; (iv) 9th week ;
(v) 12th week.
D
28. See ch. iv., 5. 29. See ch. iv., 5.
EXERCISE III.
See ch. ui. ; ch. vi., 1, 2.
A.
1. (i) W = b + 37m ; (ii) W - b + nm ; (iii) W = b + Vi ;
(iv) 1 - 8tj + 5t a ; (v) 1 - Jn, + Jna ; (vi) 1 = n^ + n a ta ;
(vii) W = n x p + n 2 s + n 3 h ; (viii) P = Ujp + 12^8 + 30n 3 h.
2. (i) S = 50 + 5t ; (ii) S = S -f it.
3. (i) s == 14 + 3t ; (ii) s = s + it.
4. (i) S = 35 - 2t ; (ii) S = S - rt.
5. (i) V = 45 - lOt ; (ii) V = V ~ rt ; (iii) V - V - r^ -f r 2 t,
6. (i) R = f + njm - n 2 r ; (ii) R = n 2 r - f -
7. (i) A = HB - hb ; (ii) A = HB - 3hb ; (iii) A - HB - nhb.
8. (i) V = 40 + l-7t ; (ii) V - 40 + 3-4t ; (iii) V - 40 - 2'lt ;
(iv) V = V + st ; V = V + 2st> ; V = V - wt.
9. In (i) 2 is added to each term to obtain the next term.
To obtain the first term 2 must be added to 1. Hence
T n = 1 + 2n.
The formulae for T n obtained in this way are :
(ii) 3 + 5n ; (iii) 17J + 3n ; (iv) 07 + 07n ; (v) ^ + f n
(vi) 100 - 6n; (vii) 20- l'8n.
10. 21, 53, 62^, 77, 4V, 40, 2.
11. (i) t = V/s; (ii) t = Y/2s; (iii) t = V/(2s + w).
12. (i) t - ( V - 40)/s ; (ii) t = (V - 40)/2s ;
(iii) t = (V - 40)/(2s + w).
13. (i) A = It '
Ib ; (ii) 1 = A/b ; (iii) b = A/1 ; (iv) W = Aw ;
(v) A = W/w ; (vi) w - W/lb ; (vii) A = bh ;
(viii) W = wbh ; (ix) W = 17 bh ; (x) h = W/17b.
14. (i) a - V/n ; (ii) a = V/(n + 1) ; (iii) a = V/(2n + 1) ;
(iv) a = V/(n - 2) ; (v) a = V/(2n - 5).
15. (i) s = (w! - w a )/150 ; (ii) s = (w l - w 2 )/n.
16. f = (21 - 5w)/n.
17. (i) n = (N + l)/2 ; (ii) n = N/2 ; (iii) n (^ - N a )/2 - 1 ;
(iv) Yes.
18. (i) c = n/6 ; (ii) c = np/12.
19. (i) b - A/n ; (ii) c = A/3n.
20. (i) b = A/n ; (ii) b = A/22n.
ANSWERS TO THE EXAMPLES 361
B.
21. (i) i = V,(v a - vi)/vj ;
(ii) V 2 = Vjva/vi or = V x + V^v, - Vl )/v,.
22. (i) d = (r a - r^Rs/r, ; (ii) Rj - R^/r,.
23. (a) S = 1/n + t/n ; (b) S - (1 + t)/n.
24. (a) 1 -= h/u + b/n ; (b) 1 = (h + b)/n.
25. (a) S - p/n - d/n ; (b) S - (p - d)/ii.
26. Si - En^nj + n a ) ; S 2 - Ena/fo + n a ).
27. (i) i = (b - d)t ; (ii) P = P + (b - d)t ;
(iii) P - PO - (d - b)t.
28. (i) d = (BX + Sa)t ; (ii) d ^ d + fa + s a )t ;
(iii) d = (sj - s 2 )t ; (iv) d - d u + (s l - s 2 )t ;
fv) d = d () - (s 1 - s 2 )t.
29. (i) r - H/(m 2 - m 3 ) ; (ii) r = R/5280(m- 2 - mO ;
(iii) r - R/52-8(m a - nij).
30. (i) r = R/(d 2 - dO ; (ii) r = R/12(d, - d^-
31. (i) I = n(r - g) ; (ii) T = ^n(r - g) ;
(iii) I = n(r - g)/c ; (iv) T - T ^n(r - g)/c.
32. (i) i = (d, - dj)T/t ; (ii) d = d, + (d a - dO T/t ;
(iii) d = d x - (dj - da)T/t ; (iv) T = d 1 t/(d I - da).
33. (i) n = 6w/5 ; (ii) n = 3w/2.
34. (i) n - 81/7 ; (ii) n = 961/7 ;
(iii) n = 41/5 ; (iv) n - 481/5.
35. (i) n - 3A/21 ; (ii) n - 4A/31 ; (iii) n = 6A/51.
36. (i) C = 30 + 2n ; (ii) C = 2j + ^n ; (iii) n - (C - 30)/2 ;
(iv) n - 6(0 - 2J).
(iii) n = m + 120(T - ~) ; (iv) n = b 4- 80^T -
38. (i) nj - 2(20T- n a ) ; (ii) n = n 2 + 2(20T - n,j) or n - 40T - n a ;
(iii) nj - (20T - n a s)/a ; n = n a -f (20T - n. 2 s)/a
or n - {20T - (s - a)i\o}/a.
39. (i) P - W + nw ; (ii) w = (P - W)/n.
40. (i) d = 24 - it ; (ii) T = ?(24 - tt) ;
(iii) d = d - rt ; T = (d - rt)/r.
EXERCISE IV.
See ch. vi., 1, 3.
A.
1. (i) 1, 3, 5, 7, 19 ; (ii) 1, 5, 9, 13, 37 ; (iii) 94, 88, 82, 76, 40 ;
(iv) 4, 16, 28, 40, 112 ; (v) 3, 6$, 85, 11, 27 ; (vi) 0, 1, 4, 9, 81
(vii) 4, 15, 30, 49, 247 ; (viii) 6, 7, 10, 15, 87 ;
(ix) 4, 44, 156, 376, 5980 ; (x) 2J, 2, If, 1|, I r 3 r ;
(xi) 1, 1, J, ^ A ; (xii) ! A, At 4 4 o, air-
2. The third, 235. 3. (ii) (a) 7J inches ; (b) 6 inches.
362 ALGEBRA
4. (i) 177 tons ; (ii) 10-23 tons. 5. (i) 0-34 Ib. ; (ii) 34:3 Ib.
6. (i) 6'1 miles; (ii) 19 '52 miles ; (iii) 24'4 miles.
7. (i) 1-6 feet ; (ii) 1-35 feet.
8. (i) 1 ft./min. ; (ii) 4 ffc./min. ; (iii) 100 ft./min.
10. (i) 771 miles ; (ii) 9-13 miles; (iii) 10'57 miles ;
(iv) 17-25 miles ; (v) 33-86 miles.
xx. 61-28 feet. 12. (i) 20'04 ft./min. ; (ii) 70-24 ft./min.
14. (i) 117 feet ; (ii) 14*1 feet. 15. 245i feet ; 115 feet.
B.
17. (i) 4-53 cwt. ; (ii) 3-07 cwt. ; (iii) 5 cwt.
18. (i) 822-8 gallons; (ii) 49572 gallons.
19. (i) 0-512 ton ; (ii) 7'2 tons ; (iii) 7'2 tons ; (iv) Yes.
21. (i) 1-812 tons ; 2-813 tons ;
(ii) 3* 16 inches; 3*94 inches.
22. (i) 10 tons ; (ii) 37 tons.
23. (i) 3-2 inches; (ii) 2'32 inches ; (iii) 27*04 inches.
24. 076 ton. 25. Oral. 26. (i) J foot ; (ii) 4 inches.
27. 259-2 Ib. 28. 150-04 feet. 29. 0-97 foot.
0.
30. (i) 20,160 Ib. ; (ii) 27,337'5 Ib.
31. (i) 4-5 Ib. ; (ii) 375 Ib. ; (iii) 2'85 Ib.
32. (i) 749-8 ; (ii) 1075-2. 33. (i) 5040 ; (ii) 4320.
EXERCISE V.
See ch. vi., 4 ; ch. vn., A.
1. Square feet : (i) 520 ; (ii) 1280 ; (iii) 990 ; (iv) 580.
2. Square feet : (i) 1500 ; (ii) 680 ; (iii) 396 ; (iv) 630 ;
(v) 1040 ; (vi) 1825.
3. Square feet : (i) 1500 ; (ii) 2600 ; (iii) 1600 ; (iv) 1540 ;
(v) 6000 ; (vi) 2200.
4. A = (a -f b + c)d. 5. A = (i) (a + b)c ; (ii) a(b + c) ;
(a + 2b)c ; (iv) (3a + b)c ; (v) a(2b + c) ;
(iii
1
a(b + 4c).
360 ; (ii) 40 ; (iii) 6s. 8d.
A = a(b - c) ; (ii) C = pa(b - c)/108.
8. (i) A = a(b - c - d) ; (ii) = pa(b - c - d)/108.
9. (i) A = a(b - c) ; (ii) C = pa(b - c) ; (iii) C = pa(b - c)/108.
10. 1750 square feet. ii. A = (a - b)c. 12. = (a - b)cl/9.
13. Square feet : (i) 4000 ; (ii) 3200 ; (iii) 4560 ; (iv) 4600 ;
(v) 4950.
14. A = (i) (2a - b)o ; (ii) a(2b - c) ; (iii) (3a - b)o ;
(iv)(}a - o)b ; (v) (a - |c)b.
15. A = (i) (a + 3b)c ; (ii) 2(a + b)c ; (iii) 2(a + fb)c or
(2a + 3b)c ; (iv) (a + b + o)d ; (v) a(a + 2b) ; (vi> a(a - 2b) ;
(vii) (a + c)b ; (viii) 2(a - b)c ; (ix) (3a - 2b)b.
ANSWERS TO THE EXAMPLES 363
16. (i) 86 ; (ii) 83 ; (iii) 39 ; (iv) 260 ; (v) 280 ; (vi) 56 ;
(vii) 209 ; (viii) 1540 ; (ix) 6 ; (x) f .
17. Exp. = (i) a 2 (b + c) ; (ii) (pr - q)q ; (iii) (p - r)q 2 ;
(iv^ (al + bin)lm ; (v) a 2 (a - p) ; (vi)(pa + q' J )a'" ;
(vii) (ab + bd - de)c ; (viii) a(2b + 3c) ; (ix) pqQp - 4q) ;
(x) ab(ab - 2) ; (xi) pq(ap - 1) ; (xii) (ar 2 - 2br + l)r.
18. (i) A - 1(41 - 5d) ; (ii) M = ap(Ja + \q) ;
(iii) V - |r 2 (4r + h) ; (iv) P - abc(a - b + c) ;
(v) Q - ab(2a a + 3b 2 ~ 1) ; (vi) T = (ap 2 - 2bpq + 3q a )q ;
(vii) W - 2rw(7rr - 2a) ; (viii) B - 3mn(4m + 3n - 1) ;
(ix) - a(2a y - 5a + 1) ; (x) V -= rac(3a + 4b)
19. W - aw(a + Sib). 20. W - (3a - 2b)bw.
EXERCISE VI.
See ch. vi., 4; ch. vn., B.
1. Square feefc : (1) 15,800 ; (ii) 5700 ; (iii) 18,400 ; (iv) 2232.
2. (i) 180 sq. cms . ; (ii) 44 sq. inches ; (iii) 52 sq. inches ;
(iv) 247 '2 sq. inches.
3. 720 c.cs. 4. (i) V - (a + b)(a - b)l ;
(ii) W - (a + b)(a - b)wl.
5. w - W/(a + b)(a - b)l. 6. A ---= (a + 2b)(a - 2b).
7. A = (a + 3b)(a - 3b). 8. A = (a -t 2b)(a - 2b).
9. (i) W - (a -!- 3b)(a - 3b)w ; (ii) W - (a - 2b)(a 4- 2b)wl ;
(iii) w = W/(a - b)(a + b).
10. c - 20C/(a - b)(a + b)l.
11. Kxpr. - (i) (p + q)(p - q) ; (ii) (2a + b)(2a - b) ;
(iii) (m + 3n)(m - 3n) ; (iv) (6a + 5b)(6a - 5 b) ;
(v) (pa + b)(pa - b) ; (vi) (u + vt)(u - vt) ;
(vii) (pu -f vt)(pu - vt) ; (viii) (ab + 4c)(ab - 4c) ;
(ix) (a + 4)(a - 4) ; (x) (9 + b)(9 - b) ;
(xi) (pq + 5)(pq - 5) ; (xii) (1 + mn)(l - mn).
12. (i) p(a + b)(a - b) ; (ii) a(2p 4- 3q)(2p - 3q) ;
(iii) 7r(ri + r a )(rj - r a ) ; (iv) a(a + b)(a - b) ;
(v) (p + t)(p - t)t ; (vi) (2p + 5t)(2p - 5t)t ;
(vii) 3(2a + 3b)(2a - 3b) ; (viii) (a + 6b)(a - 6b)b ;
(ix) 2(2a + 5)(2a - 5) ; (x) p(2p + 3)(2p - 3) ;
(xi) (ab + cXatt - c) ; (xii) a(3ab + l)(3ab - 1).
13. (i) (a + b)(a - b + 3) ; (ii) (a - b)(a + b + 7) ;
(iii) (a + b)(a - b - 4) ; (iv) (a - b)(a + b - p) ;
(v) (a + b)(a - b + 1) ; (vi) (a - b)(a + b + 1) ;
(vii) 2(a - b)(a + b + 3) ; (viii) (a + b)(2a - 2b - 3) ;
(ix) (p - q)(ap + aq + b) ; (x) (p + q)(ap - aq - b).
14. (i) (a + b + c)(a + b - c) ; (ii) (p - q + r)(p - q - r) ;
(iii) (a + b + 2)(a + b - 2) ; (iv) ( p - q + 3)(p - q - 3) ;
(v) (p + q + 3r)(p + q - 3r) ; (vi) (a - b + 4c)(a - b - 4c) ;
(vii) (2u + 2v + w)(2u + 2v - w) ;
364 ALGEBRA
(viii) (3u - 3v + 2w)(3u - 3v - 2w) ; (ix) (2a + b)b ;
(x) a(a + 21)) ; (xi) a(a - 2b) ; (xii) (2b - a)b ;
(xiii) (2a + 3b)(2a + b) ; (xiv) (6a - b)(4a - b) ;
(xv) (a + 3b)(a - b) ; (xvi) (a + 2b)(a - 4b) ;
(xvii) 4(5p + 2q)(2q - p) ; (xviii) (5p - 2q)(5p - 8q).
15. (i) (2a - b)b ; (ii) (2p - q)q ; (iii) 3(2r - 3) ;
(iv) 67(2p - 67) ; (v) aq(2p - q) ; (vi) irw(2r -- w) ;
(vii) 3b(2a - 3b) ; (viii) rt(2s - rt) ; (ix) pb(2a - pb) ;
(x) ;iq(2p - aq).
16. A - (2a 4- b)b. 17. A ~ a(a + 2b).
18. A - 4a(a + 3 b). 19. A - 4p(p + 3q).
20. A - (p 4- llq)(p - 7q). 2i. A - 4b(3a + b).
22. A = (2a - b)b. 23. 220 sq. cms.
24. (i) A - 7r(i\ + r a )(p, - p a ) ; (ii) W - TTW(P I + r a );
(iii) V = TrKPi + r a )(r, - r a ) ; (iv) W = rlw(r, + r^n) - r, 2 ) ;
( V ) w - W/7rI(r, 4- r 2 )( ri - p a ).
25. A == 7r(a + 2b)(a - 2b).
EXEKCISE VIL
See oh. vi., 5 ; ch. viii., A.
1. (i) 4-8 yards ; (ii) 7'9 inches ; (iii) 97 cms. ; (iv) 8' 485 feet ;
(v) 1-81 miles ; (vi) 5'36 feet ; (vii) 9'01 yards ;
(viii) I'Ol miles.
2. (i) 187 ; (ii) 39'5 ; (iii) 20-4 ; (iv) 0'207 ; (v) 0-0171.
3. 348-4 miles. 4. 241-6 miles. 7. 28'1 miles. 8. 89 miles.
o. 143'6 miles. 10. 10'9 ft./min. n. 18J seconds.
12. R - \/(a/7r). 13- (0 6'34 inches, (ii) 0761 foot.
14. 196-3 miles ; 136*3 miles. 15. R =
16. 042 cm. 17. 0-076 cm.
EXERCISE VIII.
See ch. vi., 7 ; ch. viii., B.
1. 2-82, 3-46, 2-44, 6'23, 9'96, 23*85, T66, 072, 071, 6-28, 1173.
2. 0705, 2-88, 2-14, 0*69, 0'479, 0-851, 0'403, 7'02.
3. 2 J2, is/5, is/3, Js/15, s/26/13,
4. (i) 8; (ii) 1-28; (iii) 17'23 ; (iv) 2'02 ; (v) 0'055 ; (vi) 11-2;
(vii) 0-576 ; (viii) 3'875 ; (ix) 5'47 ; (x) 6'31.
5. a^/2. 6. as/3. 7. as/10. 8. 2as/10.
o. (i) ^185 feet ; (ii) sV + b 2 + c 2 ) ;
(iii) s/{a 2 + (b - h) 2 + c 2 }.
10. s/{p 2 + q 2 -f (b - a) 2 }, xi. (i) |(2 - ^3) ; (ii) |(2 + s/3) .
'ANSWERS TO THE EXAMPLES 365
EXERCISE IX.
See ch. vi., 7.
Approximate equalities are indicated by the sign + or - follow-
ing* the left hand expression. The pupil may use the sign == as
instructed on p. 49.
A.
3. (i) 20 sq. inches, 4 sq. inches ;
(ii) T44 cms. 2 , 0'16 cm. 2 ; (iii) 9'8 cms. 2 ; 0'49 cm.' 2 .
4. Fraction of area = (fr. of side)' 2 .
5. 1/144. 6. 1/400. 7. A = a 2 + 2ab + = a(a + 2b) + .
8. (i) A = 40 x 46 - 1840 feet 2 ; (ii) 53 x 60 - 3180 feet 2 ;
(iii) 2175 feet 2 .
9. (i) 3000 feet 2 , (5/50) 2 - 1/100 ; (ii) 6248, 1/484.
10. (i) W - aw(a + 4b) ; (ii) 400b 2 /(a + b) 2 .
11. 314$ sq. inches. 12. (i) I - 2ab + b 2 ; (ii) I = 2ab + .
13. (i) 22-8 cms. 2 ; (ii) 1/1600 ;
(iii) b 2 /b(2a + b) = b/(2a + b) = 1/81 ; (iv) the latter.
17. A = 27rrt + . 18. (i) A - 47rr(r + 2t) + ; (ii) 87rrt +.
20. Expr. = (i) a 2 + 4ab + 46 2 ; (ii) 9p 2 + 6pq + q 2 ;
(iii) p 2 + ipq + M 2 ; (iv) a 2 + 2^5 . b + 6b a ;
(v) 1 + p + ip a ; (vi) 1 + 0'006t + 0'000009t 2 ;
(vii) 1 + 2ct + c 2 t 2 ; (viii) a 2 + 2pab + p 2 b 2 ;
(ix) I 2 (a 2 + 2ab + b 2 ) ; (x) r 2 (l + Jpt + T V>p 2 t 2 ).
21. (i) 1 = 1 + ct ; (ii) It = 4(1 + ct) ; (iii) l t = 1 (1 + ct).
22. (i) At - (1 + ct) 2 = 1 + 2ct + ; (ii) At = l a (l + 2ct) + ;
(iii) A t - A (l + 2ct) + .
23. (i) I t = 2rrct + ; (ii) 0'4092 sq. cm.
B.
24. (i) 4'5 cms. - ; (ii) 0*25 in excess.
25. ^/10 = V9 -f 1 = 3 + 2 x o = 3-J-. The other roots are :
4J, 7?, 10 J, 13 A, 61, 20J, 4-2, 9' 13.
26. (i) 2-035 sq. cms. ; (ii) 3*6 sq. inches ; (iii) 6'3 feet.
27. Expr. = (i) a + p/2a - ; (ii) a + p/a - ; (iii) a + b 2 /2a - ;
(iv) r + l/2r - ; (v) 2p + q/p - ; (vi) 4 + a/8 - ;
(vii) 9 + nt/18 - .
28. (i) J13 + a/^/13 - = 3-6(1 + a/13) - ; (ii) 2'44(2 + a/24) - ;
(iii) va + b/2^/a - ; (iv) *Jp + q/vp - ; (v) a -f
(vi)2^p + iV(q/p)-;(ra)3V
v .___.
29. (i) D = 1-22 X /(H + h) ; (ii) D - 122(\/H +
30. (i) D = 25 miles ; (ii) 0'61 mile ; (iii) d = 0-61h/^/H -.
366 ALGEBRA
EXERCISE X.
See ch. vi., 7.
2. A = a(a - 2b) -K 3. 26 '4 x 25 = 660 sq. inches.
4. 916 nq. cms.
a 2
6. Exp. = (i) a' 2 - 4ab + 4b u ; (ii) ^ - ab + b' 2 ;
(iii) P 4 - 2 P >2 q- + q 4 ; (iv) i - 2 P 2 + P 4 ; <
(v) 1 - 4p 2 + p 4 ; (vi) 1 -- 2ct + c 2 fc' 2 ;
(vii) A(l - 2ct + c >2 t' 2 ) ; (viii) r' 2 (l - 2ct + c 2 t y ) ;
(ix) a' 2 (l - 2bt 2 -I- bH 1 ) ; (x) r 4 - 2abr + Ja' 2 b' 2 e
7 . (i) V -= 27rrtl ; (u) C = 7rrfclo/6v.
8. (i) A, - A(l - 2ct) + ; (ii) d - 2ctA.
9. (i) A - arr(r + 1) ; (ii) d = 2nrct(r + 1) - .
11. (i) 3a - b/6a - ; (ii) 4'3(p ~ q/6p) - ; (iii) 4 - 2'lt' 2 /8 - ;
(iv; a - b 2 /2a - ; (v) */a - 2^/(b/a) -.
12. (i) h = a - ^(a' 2 - b' 2 ) - b <2 /2a + ; (ii) 1/3 inch.
EXERCISE XI.
See ch. vi., 7.
A.
i. a' 2 (a + 3b) +. 2. V = 10' 2 x A + = 25 cu. inches.
3. V - 1 + 3ct + . 4.1- 47rr a ul + .
6. Expr. = (i) a' 2 (a + 6b) + ; (ii) 4a 2 (2a + 3b) + ;
(iii) (1 + p) + ; (iv) a' 2 (a + 3pt) + .
8. See ch. vi., 8.
9. Expr. - (i) a' 2 (a - 3b/2) - ; () j(| ~ 3qj - ;
(iii) (1 - 6'9k) - ; (iv) a 2 (a - 3nt/2) -.
10. See ch. vi., 8.
11. Kxpr. - (i) a + d/3a 2 - ; (ii) a - d/3a 2 - .
12. S/10 - l/(W + 2) = 2 + 2/12 - = 2J -. The other roots are
approximately 3, 3Hi 10*1, 6';, 0'8085 ; -289 ; 11875.
B.
13. V = (i) 1 + 3cfc + ; (ii) 1 - 3ct + .
14. I = 47rr' ? ct +. 15. (i) 1'004 + si{. inch ; (ii) T006 + cu. inch.
16. V = 1 - 0-009 .-. (i) 1 = 1 - 0-003 = 0'997 inch ;
(ii) A - 1 - 0-006 = 0-994 sq. inch.
17. (i} L = L (l + 0-00006t) ; (ii) A - A (l + 0'00012t).
18. 2-88, 5-13, 4-32, 6'84, 2'4624, 2*9241, 4'9248, 3*545856.
19. 22/2, 3J/4, 6^/10, 102/9, 2#0'2, 6^/0*003, ^VO'7.
20. aj/b, by a, b^/a 2 , ayab, b^ab, b^/a 2 b, c^/a 2 b 2 , ab^c' 2 ,
ANSWERS TO THE EXAMPLES 367
EXERCISE XII.
See ch. vi., 8 ; ch. ix., A.
The subject of a formula, where obvious, is sometimes omitted.
1. (i) R = pq/(p 4- q) ; (ii) pq/(q - p) ; (iii) pq/(3p 4- 2q) ;
(iv) pq/(3q - 5p) ; (v) q/(q - p) ; (vi) q/(p - 2q) ;
(vii) q/(p 4- qr) ; (viii) p/(pr - q) ; (ix) qr/(pr 4- q) ;
(x) qr/(2pr - 3q) ; (xi) r/(2q - 3pr) ; (xii) r/(l - pqr).
2. (i) A = 2(b - 3a)/ab ; (ii) A - c/a(b + c) ;
(iii) P = p(r - q)/r ; (iv) D - ds/d^da 4- d 3 ) ;
(v) B = b/a(b + 1) ; (vi) R - r a / ri (l - 2r 2 ) ;
(vii) R - 3r,/ ri (2 - 3r 2 ) ; (viii) A = (a 2 4- 2)b/a ;
(ix) A - b/a(l - 3b 2 ) ; (x) A - (a 4- b)(a - b)/ab ;
(xi) V = (2u + v)(2u - v)/2uv ;
(xii) R - pq/(2p + 3q)(2p - 3q).
3 . (i) P ~= (q + l)/pq J (ii) (2 - 3q)/pq ; (iii) (p 4- q - l)/pq ;
(iv) pq/(p 4- q - 2) ; (v) pq/(bp + aq - c).
4. (i) V - uv/(u + v) ; (ii) 2uv/(v - u) ; (iii) uvw/(v - u) ;
(iv) 4au a /(2u + v)(2u - v) ; (v) av 2 /(u 4- bv)(u - bv).
5. (i) V - u 4- v ; (ii) u - v ; (iii) (v - u)/u ;
(iv) (u 4- v)(u - v)/uv ; (v) uv ; (vi) u ; (vii) u ;
(viii) 1/uv ; (ix) (u - v)/u ; (x) (u - v)/u 2 ; (xi) v 2 /(u + v) ;
(xii) uv/(u 4- v)(u - v).
6. (i) (a + b)/a' 2 b ; (ii) (a - b)/ab 2 ; (iii) (b - a)/a 2 b 2 ;
(iv) (a 4- b + c)/abc ; (v) (a 2 4- b 2 + c' 2 )/abc ;
(vi) (a + c)(a - c)/abo ; (vii) (a 4- b) 2 /a 2 b 2 ; (viii) (b - a) 2 /a 2 b 2 .
7. (i) (2 - 3p)/6p 2 ; (ii) a 2 4- b-)/pab ; (iii) (in + n)(m - n)/anm ;
(iv) (pa + qb) 2 /p 2 q 2 a 2 b a or {(pa + qb)/pqab} 2
(v)
8. (i) 1 = (A^ + A^/bjba ; (ii) (nb a +
9. (i) T = mb/(m + b) ; (ii) mb/(2m + 3b) ; (iii) nm/(ri + 1) ;
(iv) b/(n 4- 1) ; (v) nm/(pn -f q) and b/(pn + q).
10. (i) n = aq/bp ; (ii) aq/3bp ; (iii) 9aq/bp ; (iv) aqs/bpt ;
(v) aqs 2 /bpt 2 .
11. (i) V = ^q 4- bp)t/pq ; (ii) ii(aq 4- bp)/p ; (iii) T = Qp/a ;
(iv) T = Qpq/(aq 4- bp) ; (v) T = (Q - Q )q/b ;
(vi) (Q - Qo)pq/(aq 4- bp).
12. (p 2 4- q 2 )/p 2 q 2 .
13- (P + q)(p - q)/p 2 q 2 .
14. (i) d - 27rn(pR - Pr)/Pp ; (ii) t = p/R
15. (i) w = 300(mq - np)/pq ; (ii) d = 200rr(mq - np)/pq ;
(iii) d = 2507r(mq - np)/llpq ;
(iv) A = 900007r(mq 4- np)(mq - np)/p 2 q 2 ;
(v) = 1257rc(mq 4- np)(mq - np)/3p 2 q 3 .
16. (i) d = V(A 2 - A 1 )IA l A a ; (ii) V(r 9 4- r^r, - r^/Trr 2 ^ 2 ,.
17. (i) (p - 1)/ P ; (ii) h = (p - l)V/pA ; (iii) (p - l)V/rrpr 2 ;
(iv) (p - q)V/pA and (p - q)V/7rpr 2 .
18. (i) v = (p - q) 2 V/p 2 ; (ii) d = (2p - q)qV/p 2 A.
19. d = V{ap 2 - A(p - q) 2 }/aAp 2 .
20. (i) n = (p - q)A/a^ ; (ii) {(p - q)R/pr} 2 .
I. (i) 9 -
(v) 8(]
368 ALGEBRA
EXERCISE XIIL
See oh. vi., 8 ; ch. ix., B.
A.
> - a ; (ii) 23 ~ a ; (iii) a - 5 ; (iv) 2(p + 4) ;
- 1) ; (vi) 42 - r ; (vii) 6 + r ; (viii) 2;3r + 18 ;
(ix) 6(6r - 3) ; (x) p + 8b ; (xi) (2* 4- p)(2a - p) ;
(xii) (3m + 2p)(I3ra - 2p) ; (xiii) 3r - 5 ; (xiv) 20 - 2p.
3. (i) a > 9 ; (ii) a > 23 ; (vi) r > 42 ; (xi) p > 2a ;
(xiii) r<5/3.
4. (i) 3p ; (u) 4 + t ; (iii) 2(t - 1) ; (iv) 5a ; (v) a - 12b ;
(vi) 2(3q - p) ; (vii) 23n - 10m ; (viii) a(p + 4) ;
(ix) 2a(2 - p) ; (x) 4a -f b 2 ; (xi) p - 2 ^2 5 (xii) 2 J2.
5. (i) a<12bora/b<12; (ii) p > 3q ; (iii) m>2'3n.
7. (i) t - 800/q(q - 8) ; (ii) t = Qd/q(q - d).
8. 72d/(14- d). 9. 720/p(p+ 1).
10. 90(8 - p)/p(p + 1).
11. (i) 31'5/a(a + 6-3); (ii) (1-2 - p)/(8'4 - p) ;
(iii) (3-7 - p)/(p + 6-1) ; (iv) 2(11 - p)/(p - 6-1) ;
(v) 3(5-2 + a)/(3 + a) ; (vi) 3(5 -2 -f a)/a(3 + a) ;
(vii) a(07a + l)/(07a +3) ; (viii) b(3'lb - 4)/(7 - 3'lb) ;
(ix) 5(2a - l)/2-3a(7'7a - 5) ; (x) 8(96 - 7p)/3(32 - p) ;
(xi) a/(12 + a) ; (xii) (p - 21)/(30 - p).
12. (i) 2b/a(a + 2b) ; (ii) q/3p(3p - q);
(iii) b/5a(5a +b) ; (iv) pq/3(3 - p) ;
(v) pn/(m - n).
13. lps/(36 - s). 14. (i) olt/(l + ct) ; (ii) olt/(l - ct).
B.
17. (i) 1 + a + . (ii) 1 - a + .
18. (i) 1 -f pt -f ; (ii) d(l - pt) + ;
(iii) d(l - 0-0006t) + ; (iv) 1 + - 4- ;
a
(v) a + b + ; (vi) (a - b)/a 2 + ;
(vii) (a - b)/a 2 + ; (viii) (a - b)/a 2 + ;
(ix) (p + q)r/p 2 + ; (x) a(p 2 - q 2 )/p 4 +.
19. (i) 18d(14 + d)/49 -f ;
(ii) clt(l - ct)t, clt(l + ct) +.
20. 320,000(1/80 - 1/804) - 4000 {1 - 1/(1 + ^ v )} =
4000 x 1/160 = 25-.
21. 125s. x. 22. 211200Lo/l.
23. 2md/s 2 .
24. (i) (a - b)t ; (ii) (d/c 2 - b(a a )t ; (iii) 1 - 2a ; (iv) 1 + 2a ;
(v) (1 - 2pt)/a 2 ; (vi) (a - 2b)/a ; (vii) (p + 2q)/p 3 ;
(viii) 2q/p s ; (ix) 1 - a/2 ; (x) (2a 2 + p)/2a^; (xi) r/2p 2
(xii) ct/2000 ; (xiii) 2ct/P ; (xiv) 1 + b/2a ; (xv)
(xvi) 1 + 3a ; (xvii) (1 - 3cfc)/r 3 ; (xviii) 1 + 3b/a ;
(xix) l + p/3; (xx)r/3a<.
25. 4v/7ra. 26. 4v/7ra + 2v/rrr.=
(x) (p - n)(q - m)/(pm - qn) ; (xi) (p - a)(q - a)/(p - q)
(p + q - a) ;
ANSWERS TO THE EXAMPLES 369
27. 9h/IVD; 1/81 Ib. 28. 9h/Dv/D.
29. hT/2'21^/1. 30. The same.
31. 1 4- a 4- a 2 . 32. 1 - a 4- a 2 .
33- (0 -(i + - + tf) ; () -(i - - + jp J ;
(iii) 1 - ct 4- c a t' J .
C.
34. P = (i) (2p 4- 6)/(p 4- 2)(p + 3) ; (ii) l/(p + 2)(p + 3) ;
(iii) (2p 4- a 4- b)/(p 4- a)(p 4- ,b) ; (iv) (p 4- a)(p 4- b)/(a - b) ;
(v) (p - a)(p 4- a)/2p ; (vi) (2p - a)(2p 4- a)/2a ;
Q = (vii) (m 4- n - 2a)/(m - a)(n - a) ; (viii) (m - n)/(m - a)
(n - a) ;
(pm 4- a)(qn 4- b)/(pm - qn 4- a - b) ;
(p _
(p 4- q - a) ;
M = (xii) (13m - 12n)a/(2m - 3n)(3m - 2n) ;
(xiii) 5ma/(2m - 3n)(3rn - 2n) ; (xiv) (1 - pa)(l - qb)/(pa - qb).
35. (i) (a 4- b 4- l)/(a 4- b) 2 ; (ii) b/(a 4- b) 2 ; (iii) b/(a - b) 2 ;
(iv) a/(a - b) 2 ; (v) (a + b)/(a - b) 2 ; (vi) (a - b + l)/(a' 2 - b 2 ) ;
(vii) (a 4- 2b)/(a 2 - b 2 ) ; (viii) a/(a 2 - b' 2 ) ; (ix) (2a 4- l)/(a 2 - b' 2 );
(x) (2b + l)/(a 2 - b 2 ) ; (xi) 2/(a - b) ; (xii) 2/(a - b) ;
(xiii) 0. (xiv) ; (xv) 2p/(p - q)(p 4- q) a ;
(xvi) 2q/(p 4- q) a (p - q) ; (xvii) 2q/(p - q) 2 (p 4- q) ;
(xviii) l/(p - q) ; (xix) (p 4- q/(p - q) 2 ;
(xx) (p 2 4- q 2 )/(p - q)(p 4- q) a .
36. (i) 2a 2 /(a + b) a (a - b) ; (ii) 2a 3 /(a 4- b)(a - b)' 2 ;
(iii) 2ab/(a 4- b) a (a - b) ; (iv) - 2ab/(a 4- b)(a - b) ;
(v) b/(a + b) 3 ; (vi) (a 2 4- b 2 )/(a 4- b) 3 ; (vii) 2a' 2 /(a - b) a ;
(viii) 2b 2 /(a 4- b) 8 .
EXBBCISB XIV.
Ch. vi., 9 ; ch. x. Note especially in the "literal " examples
that the systematic application of the rules of ch. x. leads directly
to the simplest form of the answer. In Nos. 7 and 14 (and, if neces-
sary) in No. 23 the working may be an actual transcription of the
pupil's solution of the arithmetical case (as in ch. x., 3).
A.
i. 2*3. 2. |K 3- 127. 4- H. 5- 17-28. 6. 27 '5.
7. (i) n = (c - b)/a ; (ii) (b 4- o)/a ;
(iii) (n 4- a)b = c ; n 4- a = c/b ; n = c/b - a ; (iv) a 4- c/b ;
(v) a 4- be ; (vi) (c - b)a.
8. 3. 9. 6. 10. 23. ii. 8. 12. 5. 13. 7.
14. (i) n = (d/c - b)/a ; (ii) a 4- c/b ; (iii) (b 4- cd)/a ;
(iv) (cd - b)/a ;
(v) b/(n 4- a) = c, b == (n 4- a)c, n 4- a = b/c, n = b/c - a ;
(vi) (b + a/d)/o.
25
370 ALGEBRA
15. 4. 16. 27. 17. 1*9. 18. 3. 19. 9. 20. 7'5. 21. 3. 22. 3.
23. (i) n - (a/d + c)/b ; (ii) n = {c(d + e) - b}/a ;
(iii) n = {a/(e - d) - c}/b.
24. D = {c(d - e) + bl/a.
25. n = {(a + b)r - qW
26. n = ((p + q)a + c}/b.
27. n - [a/b(e + 1) - q}/p.
28. n - {a/(b - c) + q}/p.
29. n a(l -f q + b/p).
30. n = att> - q)/p.
31. n = p{a/(b + c) - q}.
B.
32. (i) n = (W - b)/m ; (ii) m - (W - b)/n.
33. (i) ta - (1 - n a t 2 )/n! ; (ii) t a = (1 - njtO/n,.
34. t = (S - S )/i.
35. W S = S + rt ; (ii) t., =(S - S)/r.
36. (i) 1 = kbd 2 /W ; (ii) b = Wl/kd 2 ; (iii) 6'8 feet
37. P = 21,000 H/nlD 2 ; P = 35 Ib.
38. d= JlL/7'11}; 1-2 inches nearly.
39. L = I/(10p).
40. 1 = ^/(Sds/w) ; 106 feet 8 inches.
41. (i) h = d 2 /l-49 ; (ii) d = W 2 /20A a ; (iii) D - 324/P^.
42. i = (s - s )/t.
43. (i) wj = ns + w a ; (i:) n = (wj - w 2 )/s.
44. (i) d = d - (B! - s 2 )t ; (ii) t = (d - d )/(s! - s a ) ;
(iii) B! (d - d )/fc + s a .
45. R = 19r(d g - dO. 46. The same.
47. (i) T - it/(d 2 - dO ; (u) t - T(d 2 - dj/i ;
(iii) d, = it/T -f dj.
48. h = Ld(W + P)/W.
49. W (D - 0'2) 2 /0-45.
EXERCISE XY.
See ch. vi., 9 ; ch. x
A.
x. 3'4. 2. 34-5. 3. 33. 4. 169. 5. 257*4.
6. 5'2. 7. 61. 8. (i) n = (a - c)/b; (ii) {b - a/(e - d)}/c;
(iii) {c - b/(a - e)}/d ; (iv) b(a - c) ; (v) a(b + c) ;
(vi) (b - a + d)/c ; (vii) {o - (a - e)/b}d.
9. n = {q - b/(a - c)}/p.
xo. n = b - l/p(a - c).
11. n = {b - U - q)/p}/o.
12. n - {p - (c - a a )/b}/q.
13. n = (a - q^/pVb 2 .
14. b d - (P - P)/t.
IS- s a = s, - (do - d)/t.
ANSWERS TO THE EXAMPLES 371
16. d, - d x - (d! - d)t/T == {d x (T - t) - dt}/T.
17. d a = d! - d x t/T = di(T - t)/T.
18. T = 24 - 2R = 2(12 - R).
19. d a = dj - D(h - E)/(H - h) - il foot uoarly.
20. t = (12p - 377nD)/3'15ir.
22. lj = (R! -H R 2 -1- R 3 ) - E.
23. d = #{D J - 12I/B}.
B.
24. 7'15. 25. 4.
26. (i) 3; (ii) 3; (iii) 5; (iv) 2'1.
27. (i) n = (c - b)/(a - 1) ; (ii) (b + c)/(a + d) ;
(iii) (a - o)/(b - d) ; (iv) (a + d)/(o - b).
28. 150. 29. 45 and 15. 30. 50, 70. 31. 69.
32. (i) 12; (ii) 13; (iii) 12J ; (iv) 4'1 ; (v) 11; (vi) 6; (vii) 8;
33. (i) n == bc/(a - b) ; (iii) (ab + cd)/(c - a) ;
(v) (ab + cd)/(a + c) ; (vii) (ac + df)/(do - ab).
34. 29, 32 or 104. Draw diagram. C may be (i) between A and
B, (ii) below A or (iii) above B.
(i) requires 3(41 - n) = 4(n - 20) or 4(41 - n) - 3(n - 20) ;
(ii) requires 4(20 - n) 3(41 - n) which gives no result ;
(iii) requires 4(n - 41) = 3(n - 20).
35. Four positions possible :
(i) between trees 36 yards from first ;
(ii) outside trees 120 yards from first ;
(iii) and (iv) the same positions with respect to the second tree.
36. 60. 37- (i) 22 ; (ii) 12 ; (iii) 2'5 ; (iv) 9.
38. (i) n = (ad + bc)/(b + d) ; (iv; (af + oe)/(bf + cd).
39. 408.
40. (i) 124 ; (ii) 4 ; (iii) 9 ; (iv) 5 ; (v) 8 ; (vi) 7 ; (vii) 12 ;
(viii) 3 ; (ix) 8-23 ; (x) 5.
41. 11 mls./hr.
42. (i) 49; (ii) 1; (iii) 279; (iv) 5'34; (v) 7'3 ; (vi) 5'5;
(vii) } ; (viii) 2.
43. 30. 44- 60.
45- (i) 48 ; (ii) 4}f ; (iii) ; (iv) 1J ; (v) 16 ; (vi) 4 ; (vii) 7 ;
(viii) 5'6 ; (ix) 5^ ; (x) 2^.
46. (i) n = po/(pb - qa) ; (ii) ( ap + bq)/(aq + bp) ;
(iii) (c - aq - bs)/(ap - br) ;
(iv) (a - b)/c(a + b) ; (v) c/(a - b)p 4- q/p ;
0.
47. p = q(an - b)/(a - bn) ; q = p(a - bn)/(an - b).
48. a/b = (1 + cn)/(l - en). 49. a - b = c/(pn - q).
50. h = (c - c a )/(2c 2 + c). 51. V 2 = Vj. - il/cS.
25 *
372 ALGEBRA
52. h = a - Fa/47rl ; a = 47rlh/(4trl - F).
53. n = V(2b* - Rbc/Ld).
54. Vi = V - 47rd 1 d 2 Q/S i (d 1 + da).
55. q = 1/(1 - d) - a /a.
56. r 2 = T! - ri RI/e = r,(l - Rl/e).
57- V! - V(l - ROl/t). 58. r = v/{B/(A - e)}.
59. u = V(v 2 - 2Fsg/W). 60. r == V(a a - Aev/fo - p a )}.
EXERCISE XVI.
See ch. vi., 10.
1. (i) S Sq_- mfc ; (ii) T = (So - mt)/m.
2. (i) s = Vlb ; (ii) p = 1 - Vlb J (i") q = \/Tb ~ b.
3. (i) C = ?rd ; (ii) = 27rr ; (iii) r = 0/27r ; (iv) r = l/27ru ;
(v) n == l/27rr ; (vi) A = Trr 2 ; (vii) r = ^/(A/Tr) ;
(viii) V - Trr'h ; (ix) h = V/7rr 2 ; (x) r = x/(V/7rh).
4. (i) h = V/lb ; (ii) h - V/7rr 2 ; (iii) h = 0-16V/lb ;
(iv) h = 0-16V/rrr 2 .
5. (i) d = (s! - s a )t ; (ii) d 2 =
(iii) D! = ( Sl - sOdj/S! ; (iv) D 2 = ( Sl - s,)d 2 /8 2 .
6. (i) d = d + (HI - s a )t ; (iii) Dj = do + (B! - s a )d 1 /Si ;
(iv) D a - d -f (s a - 8 2 )d 2 /S2.
7. (i) m = (s 2 - s 2 )t ; (ii) d = d - (sj - 8 2 )t ;
(iii) T = do/(i - s 2 ) ; (iv) L = d s 1 /(s 1 - s 2 ) ;
(v) t = M/( 8l - s a ) ; (vi) t = (do - D)/( 8l - aa) ;
(vii) L! = s^Ls - do)/s 2 ; (viii) L 2 = SaLi/8! ;
(ix) L a = s 2 (do - d^Si - s 2 ) + d ;
(x) L! = s^do - d)/(s! - s 2 ).
8. (i) L - Ih/w ; (ii) L = 121h/w ; (iii) L = 41h/w ;
(iv) L = 41h/21 ; (v) N - lh/63 ; (vi) C = plh/63 ;
(vii) C - plh/756.
9. (i) h = nv/A ; (ii) d n = d + nv/A ;
(iii) n = (d n - do)A/v ; (iv) n - (d - b)A/v + 1.
10. (i) W = 7rr 2 hc ; (ii) W = w + TrrMc ;
(iii) c - (W - w)/7rr'M ; (iv) - P/irrd.
B.
11. h = (i) 6 in. ; (ii) 5 '4 in. ; (iii) T92 in.
12. R = (i) 1050 ; (ii) 700'8. 13. H = (i) 176 ; (ii) 410'6.
14. d = 2 26 in. 15. d = 277 in. 16. 19 '6 Ibs.
17. D = 61-4 in. 18. s = (i) 26 22 ; (ii) 37 59.
19. w = (i) 190,000 ; (ii) 139,000 ; (iii) 77,000.
20. (i) 5 11s. ; (ii) 9 6s. 9d. ; (iii) 19 10s. ;
(iv) 67 10s. ; (v) 108 16s. ; (vi) 629 3s. 4d.
ANSWERS TO THE EXAMPLES 373
0.
21. 14 and 26, etc., for 1426 = 14 x 100 -f 26.
22. 5(20m + 12) + d = 100m + 60 + d = 578 .-. 100m + d - 518.
Birthday is 18th day of 5th month. Rule : Subtract 60 ; last
two figures give the day, the rest the month.
23. 10{10(C - 4) + 4} + n = 100 C + n - 400 + 40. Add 400 to
1289, take 40 ; answer, 1649.
24. Subtract 200 and reject the final 33 ; answer, 205.
25. Subtract 250 ; last digit = w, next = 1, others = p ; 7, 4,
124.
27. N = lOa + b = 9a -f (a + b). 9a is divisible by 9, hence if
(a + b) is also divisible the number is divisible, etc.
28. N - lOOa + lOb + c = 99a + 9b + (a + b + c). The first
two terms are divisible, therefore a + b + c is divisible, etc.
29. Yes. 30. Apply No. 28.
31. N = lOOa + lOb + c - 99a + (lOb + c + a). 99a is divisible
by 11, hence if N is divisible (lOb + c -t- a) must be divisible.
32. N - lOOOa + lOOb + lOc + d
- 990a + 99b + {(10c + d) + (lOa + b)}.
Therefore, etc., as in No. 31.
34. N = lOOOa -f lOOb + lOb + a - lOOla + HOb = ll(91a + lOb).
36. N = 99a + lib + {(a + c) - b}. Therefore last term must be
zero or divisible by 11.
38. 102036 and 151602 are divisible.
39. (i) (b - a)(l - t) ; (ii) (a - c)(t + l)(t - 1).
41. 9. 42. 594 is the difference.
43. (i) 2 and 9 ; (ii) 9 and ; (iii) 9 and 9.
44. (i) (a - d)(t - l)(t 2 + t + 1) -f (b - c)(t - l)t ;
(ii) (a - e)(t - l)(t + l)(t 2 -f 1) + (b - d)(t - l)(t + l)t.
46. 8. 47. 9 or 0.
D.
51. (i) 8 feet 5 inches ; (ii) 671, 74 j miles ;
(iii) 82J miles ; (iv) about 30 feet.
52. (i) June 21st, 18 hours 34J minutes ; (ii) 55 days.
54. (i) 51 '5 ; (ii) 27 ; (iii) between 4'5 and 5*5 minutes,
55. h) 800; (ii) 560; (iii) 460; (iv) 175.
60. (i) About 20 miles per hour ;
(ii) after 2 minutes 74 seconds, at 650 yards from A
(iii) after 1 minute 10 seconds, after nearly 5 minutes
(iv) 440 yards from A, 390 yards from B.
E.
ox. nb(a + j b). 02. w r\.
4 nb(4a + TTD)
63. b(3a 2 - iTrb 3 ). 64. *
374
ALGEBRA
65. 7rb{|a 3 - 4b 3 }. 66. hb(3a Trb). 67. whb(3a -
68. (i) 27rr(r 4- h) ; (ii) a(a 4- 2s) ; (iii) *rr (r 4- s),
69. 7rb(a 4- b).
70. a{4(a 4- b) - IT a}.
71. (i) *r(R - 3r)(R 4- 3r) ; (ii) rr(R - 4r)(R + 4r).
72. 7r(R - 4r)(R 4- 4rj.
73. (i) ?r(a 4- nb)(a - lib) ; (h) 7r(a 4- nb)(a - nb)cw.
74. (i) 47r{b - a}b ; (ii) TTW (w 4- 2r) ; (iii) ?rw(2r - w).
(i) 27rL(R 4- r) ; (ii) 27rrL(R 4- r) 4- R 2 - 2rR 4- 3r*
fi) 27rL(2r + t) ; (ii) 27r[L(2r 4- t) 4- 2r 2 4- t 2 ].
'i) 27rL(2r - t) ; (ii) 27r[L(2r - t) + 2r 2 - 4rt 4- 3t 2 ].
9 feet ; (ii) d = ^/(L - h)(L + h).
h
N/(d
3t)(d - 3t).
a 2 -f 5a + 6 ; (ii) a 2
- 13a + 40 ; (iv)
+ a - 6 ;
a 2 - 5a - i>6 ;
^niy . - 7 \- /
(v) a 2 4- 6a 4- 9 ; (vi) a 2 - 14a 4- 49 ;
(vii) Oa 2 4- 7a 4- 2 ; (viii) 12a 2 - 25a 4- 12 ;
(ix) 21a 2 -f 5a - 50 ; (x) 21a 2 - 2a - 40 ;
(xi) 180a 2 4- 81a - 77 ; (xii) 4a 2 4- 12a + 9 ;
(xiii) 25a 2 - 60a 4- 36 ; (xiv) 10a 4 - 21a 2 - 2.
82. (i) a 2 - 5ab - 14b 2 ; (d) 2a 2 - 9ab 4- 4b 2 ;
(iii) 10a 2 4- 29ab 4- 10b 2 ; (iv) 49a 2 ~ 56ab + I6b'
(v) 70a 2 4- 2ab - 12b 2 ; (vi) 26a 2 4- Slab - 35b 2
' a 3 4- 6a 2 b 4- 12ab 2 4- 8b 3 ;
8a 3 - 12a 2 b 4- 6ab 2 - b 3 ;
8a 3 - 36a 2 b -f 54ab 2 - 27b 3 ;
64a 3 4- 144a 2 b 4- 108ab 2 4- 27b s .
a 4- 4 ; (ii) a - 1 ; (iii) a 4- 4 ;
a 4- 2 ; (v) 2a 4- 3 ; (vi) 5a 4- 4 ;
2a - 3b ; (viii) 2a - 5b ; (ix) 3a 4- 7b ;
x (x) 7a - 2b.
84. (i) (a - 3Xa - 2) ; (ii) (p - 6)(p 4- 1) ;
(iii) (p - l)(p 4- 6) ; (iv) (a - 12)(a + 11) ;
(v) (a - 3) 2 ; (vi) (p - 7) 2 ;
(vii a - 4b)(a 4- D) ; (viii) (4a ~ b)(a 4- b) ;
2a 4- 3b) 2 ; (x) (2a - 5b)(3a 4- b) ;
2a - 3b)(a 4- 4b) ; (xii) (lla - 3)(5a 4- 2) ;
a - l)(b 4- 3) ; (xiv) (b - 3p)(a 4- q) ;
x _. p - 2q)(2a 4- 3b).
85. ao 4- ad 4- be 4- bd. 86. ao - ad - be 4- bd.
87. (i) + 9 ; (ii) 4- 16 ; (iii) 4- 36 ;
(iv) 4- 121 ; (v) 4- 9/4 ; (vi) 4- 49/4 ;
(vii) 4- 29 ; (viii) 4- 19 ; (ix) 4- 165/4;
(x) - 17 ; (xi) 4- 4b 2 ; (xii) - lib 2 ;
(xiii) 4- 4b 2 ; (xiv) 4- 25q 2 ; (xv) 4- 9n 2 ;
(xvi) 4- 12b 2 ; (xvii) 4- 52h 2 ; (xviii) - 11 ;
(xix) 4- 15b 2 /4; (xx) 4- 41b 2 /16.
xi
(xiii
(xv
ANSWERS TO THE EXAMPLES 375
88. (i) 2 ; (ii) i ; (iii) 13 ; (iv) 5j^ __
(v) 12 or 2 ; (vi) 5'5 ; (vii) V3a a + 2a ;
(viii) *y7a u - 6a ; (ix) v/18a + 19 ; (x) 4'6.
F.
91. 1'42 miles.
92. (i) 2
95. 30' 6 inches. 96. 18 yards.
97. About 1 miles. 98. '07 second.
99. (i) 1,006,056 sq. miles ; (ii) 2,014,628,180 cu. miles.
G.
101. m = S /(T 4- t). Monthly subscription = total amount to be
raised divided by total time.
102. b (1 - p) 3 /l. It states the breadth of an oblong of given
length which suffers a given shortening on conversion into a
square.
103. B! = s a d 1 /(d 1 - DO ; s a = S 1 d 3 /(d 3 4- D a ).
104. s a = 8i(L2 - d )/(L 3 - d) ; BI = L 1 s a /(L 1 - d -f d).
105. v = V(l*25hr/w) ; 60 mls./hr. io<5. H = nd 3 /65 ; 192.
107. (i) D = 24<v/(5d 3 /Bl); (ii) d = J /(I> 2 Bl/45).
108. B = 50 V{2H/11(10A + S)} ; 10 '8 knots. 109. a = \/A - s.
no. 4 ft. in. a = y^A + b 2 ) - b. 112. a = ^/(A. + b) + b.
113. 5 ft. 114. a = i/{A + (1 - b)*/4} - (1 + b)/2.
115. 35-8 ft., 27'9 ft. 116. 57.
117. b = || -/"-(A+ a 2 ) - a|.
118. (i) 7, 1 ; (ii) 15 ; (iii) 6 ; (iv) 3'89 ; (v) 1-42 ; (vi) 16 ;
(vii) 7, 13 ; (viii) 49-06, 2'93 ; (ix) 279 ; (x) 3-17.
U 9 . (i) 6 ; (ii) 10 ; (iii) J ; (iv) 7 ; (v) 21 ; (vi) 5 ; (vii) 1 ;
(viii) 7; (ix)0-3; (x)2.
120. (i) (c + d)/(a 2 - b 2 ) ; (ii) a + b ; (iii) 2pq/(p + q) ;
(iv) ab/(a + b) ; (v) 2c(a - b)/ab ; (vi) q ;
(vii) q(p - q)/(p 4- q) ; (viii) (a + b)/2 ; (ix) b a /a ; (x) l/(a + b).
EXERCISE XVII.
See ch. XL, 1 ; oh. xn.
A.
1. = L/3 ; = 3L/4 ; = 5L/4 ; 60 feet ; 26 feet.
2. (i) 3; (ii) 4-8; (iii) 6.
3. (i) W - 21A/40 ; (ii) A = 40W/21.
4. (i) W = 13A/20 ; A - 20W/13.
5. v = 2'5t ; 30 ft. /sec.
6. y = Jx. 7. k = * ; y = f x.
8. W - kV ; W = 62-6V,
9. W 61 -1625V.
376 ALGEBRA
B.
10. Q = 24 + 3t/2 ; t = jj(Q - 24) = JQ - 16.
11. Q - 24 - 3t/4 ; t = 32 - 4Q/3 ; empty when t = 32 seconds.
12. y = 24 + 3x/2 ; y - 24 - 3x/4 ; y = x - 16 ; y = 32 - 4x/3.
13. (i) i = 2-4W ; (ii) L = 16 + 2*4W.
15. (i) Q = 3t ; (ii) Q = 21 + 5(t - 7) = 5t - 14 ; (iii) 51 gallons.
16. If t >15, d = 2t/5 ; if t> 16, d = If + 17t/60 ; 4 miles ;
9 '4 miles.
17. t = 60d/17 - 6 T s r ; 75 minutes.
18. (i) r =(30 - 20)/5 = 2 ; (ii) Q - 20 - 2 x 3- 14
(iii) Q = 14 + 2t.
19. Note (using graph) that (30 - 17)/(18 - 8) = 1 *3 and
17 - 1*3 x 8 = 6-6 ;
hence y 6*6 + 1 3x.
20. y = 64 - 3x.
21. 0'3t + 0-4 x 5 = 8, t - 20 minutes.
22. (ii) 14 + 2t + 3(17' - t) = 52, whence t - 13 minutes ;
(iii) Q - 40 + 3t ; (iv) 1 + 3t.
23. 1980 ; 180, 300 ; 6'6 minutes. If t > 6-6, n - 1980 - 180t ;
if 8-25 > t > 6'6, n = 3960 - 480t ; 792 ; after 7 minutes.
4. When x = 10 ; y = 5 4- 8x ; y = 135 - 5x ; when x = 27.
EXEECISE XVIII.
See ch. XL, 2 ; ch. xni., A.
A.
i. 89 feet. 2. (i) 40 ; (ii) 23. 3. 46 6 feet. 4. 26.
5. (i) 781 cms. ; (ii) 21*3 cms. 6. (i) 44 ; (ii) 56.
7. 181-9 feet; 42-9 feet. 8. 16 -3 feet. 9. 691 '5 feet.
10. 429 feet. u. 173'3feet; 267 feet.
12. 33'6 feet; 2077 feet 13. 41 '2 feet.
B.
14. a = p tan a, b = p tan j8, 1 = p (tan a + tan ).
15. p = l/(tana + tan ). 16. p I/ (tan a - tan).
T8. H - htana/(tana - tan ft). 19. d = h(tana - tan/3).
20. h == d tan a tan /3/(tan a - tan). 21. 50167 yards.
22. 16,855 yards. 23. 1135'! feet ; 199*8 feet.
24. 128-2 feet ; 170*1 feet. 25. 216*5 feet above eye.
26. 190*1 feet. 27. (i) NA - d/tana; (ii) h = dtan/3/tana.
28. 721*7 feet.
EXEECISE XIX.
See ch. XL, 2 : ch. XHI., B.
A.
i. 13*8 miles N., 5*9 miles E. ; 12*6 miles N., 11-4 miles W. ; 8*2
miles S., 20'4 miles W. ; 34*7 S., 4*3 E.
ANSWERS TO THE EXAMPLES 377
2. (i) 72 E. of N. ; (ii) 247 mile*.
3. 54 W. of S. ; 10-5 miles.
4. 14'5 miles ; 8 - 3 miles. 5. 20 miles ; 17*7 miles.
6. 7'4 miles N., 28'4 miles E. 7. 75 E. of N., 28*6 miles.
8. 19 E. of S., 43-3 miles.
9. Westing = 17 sin 40 + 6 sin 60 = 16" 1 miles ;
northing = 17 cos 40 - 6 cos 60 = 10 miles .-. tana - i'61 or
course is 58 W. of N. Distance = 10/cos 58 ==> 18'9 miles.
10. Westing = 14 sin 40 - 6 sin 45;
southing = 6 cos 45 + 14 cos 40 ; d = 8 miles.
11. (i) b = c cos a ; (ii) a = c sin a ; (iii) c = b/cos a ;
(iv) c = a/sin a.
12. (i) a = b tana ; (ii) b = a/tan a.
13. (i) cos a = b/c ; (ii) sin a = a/c ; (iii) tan a a/b.
B.
14. 64, 26 ; 65-6 yards.
15. AB - 42 cos 47 -I- 54 cos 35 = 72'9 yards.
16. 42-2 yards.
17. Draw perpendicular CD. BD = 180 cos 24 = 164' 5. Hence
AD = 115-5 - 150 cos BAG. (i) 7 E. of N.E., (ii) 53.
18. 4*2 cms., 7 cms.
19. 9*1 inches, 11'7 inches.
20. 28-9 inches. See ch. v., 12.
21. 18'6 cms. 22. See ch. v., 12.
24. 29, 18 ; 20-1 cms.
25. 5'8 inches.
C.
26. 4956 yards. 27. 4 '3 miles.
28. 471 yards. See ch. v., 12.
29. 689*1 yards. 30 1*9 mile.
31. 9*5 miles, 12 '5 miles. 32. 12 - 2 miles, 13*4 miles,
33. 3 '65 miles. 34. half a mile.
EXERCISE XX.
See ch. XL, 2 ; ch. xiv., A.
A.
I. 34 36' S. 2. 20 - 7 56' = 12 4'.
3. 736 miles. 4. 6340 miles.
5. 2790 miles. 6. 21,600 x cos 60 = 10,800 mile*.
7. 3929 x cos 41 - 2966 miles.
8. 6018 cos 33 = 5049 miles.
9. 1765 miles S., 657 miles E.
10. 3155 miles S., 2716 miles W.
378 ALGEBRA
B.
11. 18-9 miles = 19' N. .-. lat. = 54 19' N. ;
14'78/cos54 = 25' W. .-. long. 34 41' W.
12. 40 miles S., 19 miles B., 25 E. of S., 43'2 miles.
13. 28 40 ; E. of N., 66 J miles.
14. Mid-lat. = 50 approx. Southing 182 miles, easting =
300 cos 50 = 193 miles. Course 47 E. of S. (approx.), dist. =
267 miles.
15. 200 x cos 67 = 78 miles - 1 18'. Final lat. - 47 5' N. ; mid-
lat. = 47 44'. 200 x sin 67 -r cos47f = 274'. Long. =
9 3' W,
16. 61 E. of S. ; lat. 45 45' N., mid-lat. 47, long. 20 55' W.
17. 47 30' W. of N., 354 miles, mid-lat. 50, long., 31 44' W.
18. 43 20' E. of N., 85 38' E.
EXERCISE XXI.
See oh. XT., 2 ; ch. xiv., B.
A.
3. I|. 4 No. 6. H. 8. V- 10. |J. 12. .
EXERCISE XXII.
See ch. xi., 3 ; ch. xv.
A.
I
. See ch.
XV.,
A,
5. (i) y = a
+ bx ; (ii) y
= bx;
2
("1
) y
y
=
3 +
bx;
2x;
s
) y =
y =
bx -
3x;
- a.
(iii)
y = 51 -
3x;
(iv
y
=
rix
- 2
3.
3
(iii
y
y
19-1
0-69
1
+ 1
7x
3x
; P\
; (iv)
V '
y =
3x - 13-2 ;
2-6x - 107.
4
. y = 1-8
+ 4
8/x 2
; y
= 6-6 ; y
= 1
848.
5. y = 5-6 ^x - 2-7 ; 42-1.
6. y = 1 + 3/(l + x) + 5/(l + x 2 ) ; 3J.
7. Ch. xv., A, 5.
8. b - (Q - q)/(P - p) J a = (Pq - pQ)/(P - p).
B.
9. (i) x = 2, y = 3; (ii) x - 1-2, y = 7'8 ;
(iii) x = 10. y = 16'6 ; (iv) x = 7'8, y = 9-2
(v) x = 24, y = 15.
10. After 1J hours, 7J miles from Charing Cross,
11. In 4 days ; 4 feet 8 inches.
12. 12*5 gms., 32 cms.
ANSWERS TO THE EXAMPLES 379
13. (i) x - 4, y = 3 ; (ii) x = 13, y - 7 ; (iii) x - 5, y - 7 ;
(iv) x - 2-1, y =, 5-3 ; (v) x - 9, y - 10 ; (vi) x = 2, y = 2 ;
(vii) x = 3, y = 5 ; (viii) x = i^~, y = 2.
16. No. 17. No.
18. (i) x = 3, y = 2 ; (ii) x = 1}, y = 2 ;
(iii) x = 2, y - 7 ; (iv) x = 3, y - 1 ;
(v) x - 5, y - 3 ; (vi) x = 6, y - 3 ;
(vii) x = 5, y = 2 ; (viii) x = 2J, y = 1.
0.
19. (i) 5x - 2y - 1 ; (ii) 14x - 15y - 1 ;
,.... u - ,. x 4 3 1 i 1 1
(in) x - 8y - 1; (iv) ~ + ^ - g or ^ -h g - y ~ fl .
20. (i) y - 4-5, z - 6-5 ; (ii) x = 6, z - 17. Yes.
21. x = 13, z 14f from both.
22. x 2 - 2xy - y 2 - 0.
(iii) ^ = 1 or x - ^ - 2x = 0.
24. y = ; cos a |, whence a = 70^.
25. 2^/2. 26. x + y = 1.
27. x = | ; a = 30 approx.
EXEEGISE XXIII.
See ch. XJ., 1 ; ch. xvi., A.
A.
2. (i) k x = 800, k a = 200 ; (ii) 160, 40 ;
(iii) 9600/7, 2400/7.
3. xy = 29-16, y = 1620, x = 1 45,800.
4. (5-4, 5-4); 6-4^/2 = 7 '64 approx.
5. (i) x = 10, y - 20 ; (ii) x = 40, y - 20.
7- (p/q> pq)-
10. ^(a/b).
!B.
xx. (h - 1)A = 34-7 ; 6 inches.
12. If inch from top ; 6f inches.
13. (i) y = k/x + 3 ; (ii) y = k/(x + 4 : 7) ;
(iii) y = k/(x - 6) - 2.
14. (i) k = 28 ; (ii) k = 3*6 ; (iii) k = 72.
15. y - 60/(x - 6).
16. (i) (y -i- 5)(x + 2) 30, i.e. y + 5 is inversely prop, to x
(ii) (y - 3)(x -f 7) - 100 or y = 100/(x -f 7) -f 3 ;
(iii) (3y 4- 2)(2x + 3) = 138 or y - 46/(2x + 3) - '
(iv) (y - 20)(5x - 2) - 40 or y - 40/(5x - 2) +
17. (i) 1 ; (iii) Ii ; (iv) 1-2.
18. (2, 4). 19. (2, 4).
380 ALGEBRA
EXEECI8E XXIV.
See ch. XL, 1 ; ch. xvi., B, C.
A.
1. 3' 75 tns. ; 3*5 inches.
2. From graph when h = 20, d = 5'45 .-. when h = 20 x 100
d = 54-5. Similarly, when d = 5 h = 16'8 .-. when d = 50
h - 1680 feet.
3. Charge is directly proportional to cube of half -thickness.
4. (i) 2-2 Ib. ; (ii) 27 feet.
5. (i) y - O'lx 2 ; (ii) y = l'9 v /lT4jJm.) y - 3 = 5ZJ^TJ.
Assume (i) y = kx 2 , (ii) y = k v /x + 4, etc., and substitute
for x and y.
6. y = 2 v /(x - 1). 7- y = 2(x + 3)*.
8. (i) x = 5, y = 15 ; (ii) x - 8, y = 1-6 ; (iii) x - 8, y = 6\/2 ;
(iv) x - 2, y - 50.
9. The curve of y = 2^/(x + 10) must be obtained by plotting
points .
B.
10. The loss of steam- pressure is inversely proportional to the
square root of the diameter of the cylinder, (i) 4*3 Ib. ;
(ii) 25 in.
xx. w = 12/d 2 .
13. (i) y = 12/(x - 3) 2 ; (ii) y - 4 = 2/^(x - 10) ;
(iii) y - 24/x 3 ; (iv) y - 24/l/x.
15. (i) The x-axis and the ordinate at x 3. (ii) The line paral-
lel to the x-axis where y = 4 and the ordinate where x = 10
16. (i) y = 48/(x + 2) 2 ; (ii) y - 21/*/(2x - 3).
17. (i) x * 4, y = 6 ; (ii) x = 10, y = 6 ; (iii) x = 26, y = J.
EXERCISE XXV.
See ch. XL, 1 ; ch. xvi., D.
2. (i) z = kx 2 /y ; (ii) z = Wi/y 2 ; (iii) z = k\Jxly ;
(iv) z = k/x3/y.
3. (i) k = 1 ; (ii) k = 48 ; (iii) k = 21 ; (iv) k = 07.
4. d = 0-0175 rl cos A. = 361 miles.
5. M - 0'J8d 3 tana = 6143.
6. d 3 - (93)"P ; (i) d = 93VI44 = SSVgSTlS - 93 x 5'26 -
489 millions of miles.
(ii) T 2 = (2780/93) 8 = (30) 3 approx. = 270 x 100.
* T = ^276 = 10 ^(IG) 2 + 14 = 10(16 + H) = 165 years.
7. y ax + b/x 2 .
8. a = ^ b = 8.
9. y = ana/(x - 1) - bxcosa/(x - 2) 2 .
10. a = 20, b = 1.
11. tan 5 = 18nCA/d" = 0'02 .-. d - 1.
12. See ch. xvi., D., 2.
ANSWERS TO THE EXAMPLES 381
EXERCISE XXVI.
See ch. xi., 4 ; and (for D) ch. iv., 7.
A.
1. (i) High ; (ii) s = v /32'2d ; (iii) 227 ft./sec. ; (iv) 34 ft./see.
2. d = s 2 /32'2 - 3h ; 26'2 feet.
3. (i) (a + 2)(b - 3) ; (ii) (p + 3q)(x - 2y) ;
(iii) (p + q)(p + q - i) ; (iv) (5 - p)(l + p).
4. (i) 18-6; (ii)4; (iii) 1J ; (iv) 8.
6. 11-3. 7. ab(p + q).
8. (i) y 2x - 3 ; (ii) y = 3 - 2x ; (1, 0) is common.
9. 32 20' E. of S.
10. 105 miles.
B.
j. (i) P = ffp(l + ^J ; (ii) 100/25 - 4 Ib. ;
(iii) greater, for ka/90 is greater for gunpowder ;
(iv) 36,000 Ib. ; yes ; (v) 61,250 Ib.
2, (i) d = j(Ri/R + l)(C a + C 3 ) ;
(ii) G 2 = 2C 1 R/(R 1 + R) - C 3 ;
(iii) Interchange C 2 and C 3 .
3. a 2 + b 2 - (4n 2 + I) 2 . Triads are 4, 3, 5 ; 8, 15, 17 ; 12, 35,
37 ; 16, 63, 65.
4. (i) (2a - 3b + 4c)(2a - 3b - 4c) ;
(ii) (3p - 2q - 5)(3p - 2q -f 5) ;
(iii) (p - q)(p + q a - b)/(pa - b)(qa - b) ;
(iv) 3(a - l)/(a - 2)(a - 4).
5. (i) J ; (ii) 6 ; (iii) 2J ; (iv) J.
6. 10 milen/hr.
7- (3, 4.)
. (p - q)x{l + (p + q)x}.
9 . y =-. 10 - 8 sin x ; (i) 10 ; (ii) 2 ; (iii) 3'368.
8. (p - q)x{l + (p +
9 . y 10 - 8 sin x ;
10. (i) 33 ; (ii) 31 ; (iii) 45'
C.
1. (i) 29-4 gals./min. ; (ii) 376 '3 gals. /in in.
2. (i) 1 = 864-4 Hd 5 /G 2 ; (ii) H - 0*001157 G a l/d 5 ;
(iii) d - 0-2586 4/(G 2 l/H).
3. 3, 4, 5 ; 7, 24, 25 ; 21, 220, 221.
4. 4n 2 (3n - 2). Expression vanishes for n = and n = .
5. (i) f ; (ii) p - q ; (iii) 7 and 2. (The value 2 does not
satisfy the given condition as it stands. It actually sutisne?
(n + l)/(3 - n) - (n + 2)/(4 - n) = 1.)
6. Tea, 2s. ; coffee, Is. 6d.
7. w = kM/d 2 ; k - 16 ; w = 0'133.
8. 218,000 miles.
9. 5/12 = tan 22J appro*. ; 5/15 = tan 18J approX-
10. (i) 73i ; (ii) 59 ; (iii) 10 feet ; (iv) 6 feet ; (v) 25 feet.
382 ALGEBRA
D.
I. Median, 59*5; mean (A.M.), 60. 2. 1'5. 3. 1'57. Former
is less.
4. (i) (a) 7, (b) 8, (c) 2-6, (d) 28 ;
(ii) (a) 16, (b)15, (o)3, (d)4;
(iii) (a) 13, (b) 13, (c) 3^, (d) 3/s.
5. Median in each case = 10 '5.
6. 13*5. Mean deviation, 21.
7. 15 ; 3.
8. Median, 13 years 8 months ; mean deviation, 7 months.
9. Median (i) t>'3 ; (ii) 17 '5. Quartiles : (i) 47, 7*2 ;
(ii) 16-1, 19-65. Quartile deviation : (i) 1'25 ; (ii) 1775.
10. The latter, because less dispersed.
11. The former, because ratio of dispersion to median is less.
12. The latter, because 2-1/822 < 2-0/824.
13. Mi. 3-133 ; Q.D. 0-0595. 14. Mi. 2-5 ; Q.D. 0-082.
15. Mi. 1-985 ; Q.D. 0-185.
17. Mi. 0-5 ; Q! : 0'45 ; Q. : 0'54. 19. Mi. 1-505 ; Q, : 1'4 ; Q 2 : 1'6.
E.
1. (i) 0-449 ; (ii) 0'2245 ; (iii) T447 ; (iv) 0'598.
2. (i) 23-319; (ii) 5-654; (iii) 24-J79.
3. (i) 5-828 ; (ii) 0'072 ; (iii) 9-898 ; (iv) 32*584; (v) 75-944.
4. (i) 7 + 5^2 = 14-070 ; (ii) 26 - 15^/3 = 0'02 ;
(iii) 48 V3 - 38/V/2 = 29-404.
5. (ii 3 ; (ii) 4 ; (iii) 1 ; (iv) 19 ; (v) 31 ; (vi) 13.
6. (i) a - b ; (ii) 4a - 9b ; (iii) ab - 1 ; (iv) pq - r ;
(v) p/2 - q/3 ; (vi) 1 - ab/5 ; (vii) a a - b ; (viii) a 2 - 9b.
7. (i) x/3 + J2 ; (ii) ^3 - J* ; (iii) (3 + ^/5)/4 ;
(iv) (J7 + 2)/3 ; (v) (2^/3 - 1)/11 ; (vi) (3^6 4- l)/44 ;
(vii) 2(6^/11 + x /6)/391 ; (viii) 14 - 2^/35 ;
(ix)(lOv/33 + 24)/227.
8. (i) x/2(a + ^2)/a(a a - 2) ; (ii) 4(3p + 4 v /3)/p(9p 2 - 48) ;
(iii) (5a - b N /5)b/5a(5a 2 - b 2 ) ; (iv) Px/2(3 + pj3)/3(3-p).
9. (i) 3-995; (ii) 12'218 ; (iii) 6'162 ; (iv)l'618; (v) 0*528.
10. 12'36 cms.
F.
1. (i) 4-8 feet ; (ii) 2 feet nearly.
2. v = N /[58-6{R/(p 2 - 1) - 0-05}].
3. The digits of the difference are (a - o - 1), 9, (10 - a + c).
These reversed give the number (10 - a + c), 9, (a - o - 1).
The sum of the two numbers is always 1089.
4. (i) (2n + l)/(2n 2 + 2n + 1) ;
(ii) 2n(n + l)/(2n 2 + 2n +1). [See C, No. 3.]
5. (a 2 + ab + b 2 )(a 2 - ab + b 2 ).
6. (i) 10/(2a - 3) ; (ii) i ; (iii) 4.
7. 4/7. 8. (i) n>2*; (ii) n>J ; .(iii) n<l/2 N /8.
9. 8 ; 1 10. 10. 36-55 yards - 109-65 feet ; 188'4 yards,
ANSWERS TO THE EXAMPLES 383
G.
1. (i) 1162 cu. ft./hr. ; (ii) 1349 cu. ft./hr.
2. (i) H = Qgl/lOOOd 8 ; (ii) d = /(Qgl/1000H).
4. (i) (4a 2 + Gab + 9b a )(4a 2 - 6ab + 9b*) ;
(ii) (a + i + I)(a - i + i) ; (0 (P + q - i)(p - q + i)-
5. (i) 26/121; (11)8^3/121.
6. 2(a* + b a ) - 12. 7. (i) 6 ; (ii) I ; (iii) 13 ; (iv) 1 ; (v) 1.
8. 700. 9. llOft./sec., lOft./sec.
10. (i) 50 Q ; (ii) 26'1 inches.
H.
1. (i) 1-44 feet ; (ii) 16'82 feet.
2. (i) B = E/d - Gs/(G + B) - R ;
(ii) G = s(R - r - B)/(s - R + r + B).
4. (i) (4n 2 - l)/(4n* + 1) ; (ii) 4n/(4n 2 - 1). [See B, No. 3.]
5. ^19/2. 6. (i) 3f ; (ii) c ; (iii) 10, 5 ; (iv) 8.
7. 132 and 228. 8. (x - 7)(y - 12)-= 64; 12\ inches.
9. 3f cubic inches.
10. (i) sin a = ^ a = 43 40 , .
(iv) 26 30', 59 30'.
SECTION II.
EXEKCISE XXVII.
See ch. xvii., 1 ; ch. xvin., A.
A.
i. 3rd floor. 2. (i) 27; (ii) 46'1.
3. - 09, - 0-21, - 56, + 1-82, - 0*54, + 0'62
- 1-51, - 0-02, - 1-62, + 0-63, + 1*80, + O'Gl.
4. 15 8s. 3d.
Jan. Feb. etc.
5. 1. + 3 min. 1 + 13A min etc.
15. + 9 min. 14. + 14 min. etc.
B.
7. (i) - 6 ; (ii) - 10-3 ; (iii) - 7'92 ; (iv) - 2 15s. ;
(v) + 4^ Ib. ; (vi) -- 4 hours 24 minutes ; (vii) + 68*25
?. (i) 20 7 ; (ii) 80 14 ; (iii) 5<JJ 21J ;
(iv) 10-15 2-45 ; (v) 1'24 + 81.
to. (i) 107 19s. 28 13s. ; (ii) 44 20 13s. ;
(iii) 5 4s. lOd. 3 10s. 2d. ; (iv) 3 crwt. 11 Ib. + 106 Ib.
'12 hours 57 minutes 1 hour 48 minutes.
44 ; (ii) 2- 6 ; (iii) 171 12s. ; (iv) 7J cu. feet ;
12. (i) - 5 ;' (ii) + 3 ; (iii) - 3 ; (iv) + 15'8 ; (v) - 20 ;
(vi) - 13-35 ; (vii) - 21-25.
13. (i) + 4 ; (ii) + 47 ; (iii) + 27 ; (iv) - 12-2 ; (v) - 26-3;
(vi) - 29; (vii) - 6'8.
34-9 3-2
39-1 4-9
31-8 5-0
26-2 2-6
24-8 2-4
27-4 3-3
287 5-4
58 correct average
31-4 1-5
31-95 1-85
33-5 2-5
34-65 4-35
31-15 7-65
30-9 7-3
34-1 7'1
0.
64-6 7'3
65-8 12-7
65-8 11 2
62-8 9-3
64-2 11 6
63-8 15-1
L t x 62-1 11-2
1 6. Correct average 47
384
ANSWERS TO THE EXAMPLES 385
17. 52. 18. 3 hours 5 f " r minutes.
19. (i) 100 ; (ii) 15 3 ; (iii) +* 5 ; (iv) - 3.
20. 30-41 ; 32-52 ; 04-16 ; fourth week in January.
21. 3-83; 4*61 ; 11-2. 23. 24 hours 51 minutes.
25. 24 hours 27 minutes.
27. 13 paces north of the starting-point.
28. (i) 50 yards N. and 20 yards W. of starting-point ;
(ti) 3 yards S. of starting-point.
29. 80 feet N., 260 feet E., 50 feet down (or 70 feet above ground).
EXERCISE XXV EIL
See ch. xvn., $ 2, 3 ; ch. xvm., B.
A.
i. Oral. 2. R = (+ 7) + (- 3) = + 4.
3. R - - 4, i.e. fourth floor. 4. R = + 47. 5. R = + 40.
6. R - + 17. 7. p - (+ <)) - (- 3) - + 12.
8. He rode back p = - 3.
9. The first component p = (+ 14) - (- 2) - (+ 6) = +10;
.*. turning is 8 miles out.
10. (i) R = + 2 ; (ii) R = + 7 ; (iii) R - - 6 ; (iv) d = + 4 ;
(v) d - + 9 ; (vi) d = - 15 ; (vii) k - - 1 ;
(viii) m = + 2.
B.
n. S - + 150 + 7'5t ; (i) 180 ; (ii) 105.
12. A = + 243 - 12t ; (i) 411 ; (ii) 3.
13. T = + 1317 - 5t ; (i) 1377 ; (ii) 1277.
14. = + 620 - 4t ; (i) 524 tons ; (ii) 716 tons.
15. See ch. xvn., 2 ; (i) 160 tons ; (ii) 30 hour* hence ;
(iii) 45 hours ago ; (iv) 48 hours ; (v) 6 days.
16. B = + 33 - 7t ; (i) balance of 61 ; (ii) deficit of 9.
17. B = - 8 + 4t ; (i) deficit of 28 ; (ii) balance of 12.
18. P - + 144 + 16t ; (i) - 240, i.e. 240 of capital not repaid ;
(ii) + 210, i.e. total profit of 216.
19. P - - 120 + 24t ; (i) - 408, i.e. 408 of capital not repaid ;
(ii) + 744, i.e. total profit of 744.
20. (i) 9 months ago, 5 months hence ;
(ii) 2 years 9 months hence.
21. 432, 840. 22. Column graphs needed (ch. iv., 4).
23. h = + 300 + 0'06d ; (i) 345 feet ; (ii) 273 feet.
24. D = + 1000 + Jd ; (i) 930 feet ; (ii) 1200 feet.
25. T - + 110 -i- 0-02d ; (i) 130 ; (ii) 94.
26. T = + 50 + 0-02d.
27. d - 50(T - 50) ; (i) 5500 feet ; (ii) 2050 feet.
29. B - + 29-8 - 0-OOlld ; (i) 30-625 inches ;
(ii) 28-623 inches.
30. B - 30-625 - 0-OOllh.
31. h = 909(30-625 - B) ; (i) 1386 feet ; (ii) 295 feet,
26
386 ALGEBRA
C.
32. (i) - 23 ; (ii) -f 11 ; (iii) - 48 ;* (iv) ,+ 57 ; (v) - 13
(vi) - 18.
33. (i) A = p + 2q -3r = + 4 ; (ii) G = lOp - 6q = + 136 ;
(iii) n = 5p + 2q + 2r = + 34 ;
(iv) E = - Ap + iq = - 1J ;
(v) B = 3p + |q - 4r = + 35 '6 ;
(vi) M = 10'3p - 6'8q - 3 -or - + 150*8.
34. (i) 18n ; (ii) 9a - 12*3b ; (iii) - l*6p - 3*9q ;
(iv) - a + 5Jb + 5jo ; (v) - 6a - 21b.
35. (i) 3a - 6b + 6c ; (ii) - 3a + 6b - 6c ; (iii) 2a - 4b + 4c ;
(iv) - l*6p + l*6q - 10-1 ; (v) 2q - r - 1.
EXERCISE XX IX.
See ch. xvn., g 2, 4 ; ch. xvin., G.
x. d = + 32 + vt ; (i) + 119J ; (ii) - 55 ; (iii) - 37 :
(iv) -i- 65 ; (v) 0. [+ means north and - south of Doncaster. {
2. d = d + vt ; (i) d = + 250 + ( - 12)(+ 72) = - 614, i.e.
614 miles west 01 St. Paul ;
(ii) d = - 120 + (- 15)(+ 15) = - 345 ;
(iii) d = - 120 + (- 15)( + 10) = + 30, i.e. 30 miles east of
St. Paul.
3. d = do + ht ; d - + 13 + (- J)( - 12) = 16 feet.
4. d = d + 4 im ; d = + 400 + 4(- 2'4)(+ 4)- + 361*6 fms.
5. h = ho + 5280m sm a ;
(i) h = + 2500 + 5280(- 2*)(- 0'123) - + 876 feet ;
(ii) + 4448 feet.
6. d = p + q sin,a ; d = - 1-6 + (+ 2'25)(+ 0-8) = - 0-5, i.e.
half a mile south.
7. B = B + pt ; (i) B = - 130 + (+ 20)t ; (ii) overdraft of 30 ;
(iii) overdraft of 250.
8. v = u + at ; v = + 27 + ( - 3)fc ; (i) 12 mls./hr.; ^
(ii) 39 mls./hr. ; (iii) at rest ; (iv) 6 mls./hr. in opposite
direction.
9. d = Vl t, - v 2 t 2 ; d - (+ 24)(- Ii) - (- 20)(+ 2J) - 14
miles.
10. t = (v - u)/a ; (i) t - {(+ 8) - (+ 28)}/( - 4) = 5 minutes ;
(ii) t = {(- 14) - (+ 28)}/(- 4) = lOJ minutes.
B.
u Oral. 12. T n = + 10 + 3n ; + 160. 13. T n - + 10 -3n ;
- 170.
14. Ta = + 10 + 3n ; (i) n = - 100, T n - 290 ; (ii) - 590.
15. Tn = + 8 - 5n ; (i) - 92 ; (ii) + 68.
!6. T n = - 4 + 4n ; (i) - 4004 ; (ii) + 3996.
17. (i) - 19, - 16, - 13, - 10 ; (ii) - 4, - 1, + 2, + 5 ;
(iii) T n = - 7 + 3n ; (iv) - 127 ; (v) + 38.
ANSWERS TO THE EXAMPLES 387
18. (i) + 24, 0, - 24, - 48 ; (ii) - 96, - 120, - 144, - 168 ;
+ 11,928.
19. Oral, (i) - 317-2, + 1667 ; (ii) - 623-5, - 50125.
20. (i) T n - - 3 + 18n ; (ii) T n - - 8'6n ;
(lii) T n = + 1} - 3f n ; (iv) T n - 0'9 + 0'07n.
21. T n - T 4- dn.
22. (1 = 6-7, To = - 117, T n = - 117 + 6-7n. 23. T n = + 68 - 5n.
24. T n = - 10-8 + l-25n. 25. T n = - 22-3 - 27n.
26. T n - -53-3n.
27. d = (u - v)/(p - q) ;
T = u - p(u - v)/(p - q) v - q(u - v)/(p - q)
- (pv - qu)/(p - q) ; T n - {(pv - qu) + n(u - v)}/(p - q).
C.
28. d - 4 ; 7, 11, ... 27. 29. - 1, - 5, - 9, - 13, - 17.
30. d = (1 - a)/(n - 1). 31. d - (1 - a)/(n + 1).
32. Oral. 33. - 47-3, + 10'3, - 40*9.
34. 8, 53. 35- 15-
36. 7th, no, 9th, 13th, no.
37. Between 32nd and 33rd.
38. Yes, 3d ; diff. = pd. 39. Diff. = d/(p + 1) ; n + (n - l)p.
40. Oral.
EXERCISE XXX.
See ch. xvn., 5, 6 ; ch. xix., A and B.
A.
2. (i) 500,000,500,000 ; (ii) n(n + l)/2 ; (iii) 10000 ;
(iv) n' 2 ; (v) n(n + 1).
3. (i) + 816 ; (ii) + 120 ; (iii) - 127 ; (iv) 178*.
4. (i) - 12300; (ii) - 903; (iii) + 20'4; (iv) + 102.
5. (i) + 1326; (ii) + 151}; (in) - 990; (iv) - 112-2.
7. 2550. 8. 496 yards. 9. 114 8s.
B.
10. (i) 150; (ii) 2'27 . . . ; (iii) 0'54 ... n. 20 gallons.
12. 20 gallons. 13. (i) 20 gallons ; 1*6 gal./min.
15. 6'72 gals./min. 16. 8*712 gallons. 18. (i) 3 feet 9 inches ;
(ii) 15 feet; (iii) v - 0'3t; (iv) s = 0'15t a .
See ch. xvn., 6 ; (ii) v = 5t ; (iii) e = ^t 2 ;
40 mls./hr ; (v) 2 mis.
20. (i) s - }at 2 ; (ii) s = 30at a . 21. (i) v = 15 + 3b ;
(ii) v = J + At ; (iii) s - it + At 2 .
22. (i) s = ut + }at a ; (ii) s = (ut + }at?)/60.
24. h) v = 40 - 8t ; (ii) s = (40 - 4t 2 )/60 = | - At a .
26. (i) After 6 seconds ; (ii) 54 feet ; (iii) 18 ft. /sec. downwards ;
(iy) at starting-point ; (v) - 42 ft. /sec. ;
(vi) 240 feet below starting-point,
26 *
19. (i)
(iv)
388 ALGEBKA
60 seconds ; (ii) 40,000 feet ; (iii) 960 ft./sec ;
25,600 feet high ; (v) 320 ft./sec. downwards ;
38,400 feet high.
B = + 7'2t + 3'2t 2 ; (ii) s = - lOt + 4'3t a ;
s = - 8'5t - l*8t a .
Speed in cms. /sec. : A, - 56'8 ; B, - 96 ; C, + 27'5.
(ii) Dist. from origin in cms. : A, + 248 ; B, + 530 ; C, - 95.
(iii) Speed in cms./sec. : A, + 0'8 ; B, - 18'6 ; C, - 4'9.
(iv) Dist. from origin in cms. : A, - 4 ; B, + 14'3 ; C, + 6' 7.
(v) A 1$ seconds ago and then moved + ly ;
B 1 T V seconds hence and will then move + ly ;
C 2 } seconds ago and then moved - ly.
C.
31. 10'4 sq. cms. (approx.). 32. 13'8 sq. cms. (approx.).
33. About 1100 cu. inches. 34. As No. 33. Areas of all equi-
distant sections assumed in A. p.
35. About 2450 cu. feet. 36. About 1300 c.cs
EXEECISE XXXI.
See ch. xvn., 7 ; ch. xx., A and B
B.
6. Expr. => (i) 6a 2 + 5ab - 6b 2 ; (ii) 8p a - 18pq + 9q 2 ;
(iii) 6x 2 - 23xy + 20y 2 ;
(iv) abx'- 5 + (a 2 + b a )xy + aby a ;
(v) abx 2 - (a 2 + b 2 )xy + aby 2 ;
(vi) abx 2 + (a 2 - b 2 )xy - aby 2 ;
(vii) ^x 2 - x + 1 : (viii) x- - (pb/a + qa/b)x -i- pq ;
(ix) 6/a a + 5/ab - 6/b 3 ;
(ix) 6/a a + 5/ab - 6/b 3 ;
(x) ab/x 2 - (a 2 - b 2 )/xy - ab/y 2 .
13. n 8 - 1. 14. n 4 + n a + 1.
c :! - 3abc.
12. n 3 + 1. 13. n 8 .
17. a 3 + b ! + c :! - 3abc.
0.
19. (i) 3766 ; (ii) 29631 ; (iii) 305206 ; (iv) 1020201 ; (v) 715 ;
(vi) 90909.
20. 3t 3 + 8t 2 + 13t + 6. 21. 8t 3 - 48t 2 + 76t - 24 - 3936.
22. (i) 6t - 13t 5 + 18t 4 - 17t 3 + 12t 2 - 7t + 2 ;
(ii) 3t 4 - 8t 3 + 14t 2 - 8t + 3 ; (iii) t 8 - t 3 - t a + 1 ;
(iv) 32t 5 + 1 ; (v) t - 243 ;
(vi) 6t - 9t 5 - 4t 4 + 3t 8 + 9t 2 - 2t - 3.
23. (i) a + b ; (ii) a 6 - a 4 b 4 + 2a 3 b 3 + b ;
(iii) x 3 + 3xy + y 3 - 1 ; (iv) I/a 4 - 1 ;
(v) 16p 4 /81q 8 + 4p 2 /9q 4 + 1.
24. a 2 + b 2 + c 2 + 2ab + 2bc + 2ca.
25. (i) a 2 + b 2 + c 2 - 2ab - 2bc + 2ca ;
(ii) a 2 + b 2 + c 2 + 2ab - 2bc - 2ca ;
ANSWERS TO THE EXAMPLES 389
(iii) 4a 2 + 9b 2 + c 2 + 12ab + 6bc + 4ca ;
(iv) 4a 2 + 9b 2 + c 2 + 12ab - 6bc - 4ca ;
(v) p 2 + 4q a + 9 - 4pq - 12q + 6p ;
(vi) 9p 2 + 4q 2 + 1 - 12pq - 4q + 6p ;
(vii) a 2 /b 2 + 4 + b 2 /a 2 + 4a/b + 4b/a + 2 ;
(viii) a 4 + b 4 + c 1 + 2a a b a + 2b 2 c 2 + 2cV 2 ;
(ix)
(x) I/p 2 +~4/q 2 + <)/r^ - 4/pq - 12/qr 4- 6/rp.
D.
26. 1, 10, 45, 120, 210, 252, 210, 120, 45, 10, 1.
27. a 7 7a"b + 2ia 5 b 2 35a 4 b< + 35a- { b 4 21a 2 b 5 + 7ab b 7 .
28. (i) 1 + 3a + 15a 2 /4 + 5a- 5 /4 4- 15a 4 /10 4- 3a 5 /l(5 I- a' ] /64 ;
(ii) 32p 5 - 80p 4 q/r + 80p'q a /r' 2 - 40p-q- ! /r i 4- iOpq*/r 4 - q 5 / i<5 -
30. (i) 4- 4536a 4 ; (ii) 4- 20p 4 ; (iii) - 2^G8ab J ; (iv) - 15pVr* ;
(v) 4- 2835/8; (vi) - 280a*b<p, + 112a-'b 4 /p.
EXERCISE XXXII.
See ch. xvn., 8 ; ch. xxi., A.
A.
1. York, acreage 37 x 10 tj , pop. 397 x 10 s ; etc.
2. (i) 8-3 x 10" acr., 5'9 x 10 pop. ;
(ii) 07 x 10" acr., 4'8 x 10 pop. ;
(iii) average acr. 2*1 x 10 e and 0'175 x 10"- or 21 x 10' and
175 x 10 5 ; (iv) pop. 1-5 x 10" and 1-2 x 10' J .
3. Oral. 4. Oral. 2-3827 x 10 4 , etc.
B.
5. 1-47 x 10 13 miles. 6. 677 x 10 20 Ib. 7. 9'2 x 10 ltf cu. inches.
8. (i) 9-21 x 10 7 ; (ii) 7'21 x 10 1 -' ; (iii) 7'9ti x 10 10 ;
(iv) 5-06 x 10 11 ; (v) 072 x 10 l .
G.
9. 8-33 mins. 10. 3 x 10 17 tons. ix. 3'6 x 10 2 cu. miles.
12. (i) 2-03 x 10; (ii) 6'92 x 10 ; (iii) 2 x 10 2 .
D.
13. (i) a 2 b 3 c ; (ii) a 5 /b 2 p 3 q 5 ; (iii) 1 ; (iv) 8p :5 q 3 r 3 /125 ; (v) a" ;
(vi) a 8 ; (vii) p 12 ; (viii) p 12 .
14. (i) a ; (ii) a 4 ; (iii) a 2 /b ; (iv) px/q- ? ; (v) abc ; (vi) a ;
(vii) p 2 ; (viii) p 2 /qV ; (ix) a 3 b 9 .
15. (i} X 2l y 2m /z 3n ; (ii> x n + 1 , y 1+m , z m+D ;
(iii) a n + 21 , b 1 -*- 2 *, c in+2n ; (iv) aP* 1 -**, b"'- 1 ), c 1 ^- 10 ) ; (v) a 1 .
16. (i) 3-84 x 10 ; (ii) 1-344 ; (iii) 0'0000686.
17. (r 2 8 x 3 5 x 5 4 x r ; (ii) (2 4 x 7 x 13 2 )/(3 x 5 x 17).
(iii) (3 a x H 4 )/(2 5 x 13 2 x 19 x 23).
390 ALGEBRA
EXERCISE XXXIII.
See ch. xvu., 8 ; ch. xxi., B.
1. 2-48 x 10 ~ 2 , 372 x 10 ~ 4 , 6781 x 10 - 5 , 6 '25 x 10 - 1 ,
1-875 x 10 - a , l'2o x 10 ~ 4 , 3-6 x 10 - 4 , 2'3 x 10 - , 2'5 x 10 - ,
4 x 10 - 1 *.
2. 0-45 x 10 - 1 , 2-31 x 10 - 1 , 1-93 x 10 - l , 611 x 10 - l , or
6*11 x 10.
3. 6'67 x 10 ~ 8 seconds. 4. 118 x 10- 6 lb. 5. 417 x 10 - 5 Ib.
B.
6. (i) 4 x 10 - 4 ; (ii) 174 x 10 a ; (iii) 111 x 10* ; (iv) 1-0.
7. (i) 2 l x 17 - 1 ; (ii) 2 - 6 x 13 - a x 23 A ;
(iii) 2 4 x 11 x 13 x 19 - 1 .
8. (i) a y c' 2 /b l ; (ii) 2b 5 /3a 7 ; (iii) 20r/pq 6 ;
(iv) {bc(b a - (r) + ca(c a ~ a a ) + ab(a a - b a )j/a a b a c a
= {a s (b - e) + b ! (c - a) + c s (a - b)}/a a b a c a .
9. (i) 4p a qr " 5 /5 ; (ii) x - 3 ; (iii) 2a - 3 b 3 ;
(iv) x- 1 " 3n > y m " 31 7> >u ~~ 3 ' n ; (v) x m(1 ~ n) y*( m - i) z l(n ~ m> .
EXERCISE XXXIV.
See ch. xvu., 9.
A.
1. (i) 6p + llq ; (ii) a + 3b ; (iii) 4m - 3n ; (iv) a - 1/b ;
(v) 2a + 3/b ; (vi) p + 4 ; (vii) p 2 - 13 ; (viii) ax- - 10 ;
(ix) x- - 8a ; (x) p/3q + 9 ; (xi) a* + 2a + 1 ;
(xii) a 2 + ab + ^b 2 ; (xiii) 4/p a - 4q/3p + q' 2 /9 ;
(xiv) 4a 2 - 6a + 9 ; (xv) p 2 /16 - p/4 + 1 ;
(xvi) a 2 p 4 + ap' 2 q 2 + q 4 ; (xvii) 1 + 5/3r + 25/9r' 2 ;
(xviii) a + b + c ; (xix) p - q + r ;
(xx) a 2 + b 2 + c 2 + ab + be + ca ;
(xxri a 2 + b 2 + c 2 + ab - be - ca) ;
(xxii) x 2 + y 2 -i- z 2 - xy + yz - zx.
2. (i) (a + 5)(a + 4) ; (ii) (x - 5)(x - 4) ; (iii) (a - 8b)(a - 7b) ;
(iv) (px 2 + 13)(px 2 + 10) ; (v) (4x* - p)(3x 2 - p) ;
(vi) (3a + 2b)(2a + 3b) ; (vii) (3a - 4b)(2a - 3b) ;
(viii) (4x - 5)(2x - 7) ; (ix) (2x + 7) 2 ;
(x) (4x 2 - 5/y) 2 ; (xi) (a + 4)(a - 2) ; ,
(xii) (a - 4)(a + 2) ; (xiii) (p + 7q)(p - 4q) ;
(xiv) (ap - 7)(ap + 4) ; Txv) (5a - l)(4a + 1) ;
(xvi) (5p + q)(4p - q) ; (xvii) (3x - 2)(2x + 3) ;
(xviii) (3a + 2b)(2a - 3b) ; (xix) (7a/p 2 - 4)(3a/p 2 + 6) ;
(xx) (6 + 7p 2 )(2 - 3p 2 ) ; (xxi) (13 + 2a/x 2 )(3 - 4a/x 2 ) ;
(xxii) (2a - 3)(4a 2 + 6a + 9) ; (xxiii) (1 + x/4)(l - x/4 + x lj /16) ;
(xxiv) (3a + 4b)(9a 2 - 12ab + 16b 2 ) ;
(xxv) (p - 2/p)(p 2 + 2 + 4/p 2 ) ;
(xxvi) (l/3a 2 + 5/b 2 )(l/9a 4 - 6/3a 2 b 2 + 25/b 4 ) ;
(xxvii) (x 2 + xy -f y 2 )(x 2 - xy + y 2 ) ;
ANSWERS TO THE EXAMPLES 391
(xxviii) (a 2 /4 + 3a/2 + 9)(a 2 /4 - 3a/2 + 9) ;
(xxix) (n 2 + n + l)(n 2 - n + l)(n 4 - n 2 + 1).
(xxx) (p + 2)(2p - l)(p 2 - 2p + 2)(4p 2 + 2p + 1).
B.
3. (i) a + b + c ; (ii) a - b - c ; (iii) p + 3q -f 2r ;
(iv) 2a + 3b - 5c ; (v) a/2 - b/3 + c/5 ; (vi) p' 2 + q 2 + }r a ;
(vii) ab - 2bc + 4ca.
4. (i) a 2 - a -f 2 ; (ii) a 2 + a - 2 ; (iii) a 2 - 2a + 3 ;
(iv) 2a 2 + 5a - I ; (v) Ja a - a + 3.
5. (i) p - 2 + 1/p ; (ii) p - 2 - 1/p ; (iii) p - 1 + 1/p ;
(iv) a/p + 1 - p/a ; (v) p a - p + J ; (vi) 2a/3b + 3/4 + 4b/5a.
EXERCISE XXXV.
See ch. xvii., 9 ; oh. XXH., A.
A.
1. (i) a 2 - 3a + 7 ; (ii) 3a a + llab - 2b' J ; (iii) a 2 - Tax 2 - llx 4 ;
(iv) a 3 - 2a a + 3a - 4; (v) 2 + 7p - p 2 + 8p 3 ; (vi) 3a 3 - 7a + 4 ;
(vii) 7 + x - 9x 3 ; (viii) a 3 + 2a a - 3a - 3 ;
(ix) 2a 4 - 3a :< - 7a 2 - 3a - 7 ; (x) 1 + 3a + 9a a + Ou- { + 18a 4
2. (i) a - 4; (ii) 2p - 3q ; (iii) n 2 - 2n + 1 ;
(iv) a :5 + 3a 2 - 2a + 3 ; (v) p 3 - 3p a + 2p - 1 ;
(vi) 2a 3 - 3a + 5 ; (vii) 2a :J - 3a a + 7 ;
(viii)
(ix) 2 - 6x a + 3x 4 ; (x) 1 + x - 2x a - 5x 3 + x 4 .
3. (i) a-' - a a + a - 1 ; (ii) a 3 - a a b + ab 2 - b 3 ;
(iii) a 4 - a 3 + a 2 - a + 1 ; (iv) p 5 - p 4 q 4- p'*q 3 - p' 2 q- J + pq 4 - q 5 ;
(v) 1 + a + a 2 + . . . + a 8 ; (vi) 1 - a + a' 2 - ... + a 1 ' 2 .
B.
6. (i) P - a - 6, Q = + 19 ; (ii) P = 2a + 9, Q = + 26 ;
(iii) P - a' 2 - 4a + 6, Q = - 7 ;
(iv) P - 2a 2 + 5a + 9, Q = + 48 ; (v) P - a' 2 + 4, Q = + 24 ;
(vi) P - a 3 - 2a 2 + 8a - 33, Q - + 133.
10. 4- 19. 12. (i) - 79/(a + 3) ; (ii) ; (iii) + l/(a + 1) ; (iv) ;
(v) + 3/8(2a - 3).
13. (i) P = a 2 + a- 1, Q= - 3a + 4;
(ii) P = a 2 + a + 2, Q = - 2a ;
(iii) P = 6a 2 + 7ab + 60b 2 , Q = 164ab 3 + 426b 4 ;
(iv) P = a 3 - a 2 + 2a - 3 ; Q = 5a - 4.
14. (i) (a 2 + a - 1) - (3a - 4)/(a 2 - 2a + 3), etc.
15. p = a 3 + 2a + 3, Q = (7 - p)a. i<5. p = 7. 17. 4.
0.
18. (i) 1 - a + a 2 - a :j + 2a 4 /(l + a) ;
(ii) 1-a + a 2 -. . . + a- 2a 7 /(l + a) ;
(iii) 1 4- a + a 2 + . . . + a 5 + 2a/(l - a),
(iv) l + a+a 2 -f. . . + a 8 + 2a 9 /(l - a^.
392 ALGEBRA
19. 2/11000. 20. (i) 1/1458 ; (ii) 3/1093.
21. Error 1/128 is 2/513 of whole value.
25. (i) 1 - a + a 2 - ... - a 5 4- a fl /(l + a) ;
(ii) 1 - a' 2 + a 4 - ... 4- a 12 - a 14 /(l + a 2 ) ;
(iii) 1 + 2a + 4a 2 + . . . + 32a 4- 64a<Y(l - 2a) ;
(iv) 1 - a' 2 /3 4- a 4 /9 - . . . 4- a 8 /81 - a 10 /81(3 + a 2 ) ;
(v) 1 - a - -a 2 - ... - a fl - a 7 /(l - a) ;
(vi) 1 - 3a + 6a 2 - 12a' 4- 24a 4 - 48a 5 /(l + 2a) ;
<Vii) 1 + -a/2 - a 2 /4 4- a^/8 - ... - a fi /64 4- ia 7 /4(2 + a) ;
(via) 1 - 5a/6 -i- 5a a /18 - 5a :{ /54 + 5a 4 /54(3 4- a).
26. (i) 2 4- 6a 4- 18,i a 4- 54a* 4- 162a 4 /(l - 3a) ;
(ii) 3 -- 3a/2 4- 3a <2 /4 - 3a J /8 + 3a 4 /16 - 3a 5 /l(2 + a) ;
(iii) + a 2 /4 + a 4 /8 4- ... 4- ,a 10 /64 + a 12 /64(2 -- a' 2 ) ;
(iv) I - 2a/9 + 4a 2 /27 - 8a'/81 + 16a 4 /81(3 + 2a) ;
(v) A - 5a/4 + 5a a /8 - 5a s /16 -i- 5a 4 /32 - 5a 5 /32(2 -I- a) ;
(vi) ^ + 5a/16 + 16a a /64 + 45a-V256 4- 135a 4 /256(4 - 3a).
29. 1 4- 3a 4- 2a' 2 - a 3 - a 4 (5 - a)/(l - a + a 2 ).
30. 1 - 2x + 3x 2 - 4x 3 4- 5x 4 - x 3 (6 4- 5x)/(l 4- x) 2 .
EXEECISE XXXVI.
See ch. xvn., 9 ; oh. xxn., B.
A.
I. (i) 20 metres ; (ii) 19 metres. 2. (i) 100 feet ; (ii) 90 feet.
3. (i) 33J inches ; (ii) between the 6th and 7th semi-vibrations.
4. 4000 ; 12 13s. 8d.
5. 5727 ; 611 11s. nearly.
B.
6. (i) 3/2 ; 1-88 . . . , 177 . . . , 2*66 . . . ;
20-25, 30-375, 45*59 ; (ii) - 3/4 ; + 26'37 . . . ,
- 14-22 . . . , + 10-66 . . . ; + 3'375, - 2-53 . . . ,
4- 1-89 . . . ; (iii) 1/3 ; 4J, 1, $ ; 1/162, 1/486, 1/1458 ;
(iv) - q/p ; - pW, p 4 /q', - p : Vq 4 ; - 1/p, q/p 2 , - q' 2 /p' { ,
(v) l/a(a - b) ; a 4 (a - b> { , a 3 (a - b)\ a 2 (a - b) ; l/a 2 (a - b)' f
l/a a (a - b) 4 , l/a 4 (a - b) s .
7. 6 x (3/2) 13 = 3 14 x 2 - ia ; 2 14 x 3 - ia .
8. 3 y x 2- 15 ; ( - )"2 2n + 1 x 3 U - J .
9. - q p-; (- )u + i p u + i q -u- a-
10. a n + J (a - b) n ; a - n + J (a - b) - n .
11. (i) 2 ; (ii) 2/3 ; (iii) 144 ; (iv) 25 ; (v) b/(a + b) ;
(vi) (a + b) 3 /2b(a - b) 2 .
12. (i) (1/2)"- 1 ; (n) 25 x (03) ;
(iii) ( - )"a n b- n + 1 (a - b) ~ l .
13. K = (i) (1/2)" - 2 ; (ii) (5/6) 11 ; (iii) (a - b)"(a + b) ".
14. S n = (i) 2 - 1 ; (ii) 5000{(1 02> - 1} ;
(iii) 32{( - )n(5/4)n _ i}/ 9 ; (iv) 8{(9/8)
(v) q 2 {(p 2 /q> -
ANSWERS TO THE EXAMPLES 393
15. (i) T n = ar - ! ; (ii) S n -= a(r - l)/(r - 1) ;
(iii) R n = a(l - r)/(l - r) ; (iv) S = a/(l - r).
C.
17. P = A x (1-03) ~ a ; P = the sum which would amount to A
in n years.
18. A = P(l + i)" ; A P(l + i) - n .
19. 10(5 4s. 20. 91 12s.
22. (i) a - Ai/{(l + i) Q - 1} ; (ii) a - Pi{l - (1 + i) - }.
23. 61 13s. 24. 44 19s. 5d.
25. 3063 6s. 2d. 26. 55 lls. lOd. 28. 200,000. 29. 100.
30. 720 ; 3^ per cent.
KXKBCISE XXXVII.
See ch. xvn., 10 ; ch. xxm.
A.
2. (i) P moves to left occupying - 1, - 2, - 4, etc.
(ii) P vibrates from right to loft, left to right, etc., with decreas-
ing swings, occupying -f 20, - 10, + 5, - 2'5, etc.
3. (i) P moves in towards O, occupying - 1, - J, - 4, etc.
(ii) P vibrates with increasing swings, occupying + 20, - 40,
+ 80, etc.
4. P occupies + 3625, - 1235, + 385, - 155, + 25, - 35, - 15,
- 214, - 19. As n rises P vibrates about - 20 with con-
stantly decreasing swings. As n sinks the swings become large
without limit.
8. Exp. = - 10 + ^ _ P starts from - 10 when x = - oo
and moves to right. It eventually reaches - 10 a^ain from
the left.
9. Exp. s + 10 + x 3 ^ 2 P starts at + 10 when x = - oo and
moves towards the right reaching -f 15 when x = 0. It then
returns to + 10.
ro. j - 4x = + oo if x = - GO . As x rises P moves to the left
crossing O when x = - J. Between x = - J and x = 0, P
traverses the whole negative scale. Directly x becomes posi-
tive P reappears at + oo and again moves down the scale reach-
ing O when x = + j. From x= -ftox= +00 P moves
along the negative scale.
394 ALGEBRA
B.
11. x= -13.
12. - 5. See No. 11.
13 . (i) - 2-48; (ii) + 1. (in) - 2-2 ; (iv) - 2; (v) + 21 '5 ;
(vi) + 21-6 ; (vii) - 7 ; (viii) + 7 ; (ix) - 4J ; (x) 8x~ - 8
.-. x = 1 ; (xi) - 3x 3 + 12 - 0, x - 2 ; (xii) ~ 1.
14. {(P + Q)x - (2P + Q)}/(x - 3x + 2).
15. 2x - 5.
16. (i) P + Q = 0, 2P + Q - - 1 ; .-. P = - 1, Q = + 1 ;
(ii) P - - 1, Q - + 2 ; (iii) P - + 1 = Q.
17. (i) - 3/2 and 0; (ii) + jj, + jj ; (iii) - y/20, - 3/20.
18. (i) 3/(x + 3) - 2/(x + 2) ; (ii) 2/(x - 2) + 3/(x + 2) ;
(iii) l/(2x - 3) - 4/(x + 2) ; (iv) 6/(3x - 2) - 5/(4x + 7).
19. (i) - 2 ; (ii) + 19/30 ; (iii) 35/99 ; (iv) + 13/309.
20. P - 2, Q - 3.
21. (i) 2/(x + 3) - 5/(x + 3) 2 ; (ii) l/(2x - ,3) - 0/(2x - 3) 2 ;
(iii) 3/(3x + 4) - 5/(3x + 4) a ;
(iv) 2/(x + 2) - 8/(x 4- 2) 2 + 10/(x f 2).
22. (i) 3 + 5/(2x - 1) - 12/(x + 2) ;
_.
3x - 6 xT"2 '
(iii) 1 + 2/(2x - 3) + 3/(2x - 3) 2 ;
(iv) 4 - 3/(x + 1) + 2/(x + I) 2 - l/(x + I) 8 .
____ 3 _ . ,.., __ 3 ___
23. (i) (i - 10")(x - 7) ; W (x'^Xx'^T)-
* 3
_ _
_____ (x __ ^^^ _ 9)
we have (x - 6)(x - 9) = (x - 10)(x - 7) ; whence x = + 8.
(x - 9)/(x - 10) - (x - 6)/(x - 7) = {1 4- l/(x - 10)}
- {1 + l/(x - 7)} = l/(x T 10) - l/(x - 7).
Similarly with the other side. Ileuce x = -f 8 as before.
24. (i)2; (ii) 4; (iii) -1.
0.
25- x> + 7/3.
28. (i) + 7/3<x< + 6; (ii) - 22/7<x< + 25/4;
(iii) x>+7or<-2; (iv) - 8 < x < 4 4 ;
(v) (x - 3)(x - 2)<0, i.e. + 2< x < -f 3 ;
(vi) (3x - 2)(2x + 7)> 0, i.e. x < - 3J or + f .
29. m = - 3, n = - 7.
30. (i) + 5, - 3 ; (ii) +1, - 1 ; (hi) - 26/15, + 26/61 ;
(iv) - 4, + 4 ; (v) - 2, - 2 ; (vi) + 9/7, - 2/7 ;
(vii) - 3, + 3, - 1 ; (viii) - 3J, - i ;
(ix) + 3, - 7 or - 7, + 3 ; (x) - 5, - 20 or - 20, - 6.
ANSWERS TO THE EXAMPLES 395
EXEECISE XXXVIII.
See ch. xvn., 11.
A.
1. -0019.
2. (i) 1 ; (ii) 7761 ; (iii) "6251 ; (iv) '625.
3. (i) - '008041 ; (ii) - '00009399.
4. (i) 1 ; (ii) 51* ; (iii) - 9f.
5. (i) 20 ; (ii) 61 25'.
B.
3a
x | x ~_ 1 ) x + 3 x +
7 ' x 1 ; 1 (xVl) + x -~TJ x + 1 " x
)i; (u); (iii) l ; (iv) 3.
= 1 + 2x ; 1'002 ;
= 1 + 2x + 3x' J ; 1 '002003 ;
= 1 + 2x + 3x 2 + 4x :i ; 1-002003004.
20x 3 + 35x 4 .
= 1 - 2x + 3x 2 - 4x 3 + . . . ( -l) r
1 + x (1 + x) 2
1 - 3x + 6x 2 - 10x :{ + 15x 4
(1 + x)< -
- 4x + 10x 2 - 20x + 35x 4 .
_
16. (i) 1-003006010015 ; (ii) -997005990015 ; (iii) - '98399968.
G.
17. (i) -3; (ii) Oor -J; (iii) -i; (iv) - 1.
18. (i) in = 10, n = 9 ; (ii) m == 4, u = 1 ;
(iii) in = V, n = - V- ; (iv) m = J, n = - 1 ;
(v) in = + 3, n = - 7.
19. (i) - 1, + 1, + 3 ; (ii) - 5, 11, 17 ; (iii) 5, 6, 7 ;
(iv) 4, - 1, ; (v) a, b, c.
;396 ALGEBRA
2 3 1 2
5 n~^~2 n - 3
..... _JL ___ 1 1
{Ul) 2(1 + n) 2(1 - n) + (1 - n) 2 '
EXEKCISE XXXIX.
See ch. XXLV., 2 ; eh. xxv., A.
A.
Nvtt. Tangents are given to two decimal places only.
1. (i) Through origin at 58 to x - axis ;
(ii) ditto, raised 4*7 units ; (in) ditto, lowered 8'2 units ;
(iv) through origin at 16f> to x - axis raised 14 '3 units ;
(v) through origin at 102 to x - axis lowered 7'8 units.
2. (i) y - 0-6x + 47 ; (ii) y - + 22 - 2'05x ;
(iii) y - 0'9x - 12'3 ; (iv) y - - 0'4x - 11.
3. (i) y = 0'7x - 2-6, inclined at 35, lowered 2*3 ;
(ii) y =- 07x + 2'5, ditto, raised 2'5 ;
(iii) y = 0'75x 4- 3, inclined at 37, raised 3 ;
(iv) y = - | X + y , inclined 127, raised 6| ;
(v) y = - l'8x 4- 8, inclined 119, raised 8 ;
(vi) y = x 4- 3, inclined 29, raised 3.
6. (i) 7x + <>y - 14 - ; (ii) x - 3y -i- 7 = ;
(iii) 2'9x 4- y - ; (iv) x - 7 = ; (v) y + 2'3 - 0.
7. (i) (0, 4- J) ; (ii) 130^ nearly, 18J nearly ; (iii) 112 nearly ;
(iv) parallel to y - axis ; (v) parallel to x - axis. 8. f.
9. x ~ - 6, function (y) =4-8; graphs all pass through (- 6, 4- 8).
10. - 6, - 1.
B.
12. (i) y = 3x 4- 12 or y = 3(r 4- 4), 4 to left ;
(ii) y = 2-4x - 7'2 or y = 2'4(x - 3), 3 to right ;
(iii) y = - Jx 4- 6 or y = - i(x - 20), 20 to right ;
(iv) y = - 3-lx - 24-8 or y = - 3'l(x - 8), 8 to righ*v.
13- (i) y = 4x 4- 5 ; (ii) y = 4(x 4- 3) - 7 or y = 4x 4- b ;
(iii) y = - l'4(x 4- 5) 4- 6'2 or y = - l'4x - 0'8 ;
(iv) y = - l'4x - 0-8 ; (v) y = f(x - 4) 4- 3 or y = fx - 3 ;
(vi) y - |x - 3.
14. (a) (i) - 8, (ii) - 16, (iii) - 9, (iv) - 28 ;
(6) (i) 4- 4, (ii) - 5, (iii) 4- 15, (iv) - 14.
15. y = 2x 4- 13.
16. (i) y = l-5x - 11-5 ; (ii) y = - 2'3x - 19'5;
(iii) y = 0'87x - 7'22 ; (iv) y = - 1'Ox + 8'3 ;
M 4x 4- 3y 4- 7 = ; (vi) 5x - 2y - 12-4 - 0.
20. (i) and (iii) yes ; (ii) no.
ANSWERS TO THK EXAMPLES 397
EXEECISE XL.
See ch. xxiv., 8; ch. xxv. , B.
A.
3. = sin a. 4. cos a-, sill a. 5. Both = - 1/tfin a.
7. - cot a, - tan a. 8. -- cot a, - tan a. 9. 14414 yards.
10. 2524 yards. n. 3'178 miles, 12.
B.
13. 144, 15, 21.
14. 104J, 29, 46J. 15. 31; 197 "6, 105-8 yards.
16. 5334 sq. yds. nearly. 18. 69'18. 21. 3o 50', 53 10', 90.
23. 1162 square yards nearly.
24. 2000^/1023 + 3000^/209 - 107,340 square feet nearly.
EXERCISE XLI.
See ch. xxiv., 7 ; xxiv., 8.
A.
1. AC - 2-348, BC - 2'244 ; CD - 3'068, AD - 3 31 ;
DE = 2-455, CE - 2-315 ; EF - 2'56, DF - 2 16 ;
FG - 2-00, EG - 2-197 ; GH - 3 662, FH - 3'589 ;
GK = 3 '499, HK - 3*31.
2. 867 feet. 3. 1432 feet. 4. 1436 feet ; 4 feet difference.
5. 312 feet ; 314 feet.
6. C from A, 140 ; D from C, 34 ; E from D, 166 ; F from E, 37 ;
G from F, 161 ; H from G, 53 ; K from H, 173.
7. Partial results : C, - 1-8, + 1'61 ; D, + 2'544, + 1716 ;
E, - 2-38, + 0-594 ; F, + 2*044, + 1'54 ; G, - 1'89, + 0'651 ;
H, + 2-204, + 2-925 ; K, - 3*285, + 0-403.
Final results : DC, - 1-8, + 1 -51 ; D, + 0744, + 3'226 ;
E, - 1-636, + 3-82 ; F, + 0'408, 4 5'36 ; G, - 1-482, 4- 6'011 ;
H, + 0722, + 8-936 ; K, - 2'563, + 9'339.
B. .
9. Sin 220= - sin 40, cos 220= - cos 40, tan 220 = + tan 40.
10. Sin310 = - sin 40, cos 310 = + cos 40, tan 3 10 - - tan 40.
11. (i) sin a = sin (180' - a), cos a = - cos (180 - a), tana =
- tan (180 - a) ;
(ii) sin a = - sin (a - 180), cos a = - cos (a - 180), tan a
= tan (a - 180) ;
(iii) sin a - sin (360 - a), cos a = cos (360 - a), tan a =
- tan (360 - a).
12. Co-ordinates : B, - 139-8, + 207'25 ; C, + 318'6, + 430 '8 ;
D, + 474-8, + 80 ; E, + 852*9, + 454*3 ; F, + 10317, - 213-1 ;
G, + 626-1, - 329-4.
13. About 9 yards.
398 ALGEBRA
14. AB47 W. of N., BC 75 E. of N., CD 34 W. of N., DE 57
E. of N., EF 5 W. of N., FG 58 W. of N.
15. A : 0, ; B : - 2647, + 246'9 ; C : + lll'l, + 347*6 ;
D : - 1517, + 737-3 ; E : -f 191'4, + 960 ;
F : +172-9, + 1171-2 ; G : 0, + 1279-3.
0.
17. cos (-48) - + 0-669 ; sin (-48) = - 0743.
18 Sines : (i)- 0*985 ; (ii) + 0'342 ; (iii) + 0'866 ;
Cosines : (i)- 0'174 ; (ii) - 0'94 ; (iii) + 0'5.
19. (i) - 0-424 ; (ii) + 1 ; (iii) - 2747 ; (iv) + 364.
EXERCISE XLIL
See ch. xxiv., 3 ; ch. xxvi., A, B.
A.
3. (i) 9 to right ; (ii) 8 up ;
(iii)
11 down, 13 to right ; (iv) 9 up, 16 to left.
10
y + 3 = - or y = 12/x
A.
(iii) y - 5 =-~'- ; (iv) y =
10 to
5- () y + 3 = - or y = 12/x - 3 ; (ii) y - -i--
A. A. ~T
(viii) y - I = -
(a) (b) ^
(i) 3 up y - ~ ;
(ii) 3 to right y = ;
7
(iii) 5 down - y
(iv) U to right y - -~ ;
(v) 3 up, 4 to right y = - ;
(vi) 3 up, 5 to right y = - ;
9*25
(vii) 3-5 up, 1*5 to right y = ;
4
(viii) % down, 2 to left. y ~ o~-
ANSWERS TO THE EXAMPLES 399.
7- (i) 2.A - 3 2V3~; (iv) - 1*
(viii) 2 2 -, 5(1 >/3)
B.
10. (i) head down (0, 7) ; (ii) head up (0, 0) ; (iii) head up (0, 7) ;
(iv) head down (0, - 12) ; (v) head up (0, 20) ;
(vi) head down (14, 0) ; (vii) head up (14, 0) ;
(viii) head down ( - 16, - 21) ; (ix) head up (13, - 21) :
(x) head up ( - 7 '8, 15-5).
11. y - k (x + 6)' 2 + 4 ; y = - k (x + 9) 2 - 11 ;
y - - k (x - 14)* - 7 ; y - kx 2 - 17.
12. (a) (b) (c)
(i) lower 4 7 ;
(ii) upper 7 - 5 ;
(iii) upper -1'8 7 '2;
(iv) upper - 9*3 ;
(v) lower - 2 -3 ;
(vi) lower - 21 '3 - 7 '8.
13- (i) y = (x - 3) 2 - 6 ; (ii) y = (x + 5) 2 - 27 ;
(iii) y - - (x - 6) 2 + 43 ; (iv) y = 3(x - 2)' - 7 ;
(v) y = - 7(x - 2) 2 + 17 ; (vi) y = ( x + ) - 71 ;
(vii) y = - 2(x - ) 2 + 27* J (viii) y = 0(x - ' h )* ~ # ;
(ix) y =* - 4(x - V-) 2 + W ; (x) y =, 2'3(x 4- 2'5) 2 - 21-575.
14. (a) (b) (c)
(i) lower - 6 3 ;
(ii) lower - 27 - 5 ;
(iii) upper 43 H ;
(iv) lower - 7 2 ;
(v) upper 17 2 ;
(vi) lower - 7i - ;
(vii) upper 27 '5 + 35 ;
(viii) lower - 8fc 1 ( J T ;
(ix) upper 10 T 9 g I'i ;
(x) lower 21-575 - 2'5.
(v) y = 3(x + 6 + 3>y3)(x + 6 - 3^/3) ;
(vi) y = - 3(x - 6 + 3/v/3)(x - 6 - 3^/3) ;
(vii) y-(5x- 3)(x+4) i _
(viii) y = 2(x - 1-6 + 2Vl-59)(x - 1'6 - 2^/1-59).
17. (i) - 7, 1 J (iii) 2 ; (v) - 6
400 ALGKBHA
(vi) 6 3 ^/S"; (vii) , -4 ; (viii) 1*6 2 X /1 7 59.
18. y - - KX - 4)2 + 7. 19. y j(x - 4) a + 7.
21. - 2x 2 - 24x.
22. y = i(x + 10) 2 - 9 ; - 10, - 9. 23. 2x 2 - 24x - 26.
0.
24. y = - yV(x - 12f + 25-5.
25. y - *V(x - 10) 2 - 5.
26. y = - Afr - 10) 2 + 20.
27- y = - *V(x - 35)2 + 36 . 85? 36
31. y + i = yo(x - 6)' J ; - !-. 32. Nofc parabolic.
EXEBCISE XLITI.
See ch. xxiv., 3 ; oh. xxvr., G.
A.
X. (i) x - or 4, 3 or 1, 3'29 or 071 ;
(ii) x = 2 or - 8, - 4*41 or - 1-59, - 3, - 3 ;
(iii) x - 374 or - 0'39, 0'49 or 3'01 ;
(iv) x = - 1-32 or 0'32, or - 1.
2. (i) 5 '5 or - 25; (ii) imaginary ;
(iii) 0*33 or - 0'67 ; (iv) imaginary.
3. (i) + 3, + 1 ; (ii) + 5, - 1 ;
(iii) - 6, - 6 ; (iv) + 8, + 5 ;
(v) + 8, - 5 ; (vi) + 12, + 3 ;
(vii) 0, + 7 ; (viii) 0, - 5-3 :
(ix) + 0-5, -f 3 ; (x) 0'6H, + 1*5 ;
(xii) -f 0'25, - 3'5 ; (xiii) + 0'2, + 0'33 ;
(xiv) -1-6, - 2; (xv) -0\33, + 2-5;
(xvi) + 2-5, + 2-5.
4. (i) x 2 - 2x - 16 = ; (ii) x 2 + x - 56 - ;
(iii) x' 2 - 7x + 12 - ; (iv) x 2 + 20x + 100 - ;
(vj x 3 - 100 - ; (vi) x 2 - 27x - 13 - ;
(vii) x 2 - 2-4x -f 0-63 - ; (viii) x 2 - 6x - 16 =-- ;
(ix) x 2 - 4'6x + 3-85 - ; (x) x 2 - lOx + 22 - ;
(xi) x 2 + 2x - 1 = ; (xii) x 2 - 2'6x - 371 = ;
(xiii) x 2 - 2ax_j- a 2 - b 2 = ; (xiv) x 2 - 2pa 2 x + pV -
(xv) x 2 - 2 N /2x - 1 - ; (xvi) x 2 - 2^5x - 12 = 0.
7- (i) ~ I. - i; (ii) 3-65, 2-35;
(iii) 2-625, - 1'56 ; (iv) - S, I ;
/ ^ a ~ k ab
W alHb^ " a + b'
8. (i) 12X 2 - 5x -_3 = 0_; (ii) 35x 2 - 29x + 6 =0;
(iii) v/lSx 2 +(>/3 - x/5)x -1=0;
(iv) abx 2 + (a + b)x + I = ;
(v) a s b'x a - (a 3 + b 3 )x + a 2 b 2 - ;
(vi) (p a - q a )x 2 - 2(p 2 -f q 2 )x 2 4- p 2 - q 2 = 0,
ANSWEKS TO THE EXAMPLES 401
9- (i) v/
(iii) J_ _____
(v) \/a_ \/a - b ; (vi) _- >/;
(vii) Ax/7 3 ; (viii) W14 is/2.
10. (i) - 4-5 ; (ii) +70; (iii) - 7'7 ; (iv) + 4'3 ;
(v) 6?, ; (vi) - 225.
11. (i) crosses; (iv) crosses ; (vi) touches x-axis.
13. (i) b a \ 4ac ; (ii) b' 2 ^ 4ac ;
(iii) b' 2 = 4ac ; (iv) b u \ 4ac ; (v) b' 2 4ao.
14. (i) Equal roots ; (ii) unequal roots ;
(iii) 110 roots ; (iv) unequal roots ;
(v) ecjual roots ; (vi) unequal roots.
15. (i) V iVTST (ii) 12 or - 1 ;
(iii) - 2 v'll ; (iv) no roots ;
(v) - 2 2y/6. (vi) no rootsj
(vii) - V- W1UJ ; (viii) v /ll
, x N - /s x
(ix) 110 roots ; (x) ----- -- .
4
16. X >-i - xm(a + /a) 4- ur a# -0.
17. (i) x a - 8x - 48 = ; (ii) x u - 13x - 714 - ;
(iii) x- 4- x - 6.
18. (i) x' J H- 4x - 21 = 0, x =- 7 or - 3 ;
(ii) x'- 2 - 19x + 84 - 0, x - 12 or 7 ;
(iii) x 2 - llx -- 102 - 0, x - 17 or ;
(iv) x' 2 + 9x - 11)0 - 0, x - - 1 ( J or 10.
/x b i ,~-\ 1 1
19- (i) x - - or - 1 ; (111) x - - or - - ;
(iv) x = - - or - ; (vi) x = a b ;
' a b x '
, ... a a + 2b , ... N a' J -f b 2 ab
(vii) x = - , or TT J (viii) x ~ -r or c ;
vy a-b a-b' v ' b-a b-a*
20. (i) x - 1'25 ; (ii) none ; (iii) none ; (iv) x - 4'302, 0'697 ;
(v) none ; (vi) none ; (vii) none ; (viii) none.
EXEEC1SE XLIV.
See ch. xxiv., 6.
A.
1. x 2 + 2bx + 4ac = 0.
2. (i) x - ~ ; (ii) x = 5'303 or 1*698 ;
v2
27
402 ALGEBRA
(iii) not possible; (iv) x = 1*39 or - 2'89;
(v) x = 2-309 ; (vi) x = 1-414 ;
(vii) x = - 1*6 ; (viii) not possible.
3. 4 464.
4- (i) + 4, - *; tii) x= - or - 5;
(iii) x = - 0*64 or 0'39 ; (iv) no values.
B.
6. (i) x 3 - 2x 2 - x + 2 = ; (ii) x 3 - 6x 2 + llx - 6 = ;
(iii) x 3 - 4x - ; (iv) x 3 + 3x 2 - lOx - 24 = ;
(v) x 4 - 5x a + 4 = 0; (vi) x 4 + 4x :j + x 2 - 6x = ;
(vii) x 4 - 2x 3 - 13x a + 14x + 24 = ; (viii) x 3 + 3x 2 = ;
(ix) x 4 + x 3 - 30x 2 = ; (x) x 8 - 7x 3 - Ox 2 - 0.
7. (i) 2, 3 ; (ii) + 1, + 9, + 1, + 9 ; (iii) + 1, + 16 ;
(iv) + 49, + 25, + 49 + 25 ; (v) + - 1, + 3 ;
(vi) (x + 2) a (x - 2), + 2, - 2, - 2 ; (vii) 0, + 3, + 4 ;
(viii) x 2 (x + 8)(x - 5), 0, 0, - 8, + 5 ; (ix) (x - l) 2 (x - 2) 2 , + 1,
+ 1, + 2, + 2 ; (x) (x - *y, + 2, + 2, + 2 ;
(xi) - 2, - 3, - 1, + 6 ; (xii) 1, ( - 1 V- 3)/2 ;
(xiiH N/3, v/13 ; (xiv) - 1, + 4, - 2, + 5 ;
(xv) (x + l)'(2x - 5), - l f - l, - 1, + 5/2.
8. (i) x - + 5, - 1 ; (ii) x - - 2, - 7 ; (iii) x - - 3, 1/2 ;
(iv) x = - 3, 1/3.
9. (i) 0, - 120 ; (ii) 0, + 48 ; (iii) 0, 0.
n. y = x 3 - 7x - 6. 12. Latter = former + 1.
13. - 3, - 2, + 2. 14. 0-5 to left ; - 1'5, + 1, + 2.
15. - 4, - 2, + 2-5 ; - 0'8, + 0'6, + T2.
EXEKCISE XLV.
See ch. xxiv., 6 ; ch. xxvi., D.
1. (i) S = 160 - 20t + t 2 ; (ii) S - 100 - 21t 4- 3t a /2 ;
(iii) S = 192 ~ 48t + 3t 2 ; (iv) S - 150 - 40t + 5t 2 /2 ;
The highest points reached are given by (i) t = 10, S = 60 ;
(ii) t = 7, S 26-5 ; (iii) t = 8, S = ; (iv) t = 8, S = - 10 ;
In (iii) the marble just reaches top of board while in (iv) it
shoots off the board.
At a turning point v = 0.
2. (i) t - 10 N/S_- 60 ; (ii) t 7 [(^68 - 169)/3] ;
(iii) t 8 (V'SS/S) ; (iv) t = 8 (^/10S + 100/5) ;
Mid-line crossed when t = (i) .10 2-/10 ; (ii) 0, + 14 ;
(iii) 8 (10^/3/3) ; (iv) 8 - 2^11.
Falls over end when t = (i) 10 4- ^140 ; (ii) 7 4-
(iii) 8 + 10>y/6/3 ; (iv) + 6.
ANSWERS TO THE EXAMPLES 403
3. Greatest height of ball = 72 inches at a distance of 4 feet from
bowling crease. Height on leaving hand 71 '68 inches.
Height when ball reaches pLme of wickets = - 4 '88 inches.
Hence ball falls short.
4. d 4 \/3600 - 50h feet where the negative sign is to be
taken from h = 71*68 to h = 72 and after this value the + sign
is to be taken.
63-39 feet.
The batsman hits ball 7 '23 feet from wicket.
5. Greatest height = 30 feet.
6. d = 99 ^/lOOO (30 - h)/3, where the - sign is taken from
h = -597 feet to 30 feet and after this value the + sign is taken.
The fieldsman is on the circumference of one of two circles of
radius 188*4 feet and 9 '6 feet respectively from batsman.
If ball had not been caught its range would = 199 feet from
batsman.
7. S = 30n - ri j ; S is maximum, when n = 15.
8. n - 15 x/ 225 - 8 ;
(i) n - + 10, + 20 ; (ii) n - - 2, n - + 32.
9. (i) n - (17 x/289 - 4S)/2 ; (ii) n - 8, 9. The 9th term is 0.
10. (i) 72 ; (ii) S is an integer.
11. S = 70, n = 7.
12. (i) 12 ; (ii) n = 1 gives - 22 and this is the value of the
sum obtained by continuing the scries in opposite direction,
n = | ; n V, are neglected.
13. 11 = 21 terms ; 821 = - 861.
14. n - (83 V6889 + 8S)/4 ;
(:
(i) n ~ 3 ; (n) n --= 30 ; (ui) n 42 ;
iv} If n = - 2, S_ 2 - 174.
B.
16. (i) y = (x -h 2)/3 ; (ii) y ^ (2-8 - *)/-7 ;
(iii) 7y - 4x + 8 - ; (iv) 3y + t2x 5 - ;
(v) y - 2 + 3/x ; (vi) y = 2/x - 7 H- 3/2 ;
(vii) y - (15x + 21)/6x + 8 ; (viii) 8xy + 2y - 20x - 5 = ;
(ix) x = 3y 2 + 7y - 1 ; (x) x - - 2y- + y - 8.
17. x = ay -h b.
19- (i) y 5 s " ~ f i > (ii) y <C - 63/8. For inverse functions write
x for y.
20. (i) x = 2y 2 - 3y + 4 ; (ii) x = 7 + 3y - 5y a ;
(iii) x = ay a + by + c ; (iv) y = t -~ -^ ^ i
ex m
(vii) y = (6x + >/8x 31x' J )/4x ;
27 *
404 ALGEBRA
(viii) y - (2x + 1 VI + 4x)/2x ; (ix) y - (x 2 - 3)/2 ;
x 3 + 2
22. (i^ x cannot lie between and 5 ;
(ii) x cannot lio between - 3 and + J ;
(iv) x cannot lie between - 7 and + 1 ;
(v) x>3or< - 7;
(vi) x <^ - 2 or | > x > 2. No limits for values of function.
24. (i) Negative in both cases ;
(i) x = is an upper turning-point ;
(ii) x. = 1 is a lower turning-point.
26. x = - 1 gives a lower turning-point. The inverse function
has no finite turning-points.
27. (i) (x = 0, y = - 3); (x = 0, y ) are respectively lowei
and upper turning points.
(ii) (x = 0, y ~ 7), (x = 0, y = f ) are turning-points.
28. y = (7 + 3x 2 )/(x 2 - 1). The inverse function has turning
values (x = U, y = - 7).
29. y - [ - 3x a ^/9x 4 + 32x a + 16J/2.
30. Upper turning val. 8/31 when x = 3/4.
EXEECISE XLVL
See ch. XJLIV., 6.
A.
3. (i) b = c cos a >/(a 2 - c 2 sin 2 a) ; (ii) sin a not > (a/c) 2 ;
(iii) two ; (iv) sin a a/c.
4. 148| yards, 29* yards. 5. 1'6 or 0'8 miles approx.
7. 11-4 or 129-6 fB - 94 or 13 ; C - 50 or 130.
9. OP . OP 1 - d 2 - r 2 .
13. cos a = x/U - (5/13) 2 }, etc. a - 22 37' or 157 23'.
14. 14 15' or 194 15'. 15. 323 8'.
EXEECISE XLVII.
See ch. xxiv., 7 ; ch. xxvn., A,
A.
T 3
x- (i) ^ ; (ii) -6x 3 ; (iii) - ** ;
(iv) 4x' ; (v) x^ ; (vi) 3'4( - x)*.
2. 144 ; 300 ; 500 ; 3400 ; 307 '2.
3. 480|. 4. 729-6. 5. 335.
6. (A) 542 -9; (B) 66f ; (C) 413 '2.
7. 296^3/3 = 171-2. 8. 76>/10/3 = 80.
9. -04; 106}. 10. 266f. ii. -04x 2 +tx.
ANSWERS TO THE EXAMPLES 405
12. (i) Jx 2 + 7x 3 ; (ii) -8x a + '3X 3 ; (iii) 12x + 2x 2 + ^ ;
o
(iv) 10 -8x - T2x 2 + -Sx 3 ; (v) fr 1 - *x 2 + 7x ;
(vi) + -x 2 - 8x ; (vii) x -f 4x^ ; (viii) 2'6x 2 - 3xi
(ix) *5x' - '8x^ ; (x) '6x ~ '8x^ -f l'2x 2 - T6x 3 .
13. 1499-3; 100; 2656; 1347.
14. y = 20 + 4x - ix 2 ; 533J-.
B.
2n +J. m 385n>
16. Volume of cone = ^?rr 2 h.
17. Trh 2 tan 2 35 ; >h 3 tan 2 35 ; \{(3 + 2h) a (h + lj) - 13J} ; 2023J.
18. 3 + 2h ; (3 + 2h) 2 .
s
2O. 7r(12 - \/3x) 2 ; 7r(144x - 48x* + ix 2 ) ; 384?r.
3k A
24. T x f.
26. (i) *Hx 4 ; (ii) - '8x 4 ; (iii) 8x^ ; (iv) - 6x* ; (v) x - *x 4 ;
(vi) x + x 2 -f x 3 -f x 4 ; (vii)3'2x - 7x 2 -f '9x 3 - "8x 4 ;
(viii) 4x - 9x* ; (ix) x-~+~;(x)x+^+^ + ~-;
(xi) x - ~ + ~ - ~ ; (xii) 8x + 6x 2 + 6x- { + Vx 4 .
27. 777-6 ; 8000 ; 120V10 ; 136 ; 93^ ; 244^.
BXEECISE XLVIIT.
See ch. xxiv.^ 7 ; ch. XXVIL, B.
A.
1. (i) 3h, y/ax - 3 ; (ii) (a - b)h ; ^y/^x = (a -- b) ;
(iii) 41(2xh + h' 2 ) ; 8y/fix - 8'2x ;
(iv) 4'l(2xh + h 2 ) - 37h ; Sy/Sx - 8'2x - 37 ;
(v) 7(3x 2 h + 3xh 2 + h : <) ; 5y/Sx = 21x 2 ;
(vi) -h/(2x 2 + 2hx); Sy/Sx = - l/2x 2 .
2. (i) 3y/5x = 1/4 ; (ii) fiy/5x = - 3'5 ; (iii) Sy/x = 6'4x ;
(iv) Sy/Sx - 8'6x - 1-2 ; (v) 5y/Sx = 12 - lOx ;
(vi) dy/8x - 6x 2 ; (vii) Sy/Sx - 3x 2 - 4 ;
(viii) 8y/5x = 12x 2 + 2x - 7 ; (ix) dy/8x = 3 - l/3x 2 ;
(x) dy/dx = 5 + 7/x 2 .
(i) y = 2x + p ; (ii) y = - ix + p ; (iii) y = 4x 2 + p ;
406 ALGEBRA
3. (iv) y = x 2 - 3x + p ; (v) y - 5x - xx' 2 -f p ;
(vi) y = 2x" - 2x 2 + x + p ; (vii) y = - 3/x + p ;
(viii) y = x + 1/x + p.
4. (i) 19 ; (ii) * ; (iii) ; (iv) - 5 ; (v) 1 ;
(vi) ; (vii) 2 ; (viii) 0.
5. (i) S 2 y/Sx 2 - } ; (ii) ft a y/ax a = 30x - 6 ;
(iii) a 2 y/Sx 2 = 12 (x - 1).
6. (i) y = - fx a - 5x + 1 ; (ii) y = x-x 9 + x-l;
(iii) y = 2x 3 - llx 2 .
8. (i) y = -}ax 4 + p ; (ii) y = x 4 - x 3 + x' 2 + px + q ;
(iii) y = ' 4 x 4 + px 2 + qx + r ;
(iv) y = - -2*4 x 1 + px 3 4- qx 2 + rx + s.
B.
ii. (i) 2 x 10- 5 ; (ii) \ ; (iii) 2 x 10~ 7 ; (iv) 2 x 10 - 10 .
13. (i x 10~ 4 .
15. (i) I ; (iii)' 1-5 ; (iv) 2, - 3 ; (v) f, 3.
16. (iv) - i ; (v) V-
IflXEEGISB XLIX.
Seo ch. xxiv., 7.
A.
1. (i) 9 ; (ii) 0, max. ; (iii) - 3 ; (iv) 0, min. ; (v) 9.
3. (i) V- ; (ii) 0, max. ; (iii) - 36 ; (iv) - 36 ; (v) - 24 ;
(vi) 0, min. ; (vii) *-.
6. q 2 >3pr.
7- (0 (I, V)5 (ii) (- -25,7-3675);
(iv) ( - 4, V), (3, - 25) ; (v) (0, - 1).
B.
8. (ii) Turning-point ; (iii) point of inflexion ; (iv) turning point.
9. (ii) Turning-point ; (vi) turning-pomt.
10. Where x = , x = 2.
n. (iii) Where x = - 1 ; (iv) Where x = -|.
C.
19. No max. Miti. 9727r cubic inches. 21. 10.
22. 2 feet x 1 foot x 1 foot.
EXEECISE L.
See ch. xxiv., 9 ; ch. xxvm., A and B.
A.
2. C - 2,0, N/2. 4. (i) 2; (ii) 2r. 6. 1-158.
7. 3 L v L 3-474. 9. 3'103 L TT L 3-215.
10. 8v/(2 - /s/2), 16 ^ {(2 -
ANSWERS TO THE EXAMPLES 407
n. 3-1012 Z.TT L 3-3136.
12. G - J {(4 - COCj}. 13. J3.
14. 2^3; 2-598 L IT L 5-196.
B.
17. sin - x/J {1 - ^(1 - sin^a) = *J$ (1 - cos a).
18. 0-065.
SECTION III.
EXMRC1SK LI.
See oh. xxix., 2.
A.
1. 2-1 inches ; 2'1 inches ; 2'1 inches ; 1-04S ; 1*038 ; 1-040 ;
the ratio.
2. 1-052 ; (i) 58-3 inches ; (ii) 63'1 inches.
3. 58 '2 inches.
4. Boy : 1-052, 1'038, 1-033, 1-02.
Girl : 1-048, T040, T043, 1-020.
5. 517, 53-8, 55-9, 58'1, 60-4, 62'9 inches.
6. 44'2 inches, 42'5 inches.
7. 1'02. 8. 50719 ; 46857. 9. 0'99. 10. 4386 ; 4566.
12. P - P x (1-02) ; P - PO x (0'99).
13 and 14. The factors are given in the following table :
Time. -5 -4 -3 -2 -1 +1 +2 -f3 +4 +5
620 -683 -751 -826 -909 1-000 1-100 1-210 1-331 1-464 1-610
268 -349 -454 -591 -769 1 000 1-300 1 690 2-197 2-856 3'712
3-050 2-440 1-95;M -562 1-250 1-000 -800 -64 -512 -409 -327
B.
15. ,J2 = 1*414. 16. */2 - 1'189.
17. Growth factor constant for all equal intervals.
18. (i) 1-1; (ii)7237; (iii) 3371.
19. 1-3. 20. 0-8; 630.
21. 51-8, 53-8, 55-8, 57'8 inches.
EXEECISE LIT.
See ch. xxix., 2 ; ch. xxx.
A.
2. 25/ . 3. (i) 69-80 ounces ; (ii) 74 -97 ounces ; (iii) 35-82 ounces
4. 2629. 5. 15822 ; 33861.
ANSWERS TO THE EXAMPLES 409
6. (i) 1 5s. 3d., 1 18s. 6d., lls. 5d., 1 Us. lid. ;
(ii) 1 Is. 6d., 1 4s. 4d., 16s. lid., 1 3s. 8d. ;
(iii) 14s. 8d., 8s. 6d., 2 Is. 4d., 9s. 8d. ;
(iv) 6*27 years time, 1*64 yoars time, 4*11 years ago,
2 '11 years time.
7. (i) 182*8 fathoms ; (ii) 257 '0 fathoms ; (iii) 300 '9 fathoms ;
(iv) 83*23 fathoms ; (v) 62*37 iathoms ; (vi) 29*54 fathoms.
8. (i) 878*6 fathoms ; ;n) 68*16 fathoms.
9. (i) 21 6; (ii) 15*4; (iii) 30'3 ; (iv) 26*4.
10. (i) 2 18s. 5d. ; (ii) 3 8s. 3d. ; (iii) 6 10s. 6d. ;
(iv) 2 2s. 8d. ; (v) 1 3s. lid.
n. (i) 3451; (ii) 2852 ; (iii) 4046; (iv) 4243.
12. (i) 2925 ; (ii) 2430.
13. (i) 1*3 ; (ii) 30% ; (iii) (a) 5*29 years ago, (b) 6*14 years ago.
14. (i) 21-21% (ii) 42*76%.
15. The first one ; 30%, 25%.
B.
16. (i) 51 ; (ii) 72 ; (iii) 130 ; (iv) 7 years ago.
17. (i) 54 ; (ii) 83 ; (iii) 4*6 years ago.
18. (i) 152 ; (ii) 395 ; (iii) 283.
19. 214. 20. 14% ; 11-8%.
EXERCISE LTII.
See ch. xxix., 3 ; ch. xxxi.
A.
I. 1*26 ; 1*52 ; 3'5. 2. x = l*26t ; x --= T52t ; x = 3'5t.
3. 3*09. 4. (i) 1-09; (ii) 1'09.
5. (i) 12*6 ; (ii) 6*1 ; (iii) 3'8 hours ago ; (iv) in 2*25 hours.
6. 6 hrs. approx. 7. (i) 166 ; (ir> 250.
8 (i) 3*24 inches ; (ii) 3*6 inches ; (iii) 2*67 inches.
9. (i) 1*21; (ii) 1-22; (iii) 0*89. 10. 1*54 n. 1*16.
5x ,... 5x ,..., llx ,. x px f 3px t ax
12. 0) 8 - ; (") T ; (HI) - -g- ; (w) -- ; (v) - ; (n) q-.
B.
13. (i) 4*84; (ii) 7'3; (iii) 4*71 ; (iv) 2'61 ; (v) 2-86;
(vi) 1*43 ; (vii) 1*69 ; (viii) 1*77 ; (ix) 1'79 ; (x) 1 79 ;
(xi) 5*75; (xii) 5 75 ; (xiii) 2-32; (xiv) 3*4; (xv) 166.
(i) 3*32; (ii) 2-46; (iii) 1*53^ (iv) 1'37 ; (v)5'89; (vi) 8'43.
(i) 387; (ii) 8*52; (iii) 32; (iv) 144.
(i) 59*8; (ii) '198; (iii) 2-8; (iv) 17*25.
(i) 2*1; (ii) 1'46; (iii) 13-1.
(i) 1-53; (ii) 4*41 ; (iii) 207 ; (iv) *36.
12*9. 22. (i) 178 ; (ii) 0*46.
410
ALGEBRA
EXERCISE LEV.
See ch. xxix., 3, 4 ; ch. xxxn., A and B.
A.
(iii) 073 ;
(iv) 0-33.
(ii) 1-375 : , ,
(ii) 1-04 ; (iii) 1-02.
(ii) 1-26 ; (iii) 1-35 ; (iv) 7'32.
i (ii) 0-000129; (iii) 0-00000945 ; (iv) 128.
(ii) 4-28 ; (iii) 4- 28.
(ii) 5-46 ; (iii) 5'69 ; (iv) 6'58 ; (v) 3'69 ; (vi)475.
(ii) 1-89 ; (iii) 6'58.
(ii) 498 ; (iii) 2*97 ; (iv) 155000 ; (v) 431.
; (ii) 0-442; (iii) 0'663 ; (iv) 712; (v) 0'0821.
; (ii) 50-82 ; (iii) 16560 ; (iv) 0'0003654 ; (v) 630900.
B.
Logarithms for base 1'25.
i
logn
n
0-5
- 311
0-9
0-6
- 2-29
1-0
07
- 1-60
11
0-8
- i-oo
1-2
n
logn
n
logn
n
logn
3-9
0*47
1-3
1-18
17
2-37
I/O
o-oo
1-4
1-51
1-8
2-63
LI
0-43
1-5
1-82
1-9
2-88
L-2
0-82
1-6
2-11
2-0
311
14.
Anti-logarithms for base 1*25.
1
antilog 1
I
antilog 1
1
antilog 1
1
antilog 1
- 3-0
0-51
- 1-4
0-73
0-2
1-05
1-8
1-49
- 2-8
0-54
- 1-2
076
0-4
1-09
2-0
1-57
- 2-6
0-56
- 1-0
0-80
0-6
114
2-2
1-63
2-4
0-59
- 0-8
0-84
0-8
119
2-4
171
- 2-2
0-61
- 0-6
0-87
1-0
1-25
2-6
179
- 2-0
0-64
- 0-4
0-91
1-2
1-31
2-8
1-87
- 1-8
0-67
- 0'2
0-96
1-4
1-37
3'0
1-95
- 1-6
070
o-o
1-00
1-6
1-43
15. (i) - 1-93; (ii) 0'93 ; (iii) 2-43.
16. (i) 1-8; (ii) 1'2 ; (iii) 07.
17. (i) 117; (ii) 0-88; (iii) 014,
ANSWERS TO THE EXAMPLES
411
1 8. (i) 0*69 ; (ii) 1*25 ; (iii) T85 ; (iv) 0'58.
19. (i) logaPQ; (ii) logk|; (iii) nlog a P; (iv) j~log a P.
20. (i) antiloga (P + Q) ; (ii) antilog. (P - Q) ;
p
(iii) antilog a nP ; (iv) antilog - .
EXEKCfSE LV.
See ch. xxix., 4 ; ch. xxxm., A.
A.
1. (i) 1-45 ; (ii) 2'87 ; (iii) 3-82 ; (iv) 4 -54 ; (v) 5 00 ; (vi) 6 -68.
2. (i) 1-23; (ii) 2-54; (iii) 3'38 ; (iv) 673; (v) 8-24.
3. (i) 2-10; (ii) L>'91; (iii) 3 '32.
4. (i) 174 ; (ii) 2'64 ; (iii) 6 '96.
5. (i) 170; (ii) 3-53 ; (iii) 8-34.
6. (i) 2-21; (ii) 6-19; (iii) 2 -91.
B.
7. 11, 1-21, 1-33, 1'40, 1-61, 177, 1-95, 214, 2-36, 2 -59.
9. (i) 4177; (ii) 4-51)5; (iii) 673; (iv) 7'40.
10. 161. II. (i) 1-4; (ii) 1'8; (iii) 1'46 ; (iv) 2'14 ; (v) 4'50.
12. (i) 0-80 ; (ii) l'4(i ; (iii) 3*93.
13. 2-59. 14. (i) 0-4; (ii) 0'9 ; (iii) 073; (iv) 1*33; (v) 7'4().
15. 65(i
16.
antilog n
o-oo
0-25
0-50
075
1-00
1-25
n
1-0000
0-9000
0-8100
0-7290
0-6561
0-5905
antilog ii
1-50
175
2-00
2-25
2-50
n
0-5315
0-4784
0-4306
0-3875
0-3488
17. (i) 1-75; (ii) 075; (iii) 218.
18. (i) 0-5905 ; (ii) 0'4306 ; (iii) 07776.
19. (i) 1'46; (ii) 1'25 ; (iii)rei; (iv) 0'125,
20. (i) 0-6; (ii) 0'4784 ; (iii) 0'3488,
4:12 ALGEBRA
EXEECISE LYI.
See ch. xxix., 6 ; ch. xxxm., B.
A.
x. (i) 0-301; (ii) 0'531; (iii) 0'021 ; (iv) 0'940 ; (v)J>968.
2. (i) 3-301 ; (ii) 6-531 ; (iii) 3*531 : (iv) 4'021 ; (v) 1'021 ;
(vi) 5-940; (vii) 2-968.
14-29 ; (v) 2958 :
7. (i) 0-357; (ii) 0-000935; (iii) "00011 5 ; (iv) 11-5;
8. (i) 0-0000748; (ii) 0'305 ; (iii) 0*(>22 ; (iv) 0-0542 ; (v) 1*02.
9. (i) 112-3; (ii) 34810; (iii) "00033 8 ; (iv) 0*0363.
10. (i) 60-46 ; (ii) O'llO ; (iii) 7*59.
EXEECISE LVIL
See ch. xxix., 6 ; ch. xxxin., C.
A.
1. (i) 0-63175; (ii) 463175; (iii) 4-D3175 ; (iv) 3-30856;
(v) 1*30856; (vi) 1-88098; (vii) 272444; (viii)_3-00290 ;
(ix) 3-00290; (x) 0-90309; (xi) 4*00000; (xii) 5-00000.
2. (i) 17498; (ii) 17498- ; (iii) '017498 ; (iv) 8173*4;
(v) 1-0778; (vi) 0-0010778; (vii) 0*10109; (viii) 1-0014;
(ix) 1001*4; (x) 0-000015849.
3. (i) 1*32319 ; (ii) 2^31146 ; (iii) 6*09269 ; (iv) 3'99710 ;
(v) 3-99710; (vi) _6'55112 ; (vii)_l 64294 ; (viii) 12-01160 ;
(ix) 0-92635; (x) 1*32635; (xi) 1 '960326 ; (xh) 1 "250725;
4. (i) 0-210475 ; (ii) 0*020486 ; (iii) 1237870* ; (iv) 9933 '5 ;
(v) 0-0099335 ; (vi) 0'00000355726 ; (vii) 0126207 ;
(viii) 0-0000000000010271 ; (ix) 8*4401; (x) 0*21201 ;
(xi) 0-912706 ; (xii) 0-17813.
5. (i) 103-2 ; (ii) 0*000003474 ; (iii) 0*5623 ;
(i-^ 0-3374 ; (v) -01721.
B.
6. (i) 62801733 cos. ; (ii) 2*2576 feet.
7. (i) 636 148. lOd. ; (ii) 181 11s. Od.
8. 97'9 years. 9. 4*51/ .
10. (i) 423900000 ; (ii) 324200000.
XI (i) 4*993816; (ii) 0-00614619; (iii) 0-00003831.
ANSWERS TO THE EXAMPLES 413
12. (i) 98-28/ j (ii) 99-97%.
13. n. 14. n.
15. (i) 31 ; (ii) 785 ; (iii) 33.
16. (i) 30 ; (11) 784,
17. 10. 19. 10. 20. 2. 21. 2. 23. 9. 24. 16.
C.
25. 636-46. 26. 64-015. 27. 642-6.
28. (i) 545-37 ; (ii) 652-055.
EXERCISE LVIII.
See ch. xxix., 7 ; ch. xxxiv.
A.
5. (i) 1-57; (ii) 2-44; (iii) 0'54.
6. (i) 0-75; (ii) 0'54 ; (iityO'44.
8. (i) '78368, 6 077 ; (ii) 4' 64494, 0*0004406;
(iii) 5-93542, 861820.
10. (i) 229-1; (ii) 0' 000191 ; (iii) '000003 16 ; (iv) 1959'4 ;
(v) 0-1316; (vi) 1273 x 10".
12. (i) 1-28 ; (ii) 1-38 ; (iii) 0'64.
13. (i) 0-000007943 ; (ii) 5 749 ; (iii) 11-22.
14. (i) x> ; (ii) x* ; (iii) x' ; (iv} x*.
B.
*5- (i) y = 3 * ; 94 ' 8( > ; ( ]i ) y = 3x ^ 13-925, 13-925 ;
(iii) y = 2x*, 11 -247 ; (iv) y - 256x- J , 25 '6, - 25 -6.
17. y = ax m , where a = nut i log c. 18. y = 30'2x 024 .
19. (i) y - 6-31x- 2 ; (ii) y - 70'8x-- 74 .
20. a = 2-21, in = 0'421. 21. p = 6-46u-o-7!>'
C.
23- (1 + -) n . 24. (i) 2-5936; (ii) 2 '704 ;
(iii) 2-716; (iv) 2-717.
28. (i) aH"^ ; (ii) a~ J ^ b^ ; (iii) (x* - y*) ; (iv) x - y4.
EXEEGISE LIX.
See ch. xxix., 8 ; ch. xxxv,
A.
1. (i) 0-0506, 5-06; (ii) 0-05095, 5'095; (iii) 0-05117, 5-117.
2. The former with an effective interest of 2*5156 per cent per
annum is more profitable than the latter with an effective
interest of 2*269 per cent per annum.
414 ALGEBRA
3. A has 7*75 more.
5. 20. 6. 6. 7. V = P (1 + j/p)-*.
B.
9. (i) 3-56 ; (ii) 5-L'J. n. 30.
12. P - ? (1 -- e -J). 13. (i) 10-25 ; (ii) 10'3813 ;
(iii) 10-4718; (iv) 10-5093.
SUPPLEMENTARY EXERCISES.
EXERCISE LX.
See ch. xxxvi., 2.
A.
I. 4. Oral.
5. (i) 5-932 ; (ii) - J1148 ; (iii) 103*35 ; (iv) 2'924 ; (v) '0154.
6. (i) 1-01940; (ii) 1 '87778 ; (iii) 2*26270; (iv) '93306.
B.
8. (i) y = 77, b = 144, c - 158*9 ;
(ii) y = 59 18', b - 16'98 ? o - 18' 19 ;
(iii) y - 44, b - 93'0, c - 188'02 ;
(iv) y = 109 49', b - 712-11, c - 742'58.
9. (i) C = 55, b = 498'5 yards, a - 410'77 yards ;
(ii) C = 46, b = 816-97 feet, a - 1376' 56 feet ;
(iii) A - 13 57', a - 401-83 yards, c - 1408'9 yards.
12. 28 U 58', 104 28', 46 34'.
/\ i. tt /( 8 - b)(s - c)
M- (')"" g V- (-) '
2
(ii) sin a = , \/s (s - a)(s - b)(s - c).
16. (i) a - 32 35', ft = 37 24', y = 110 1', A = 1170 ;
(ii) a - 58 57', ft - 67 55', y - 53 18', A - 16619 ;
(iii) a - 16 48', /i - 113 56', y - 49 16', A - 6697 ;
(iv) a 85 10', ft = 39 10 V , y ^ 55 40', A = 37951.
C.
21. (i) ft = 104 61', y = 33 9', a = 19*8 ;
(ii) j8 - 30 17', y = 43 13', a - 397-2 ;
(iii) a - 93 37', - 36 3', c = 121-09 ;
(iv) a = 24 12', y - 33 48', b = 1250 "6 ;
(v) a = 167 32', y = 4 2', b = 83*12.
22. (i) There is no triangle satisfying the conditions,
(ii) a - 25 18', y = 116 42', c - 298-92 ;
(iii) a = 33 33', ft = 50 27', a = 170 "61.
(iv) There are two possible triangles :
a - 90, y - 60, a-
a = 30, y- 120, a
416 ALGEBRA
EXERCISK LXI.
See ch. xxxvi. , $ 3.
EXERCISE LXn.
See ch. xxxvi., ^ 4.
6. (i) 2 cos 30 . cos 7 ; (ii) 2 sin 30 . cos 7 y ;
(iii) 2 cos 60 . cos 12 ; (iv) 2 BUI 804 . cos 23* ;
(v) 2 sin 58 . sin 24 ; (vi) 2 cos 70 a . am 54 ;
(vii) 2 sin 55 . sin 73 ; (viii) 2 cos 55 . win 73.
7. (i) cos 47 + cos 17 ; (ii) cos 52 + cos 58 ;
(iii) sin 70 + sin 16 ; (iv) - sin 1 - sin 75 ;
(v) - ^(sin 31 + sin 35) ; (vi) 14(cos 37 - cos 71).
8. (i) tan 52 ; (ii) cot 52i ; (iii) tan 4<x, ; (iv) tan 2a ;
(v) tan ( - ft) ; (vi) tan \(a + ).
II. (i) cos 2a ; (ii) tan $a ; (iii) cot Aa ; (iv) cos a -\- sin a ;
(v) siu 2a ; (vi) tan 2a ; (vii) tan 2 a ; (viii) tana.
EXERCISE LXII1.
See ch. xxxvi., g 5.
A.
1. (i) 48 '42 feet, 4*035 seconds ; (ii) 57 foot, 4*75 seconds ;
(iii) 86 feet.
2. (i) 54-52, - 3-18 ; (ii) 54'55 feet.
3. (i) 55 15 ieet ; (ii) 4 '59 seconds.
4. (i) d - 20b, h - 4d - g ;
(ii) 2 seconds, at height of 100 feet,
(iii) Range is 100 feet.
5. 97-98 feet, 4'875 seconds. 6. 103 feet, 5'122 seconds.
8. x = 1-75.
. m v = _?-
2 - 3(z + 3)
(ii) x = - 5, z = - 1 or x = ~ f, z
ANSWERS TO THE EXAMPLES 419
3a 2 hi 2
21. -- or from the edge of square cut away.
oa on
22. On line joining centres 2*5 inches from centre of larger circle.
23. On line joining centres 13 '5 inches from point of contact-
24. (i) TT (10 -f r:)' J ; (ii) 28*7 inches along axis from smaller surface.-
25. Fig. 11 : A~-o7 i \ from upper edge ; a/2 from left edge.
. . 4c(a -f 3) + 3h , ,
tig. 12 : ~'--. 7j- b y from upper edge ;
2a(a + 2b) ~ 3bc - , C1 _ ,
rt /rt- ot. x f rom toft edge.
2(2a + 3b) 6 .
C.
27. 2?rx . ftl (i.e. frustum of a cone). 30. 2?rx x 4h 2 2?rx . ^A.
32. (i) A - 47r 2 (K - r)r ; (ii) V == 27r' J (R - r)r' J .
33. A = 620 square incheH ; V 1)30 cubic inches.
34. A = 3323 5 \ square inches V = 7542? cubic inches.
35. <A'-= 27rr(2a + TTD) ; V - 27rrb(a + 7rb/4). 36. iOO'8 pounds.
EXEECISE LXVIL
See ch. xxxvn., 2.
A.
i. '867. 2. (i) 72 feet ; (ii) '66 feet.
3. '97 feet. 4. I'O feet.
5. (i) A, 5-4 ; B, 5'46 ; (ii) A, 5105 ; B, 4'912.
B is the best.
B.
-. 13. 5-95 inches. ,5. r+. 16.
18. 3 (r being the radius). 20. 0'882 ; 1'155.
A
EXERCISE LXVIIL
See ch. xxxvn., 3.
A.
I. (i) 1 + 5i + 10i 2 -i- 10i :5 -f 5i 4 -f i* ;
(ii) 1 -f 8i -f 28i 2 -f 66i 3 -f 70i 4 + 56i 8 + 28i -f 8i 7 4- i ;
(iii) 1 + lOi -f 45i 2 + 120r ; + 210i 4 -i- 2701 + 210i + 120i 7
+ 45i 8 + 10i a + i 10 .
28*
420 ALGEBRA
- n(n ~ 1)(a ~ 2) (n ~
4. lV(n+l)i + ^"i* 4- (5LiM
4. (n + l)n(n - l)(n - 2). ... (n - r + 2)
+ ^ f !
+ . . . + (n + 1) i + 1-+ 1 .
7. (i) 1480 4*i. ; (li) 1806 2s. ; (iii) 2281 Is. 8. 635.
9. (i) 1 20x + 180x a - %0x 3 + 2960x 4 ;
(ii) 256 -f 1024x + 1792x a + I792x a f 1120x 4 ;
(iii) 720a (; - 5832a : >x + 19440ax a - 34560a :! x ! -
(iv) 1 - 20x 1 2 4- 190x - 1140x + 4840X 1 - 5 ;
(v) 128x 7 + 1344x B -1- 6048x 5 + 16120x 4 -f
10. (i) - 204720x :{ ; (ii) 70a 4 b 4 x ; (iii) 70 ;
(iv) 15a 3 b 4 ; (v) - 1959552ft 1 b 5 x*.
XI.
n
2
. . . 1,
1
2,
3
B.
3, 4, 5, 6, 7.
6 10 15 21 28
n
4
. . . 1,
4
10, 20, 35, 56, 84
12.
n
n =
n =
(i) 1
(ii) 1
1
5
6
7
. . . i,
. . . 1,
. . . 1,
4i + K.
7i + 2*
5,
6,
7,
)i~
Ji a
- 1
16, 35, 70, 126, S
21, 56, 126, 252,
28, 84, 210, 462,
- 20i 3 + 35i 4 - 6
- 84i 3 + 210i 4 -
) n(n + l)(n
J10.
462.
924.
ier +
462i r>
+ 2)*
1 2
n(n H
i
h 1
31
.)(n+2)(n + 3). 4
_u
84i"
_
17. 0-08. 20. 82214n. 21. (i) '00103; (ii) 0097.
22. 86 7s. Maximum error = 19s. 4fd.
24. (i) 672x 3 ; (ii) 330a~ 12 b 7 x 7 (iii) 1215 x 10~ 7 x 4 .
C.
28. (i) 1-10302; (ii) 1 144916444 ;
(iii) 1 + 3-2x + 3-52x 2 + l'2906x 8 ;
(iv) 1 + 4'8x + 13'92x 2 + 31'552x $ .
31. (i) 51 Ibs. ; (ii) 22 Ibs. 32. 202 C.
D.
34. 2-718.
ANSWERS TO THE EXAMPLES 421
EXEKCISE LXIX.
See oh. xxxvm., 4.
A.
4. (i) 5x 4 ; (ii) - 5/x 8 ; (iii) 28x fl ; (iv) - 40/x' ;
(v) 20x 3 - 36x 2 + 4x - 10 ;
(vi) 6x 5 - 8x :{ + 6x - ti/x s + 8/x r > - 6/x'.
5. (v) 60x 2 - 72x -h 4 ;
(vi) 30x 4 - 24x 2 + 6 + 18/x 4 - 40/x* + 42/x 8 .
6. (i) 60x 2 ; (it) - 210/x s ; (iii) 840x 4 ; (iv) - 3600/x 11 .
8. (i) l/2x* ; (ii) 2/3x* ; (iii) - 3/5x* ; (iv) 24X 1 -* ; (v) - 7'2x 3 ".
10. (i) Sy/Sx - 3(1 + n) u ; (a) dy/dx = 4/(2 - x) 5 ;
(iii) 8y/ftx = 7'2(x - 2*5)^ ; (iv) $y/8x 6f2x + o) a ;
(v) ma(ax + b) 10 " 1 ; (vi) y/8x = - l/^/(3x - 4) 4 .
B.
n. (i) A - 3x s /5 ; (ii) A = ix /3 ; (iii) 3x 3 ' 4 /3'4.
12. 54991-2. I3 . (i) A - 1(2 + x) 3 / 2 ; (ii) A - ,' ! (2x - I) 6 /* ;
(iii) A = j/(l - 3x) 3 ; (iv) A - - l/7'2(3x + I)- 1 - 4 .
C.
15 . (i) T = - 2/(t + I) 9 ; - 2, - ^ ; (ii) v - 9t' 8 ; 0, 4- 9 ;
(iii) v ~ - 10 + lOt 1 ' 5 ; -10,0;
(iv) v = 16/(2t + 3) 7 , /4 ; 15/3 7/1 , 15/5 7 ' 4 .
16. (i) B = 2t - Vfc 5 / 3 ; s - - | ; (ii) s - i -I- i 2 + t 3 ; s - 3 ;
(iii) s - 40JVO. + 6t) - 1] ; 20(^6 - 1) ;
(iv) s = 10[(2 - 3t)* - 2] ; s - 10(1 - 2').
17. (i) a = 4 /(t + !) ; (ii) a = 7'2t-' 2 ;
(iii) a - - 8/3t^ ; (iv) a = 2 + 6t.
18. (i) v = lOt - Jt a ; (ii) v = - 5b 4- 3fc 3 * ;
(iii) v - A(l -f 4fc) 8 - 5 - A ; (iv) v = (1 4- 2fc)- a ~ 1.
19. (i) s - 5t 2 - -jU 3 ; (ii) s - - ^t 2 + At 1 / 8 ;
(iii) a -ifcKl +H) 1 -* - At - T^;
(iv) 8 = - J(l+ 2b)- 1 -t+4.
D.
21. (i) ay/Ax - 12x' ; (ii) 5y/^x = ^x~ ; (iii) 7-2X 1 - 4 .
22. (i) 85 ; (ii) 33 ; (iii) 82.
23. (i) 3 ; (y - 3) = 12(x - 1) ; (ii) 1 ; <y - 1) !(* - 1) ;
(iii) 3 ; (y ~ 3) = 7 2(x - 1).
24. (i) i ; J ; (ii) - J ; * ; (iii) A ; T'J.
28. (i) 12(y - 3) + (x - 1) = 0; (ii) |(y - 1) 4- (x - 1) = ;
(iii) 7'2(y - 3) + (x - 1) = 0.
29. (i) 36; (ii) 1; (iii) 21 '6.
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