> CD
166268
OSMANIA UNIVERSITY LIBRARY
Call No. S?Q C $& Accession No.
This book should be returned on or before the date
last marked below.
ELEMENTS OP
APPLIED MATHEMATICS
BY
HERBERT E. COBB
PROFESSOR OF MATHEMATICS, LKWIS INSTITUTE, CHICAGO
GINN AND COMPANY
BOSTON  NEW YORK CHICAGO LONDON
ATLANTA DALLAS COLUMBUS BAN FRANCISCO
COPYRIGHT, 1911
BY HERBERT E. COBB
ALL RIGHTS RKSKRVKD
PRINTED IN THE UNITED STATES OF AMERICA
726.7
fltfrenaeum
GINN AND COMPANY PRO
PRIETORS BOSTON U.S.A.
PREFACE
This book of problems is the result of four years' experimen
tation in the endeavor to make the instruction in mathematics
of real service in the training pf pupils for their future work.
There is at the present time a widespread belief among teach
ers that the formal, abstract, and purely theoretical portions
of algebra and geometry have been unduly emphasized. More
over, it has been felt that mathematics is not a series of dis
crete subjects, each in turn to be studied and dropped without
reference to the others or to the mathematical problems that
arise in the shops and laboratories. Hence we have attempted
to relate arithmetic, algebra, geometry, and trigonometry closely
to each other, and to connect all our mathematics with the work
in the shops and laboratories. This has been done largely by
lists of problems based on the preceding work in mathematics
and on the work in the shops and laboratories, and by simple
experiments and exercises in the mathematics classrooms, where
the pupil by measuring and weighing secures his own data for
numerical computations and geometrical constructions.
In high schools where it is possible for the teachers to depart
from traditional methods, although they must hold to a year
of algebra and a year of geometry, this book of problems can
be used to make a beginning in the unification of mathematics,
and to make a test of work in applied problems. In the first
year in algebra the problems in Chapters I VII can be used
to replace much of the abstract, formal, and lifeless mate
rial of the ordinary course. These problems afford a much
needed drill in arithmetical computation, prepare the way for
geometry, and awaken the interest of the pupils in the affairs
iii
iv APPLIED MATHEMATICS
of daily life. By placing less emphasis on the formal side of
geometry it is possible to make the pupil's knowledge of alge
bra a valuable asset in solving geometrical problems, and to
give him a working knowledge of angle functions and log
arithms. Chapters IX, X, and XII furnish the material for
this year's work. The problems of the remaining chapters can
be used in connection with the study of advanced algebra and
solid geometry. They deal with various phases of real life, and
in solving them the pupil finds use for all his mathematics,
his physics, and his practical knowledge.
For the increasing number of intermediate industrial schools
there are available at present few lists of problems of the kind
brought together in this book. The methods adopted in the
earlier chapters, which require the pupil to obtain his own data
by measuring and weighing, are especially valuable for begin
ners and boys who have been out of school for several years.
The large number of problems and exercises permits the
teacher to select those that are best suited to the needs of the
class. In Chapters IX and XIII many of the problems contain
two sets of numbers. The first set outside of the parentheses
may give an integral result, while the second set may involve
fractions ; or the first set may give rise to a quadratic equation
which can be solved by factoring, while the equation of the
second set must be solved by completing the square.
Each pupil should have a triangle, protractor, pair of com
passes, metric ruler, and a notebook containing plain and
squared paper. Inexpensive drawing instruments can be ob
tained, and the pupils should be urged to use them in making
rough checks of computations. They should also form the
habit of making a rough estimate of the answer, and noting
if the result obtained by computation is reasonable.
In the preparation of this book most of the works named in
the Bibliography have been consulted. The chapter on squared
paper aims to emphasize its chief uses, the representation of
PREFACE v
tables of values, and the solution of problems ; and to show
that the graph should be used in a commonsense way in all
mathematical work.
The cooperation of the members of the department of mathe
matics in the Lewis Institute in the work of preparing and
testing the material for this book has rendered the task less
burdensome ; acknowledgments are due to Assistant Professor
D. Studley for the problems in Chapters XIV and XV; to
Assistant Professor B. J. Thomas for aid in Chapters I, VIII,
XII, and XIII ; to Mr. E. H. Lay for aid in Chapters II and VI ;
and to Mr. A. W. Cavanaugh for aid in Chapter IX. Especial
acknowledgments are due to Professor P. P>. Wood worth, head
of the department of physics, Lewis Institute, for his helpful
cooperation with the work of the mathematics department.
CONTENTS
CHAPTER PAGE
I. MEASUREMENT AND APPROXIMATE NUMBER ... 1
IT. VERNIER AND MICROMETER CALIPERS 9
III. WORK AND POWER 16
IV. LEVERS AND BEAMS 27
V. SPECIEIC GRAVITY 42
VI. GEOMETRICAL CONSTRUCTIONS WITH ALGEBRAIC
APPLICATIONS 52
VII. THE USE OE SQUARED PAPER 65
VIII. FUNCTIONALITY; MAXIMUM AND MINIMUM VALUES 91
IX. EXERCISES FOR ALGEBRAIC SOLUTION IN PLANE
GEOMETRY 97
X. COMMON LOGARITHMS 120
XL THE SLIDE RULE 128
XII. ANGLE FUNCTIONS 134
XIII. GEOMETRICAL EXERCISES FOR ADVANCED ALGEBRA 153
XIV. VARIATION 164
XV. EXERCISES IN SOLID GEOMETRY 177
XVI. HEAT 195
XVII. ELECTRICITY 212
XVIII. LOGARITHMIC PAPER 243
TABLES 258
BIBLIOGRAPHY 261
FOURPLACE LOGARITHMS 265
INDEX 273
VII
APPLIED MATHEMATICS
CHAPTER I
MEASUREMENT AND APPROXIMATE NUMBER
Exercise. Make a sketch of the whitewood block that has
been given you ; measure its length, breadth, and thickness in
millimeters and write the dimensions on the sketch. Find the
volume of the block. Have you found the exact volume ?
Were your measurements absolutely correct ?
1. Errors. In making measurements of any kind there are
always errors. We do not know whether or not the foot rule,
the meter stick, or the 100foot steel tape we are using is abso
lutely exact in length and graduation. Hence one source of
error lies in the instruments we use. Another source of error
is the inability to make correct readings. When you attempt
to measure the length of a whitewood block, you will probably
find that the corners are rather blunt, making it impossible to
set a division of the scale exactly on the corner. Moreover, it
is seldom that the end of the line you are measuring appears
to coincide exactly with a division of the scale. If you are using
a scale graduated to millimeters and record your measurements
only to millimeters, then a length is neglected if it is less than
half a millimeter, land called one millimeter if it is greater than
half a millimeter.
To make a reading as correct as possible, be sure that the eye
is placed directly over the division of the scale at which the
reading is made. Note if the end of the scale is perfect.
1
2 APPLIED MATHEMATICS
2. Significant figures. A digit is one of the ten figures used
in number expressions. A significant figure is a digit used to
express the amount which enters the number in that parti ?,ular
place which the digit occupies. All figures other than zero are
significant. A zero may or may not be significant. It is sig
nificant if written to show that the quantity in that place is
nearer to zero than to any other digit, but a zero written merely
to locate the decimal point is not significant. A zero inclosed
by other digits is significant, while a final zero may or may not
be significant.
For example, in the number 0.0021 the zeros are not signifi
cant. In the number .0506 the first zero is not significant, while
the zero inclosed by the 5 and 6 is significant. If in a measure
ment a result written as 56.70 means that it is nearer 56.70
than 56.69 or 56.71, the zero is significant. In saying that a
house cost about $6700, the final zeros are not significant be
cause they merely take the place of other figures whose value
we do not know or do not care to express.
3. Exact numbers. In making computations with exact
numbers, multiplications and divisions are done in full, accord
ing to methods which are familiar to all students.
4. Approximate numbers. In practical calculations most of
the numbers used are not exact but are approximate numbers.
They are obtained by measuring, weighing, and other similar
processes. Such numbers cannot be exact, for instruments are
not perfect and the sense of vision does not act with absolute
precision. If the length of a rectangular piece of paper were
measured and found to be 614 mm., the 6 and the 1 would
very likely be exact, but the 4 would be doubtful.
5. Multiplication of approximate numbers. This contracted
method of multiplication gives the proper number of significant
figures in the product with no waste of labor. Moreover, by
omitting the doubtful figures it avoids an appearance of great
accuracy in the result, which is not warranted by the data.
MEASUREMENT AND APPROXIMATE NUMBER 3
Exercise. Measure the length and width of a rectangular
piece of paper and find its area.
Suppose the length is 614 mm. and the width is 237 mm.
Let us proceed to find the area of the piece of paper, marking
the doubtful figures throughout the work.
237
614
948
237
1422
145518
The final three figures in the product are doubtful and may
as well be replaced by zeros. Hence the area is approximately
145,000 sq. mm., or, as we sometimes say, about 145,000
sq. mm. Since many calculations are of this kind, it is a waste
of time to carry out the operations in full. It is desirable to
use methods which will omit the doubtful figures and retain
only those which are certain.
Problem. Multiply 24.6 by 3.25.
First step Second step Third step
24.6 24.0 2W
3.25 3.25 3.25
738 738 738
49 49
12
79.9
First step. Start with 3 at the left in the multiplier and
write the partial product as shown.
Second step. Cut off the 6 in the multiplicand and multiply
by 2. Twice 6 (mentally) are 12 (1.2), which gives 1 to add.
Twice 4 are 8, and 1 to add makes 9. Twice 2 are 4.
Third step. Cut off the 4 in the multiplicand and multiply
by 5. 5 times 4 (mentally) are 20 (2.0), which gives 2 to add.
5 times 2 are 10, and 2 to add makes 12.
4 APPLIED MATHEMATICS
Fourth step. Add the partial products.
Fifth step. Place the decimal point by considering the num
ber of integral figures which the product should contain. This
may usually be done by making a rough estimate mentally.
In this case we see that 3 times 24 are 72, and by estimating
the amount to be brought up from the remaining parts we see
that the product is more than 75. Hence there are two inte
gral figures to be pointed off.
Problem. Multiply 84.6 by 4.25.
First step Second step Third step
84.0 SW M#
4.25 4.25 4.25
338 338 338
17 17
4
359
In this case 6 is cut off before multiplying by 4 in order to
keep the product to three figures. The two given numbers are
doubtful in the third figure, and usually this makes the product
doubtful in the third figure.
Problem. Find the product of TT x 3.784 x 460.2.
SOLUTION. 3.WJZ IJL.W
3.784 460.2
9426 4756
2199 713
251 2
12 5471
11.888
6. Measurements* In making measurements to compute
areas, volumes, and so on, all parts should be measured with
the same relative accuracy ; that is, they should all be expressed
with the same number of significant figures. The calculated
parts should not shoiv more significant figures than the meas
ured parts. Constants like TT should be cut to the same number
of figures as the measured parts.
MEASUREMENT AND APPROXIMATE NUMBER 5
EXERCISES
1. Find the area of the printed portion of a page in your
algebra.
2. Find the volume of your algebra.
3. Find the area of the top of your desk.
4. Find the area of the door.
5. Find the number of cubic feet of air in the room.
6. Find the area of one section of the blackboard.
7. Find the surface and volume of brass cylinders and
prisms, and of wooden blocks.
8. Find the area of the athletic field.
9. Find the area of the ground covered by the school build
ings and also the area of some of the halls and recitation rooms.
Compare your results with computations made from the plans
of the buildings, if they are accessible.
7. Division of approximate numbers. In dividing one ap
proximate number by another, we shorten the work by cutting
off figures in the divisor instead of adding zeros in the dividend.
The principles of contracted multiplication are used in the
multiplication of the divisor by the figures of the quotient.
No attention is paid to the decimal point in the dividend or
divisor till the quotient has been obtained. In checking multi
ply the quotient by the divisor. (Why ?)
Problems. 1. Divide 83.62 by 3.194.
3.;^ 1 83.62 1 26.18 Check
6388 20.jp
1974 3.194
1916 7854
58 262
32 235
26 10
25 83.61
1
6 APPLIED MATHEMATICS
2. Divide 41.684 by 98.247.
PE.#7 1 41.684 1.42428 Check
39299 .?
2385 98.247
1965
420
393
27
20
7
7
The decimal point in the quotient can usually be placed
quite easily by considering the number of integral figures in
the divisor and dividend. In the first problem we see that 3 is
contained in 83 about 26 times ; in the second problem 98 is
contained in 41 about .4 times.
PROBLEMS
Check the results obtained :
1. 2.142 x 3.152. 10. 86.66 * 41.37.
2. 78.14 X 1.314. 11. 316.4 + 18.74.
3. 6.718 x 86.42. 12   916 +  314 
4. 3.142 x .7854. 13 14.16 x 5.873
14. 3.142 6 x(1.666)>.
1K 36.5 x 192
7 (3142)'. 15  4.12x6.33
8. (5.164) 8 . 4 x 3.142 x (6.023) 8
9. (.6462)'. 3
17. An iron bar is 9.21 in. by 2.43 in. by 1.12 in. Find its
weight if 1 cu. in. of iron weighs .261 Ib.
18. Find the weight of a block of oak 5.62 in. by 3.92 in. by
3.15 in. if 1 cu. in. of oak weighs .0422 Ib.
MEASUREMENT AND APPROXIMATE NUMBER 7
19. Find the weight of an iron plate 125 in. long, 86.2 in.
wide, and .562 in. thick.
20. The diameter of a piston is 16.4 in. Find its area.
(TT = 3.14.)
21. The radius of a circle is 12.67 in. Find its area.
(TT = 3.142.)
22. The diameter of a steam boiler is 56.8 in. What is its
circumference ?
23. The area of a rectangle is 25.37 sq. in. Find the width
if the length is 11.42 in.
24. What is the length of a cylinder whose volume is 1627
cu. in. if the area of a cross section is 371.5 sq. in. ?
25. A cylindrical safetyvalve weight of cast iron is 15 in.
in diameter and 3^ in. thick. Find its weight if 1 cu. in. of
cast iron weighs .261 Ib.
26. A cylindrical safetyvalve weight of cast iron weighs
82.5 Ib. What is its diameter if it is 1 J in. thick ?
27. The diameter of a spherical safety valve of cast iron is
9.3 in. Find its weight.
28. Find the weight of a castiron pipe 28.5 in. long if the
outer diameter is 10.9 in. and the inner diameter is 9.2 in.
29. A cylindrical water tank is 49.6 in. long and its diameter
is 28.6 in. Find its volume. How many gallons will it hold ?
30. A steel shaft is 68.8 in. long and its diameter is 2.58 in.
Find its weight if 1 cu. in. of steel weighs .283 Ib.
31. Find the weight of the water in a full cylindrical water
tank 12.8 ft. in height and 6.32 ft. in diameter if 1 cu. ft. of
water weighs 62.4 Ib.
32. The diameter of the wheels over which a band saw runs
is 3.02 ft. and the distance between the centers of the pulleys
is 3.58 ft. Find the length of the saw.
33. A pulley 11.9 in. in diameter is making 185 revolutions per
minute (r. p. m/). How fast is the rim traveling per minute ?
8 APPLIED MATHEMATICS
34. A milling cutter 4 in. in diameter is running 150 r. p. m.
What is the surface speed in feet per minute ?
35. It is desired to make a 12in. emery wheel run at a speed
of 5000 ft. per minute. How many revolutions per minute must
it make ?
36. If we wish a milling cutter to run at a cutting speed of
266 ft. per minute, and the machine can make only 82 r. p. in.,
what must be the diameter of the cutter ?
CHAPTER II
VERNIER AND MICROMETER CALIPERS
8. The vernier calipers have two jaws between which is placed
the object to be measured. One jaw slides on a bar which has
scales, on one side centimeters and on the other side inches.
FIG. 1
The movable jaw has two small scales called verniers, one for
each main scale.
Write the following questions and their answers in your
notebook. Use the centimeter for the unit and write the results
as decimal fractions.
1. (a) How many centimeters are marked on the main
scale? (&) Verify 'by measuring with the ruler, (c) What is
the length of the smallest division of the main scale ?
2. (a) What is the length of the centimeter vernier ?
(i) Measure the length of the vernier with the ruler, (c) Verify
by counting the divisions on the main scale.
3. (a) Into how many divisions is the vernier scale divided ?
(6) What is the length of each division ?
10 APPLIED MATHEMATICS
4. Bring the jaws of the calipers together. At what point
on the main scale does the first line of the vernier fall ?
Make a drawing of the vernier and the scale as suggested
by Fig. 2. Number the points of division as in the figure.
A*MV ._
Z 3 4 <S 67 6 9 I/ IZ G /4 &
I I I I I
I I.I I 1.1,1.1
/' f 3' 4' 3 67 Q' &
I I I I ) I
FIG. 2
Slide the vernier of the calipers along until O f coincides with 0.
5. (a) Do l f and 1 coincide ? (b) What is the distance
between 1' and 1? (c) between 2' and 2? (d) between 3'
and 3?
Now slide the vernier along until V and 1 coincide.
6. (a) What is the distance between and 0' ? (b) t>etween
2' and 2 ?
Make 2' and 2 coincide.
7. What is the distance between and (V now ?
8. What is the distance between and 0' when the follow
ing points coincide ? (a) 3' and 3 ; (ft) 4' and 4 ; (c) 7' and 7 ;
(d) 9 f and 9 ; (e) 10' and 10.
Move the vernier until O r coincides with 10.
9. How far apart are the jaws ? Check with the ruler.
10. When O r coincides with 20, how far apart are the jaws ?
Check.
11. When 0' coincides with 21, how far apart are the jaws ?
12. What is the distance between the jaws when the follow
ing points coincide ? (a) 1' and 22; (b) 2' and 23; (c) 6' and 26;
(d) 8' and 29; (e) 1' and 23; (/) 7' and 29; (g) 2 f and 26;
(h) 3' and 36.
9. To measure with the vernier. Count the number of whole
centimeters and millimeters to the zero line of the vernier. Then
VERNIER AND MICROMETER CALIPERS 11
notice which vernier division coincides with a scale division ; the
number of this vernier division is the number of tenths of a
millimeter.
10. Observe carefully the following directions for making
measurements. Unlock the movable jaw by means of the screw
at the side. Place the object between the jaws, press these
together gently but firmly witli the fingers, and lock in position
with the screw. Care should be taken in pressing upon the
jaws as too strong a pressure may injure the instrument. If
not enough pressure is applied, the reading will not be accurate.
EXERCISES
1. Place your pencil between the jaws of the calipers and
measure its diameter.
2. Get a block from your instructor and measure its dimen
sions. Make a record of them together with the number of the
block, and let the instructor check the results.
3. Get a second block. Make measurements of the length
at three different places on the block and record them. Take
the average of the three readings. Find dimensions in the
same way. Let the instructor check the record.
4. Draw an indefinite line A B. With a point R about 1 cm.
from AB as a center, and with a radius of 3 cm. draw a circle
intersecting AB at C and D. Measure CD, making the measure
ment with the pointed ends of the jaws. Check your reading.
5. Take a sheet of squared paper and fix the vertices of a
square centimeter with the point of the compasses. Measure
the diagonals and take the average. Check.
6. On the same sheet of squared paper locate the vertices
of a rectangle 4 cm. by 2.5 cm. Measure the diagonals and
check the results.
7. Apply the sets of questions in these exercises to the inch
scale and its vernier, inserting the word u inch " for " centi
meter " in your record.
12 APPLIED MATHEMATICS
8. Measure the length of a block in inches and in centi
meters and find out the number of centimeters in one inch.
9. On a sheet of squared paper mark out a right triangle
with the legs 3 in, and 4 in. respectively. Locate the vertices
with the point of the compasses and measure the hypotenuse.
Show that the square of the hypotenuse is equal to the sum of
the squares of the other two sides.
10. Move the zero line of the vernier opposite 1 in. on the
main scale. Make the reading in centimeters. Compare the*
result with that obtained in Exercise 8.
11. Find the volume of a block in cubic inches and also. in
cubic centimeters. Check by changing the cubic inches into
cubic centimeters.
12. Devise other exercises in measurement.
11. The micrometer calipers. With the micrometer calipers
the object to be measured is placed between a revolving rod
called the screw, and a fixed
stop. The movable rod is
turned by the barrel, which
moves over a linear scale. The
edge of the barrel is gradu
ated into a circular scale. ._ _
Hi f ^
12. Use of the micrometer
calipers. Turn the barrel so that the screw approaches the stop
and finally comes in contact with it. Now turn in the opposite
direction and the screw moves away from the stop; at the
same time the edge of the barrel moves over the linear scale,
which shows the distance of the opening. When an object is
placed in the opening between the stop and the screw, its
measurement is obtained by reading the length of the linear
scale exposed to view.
VERNIER AND MICROMETER CALIPERS 13
EXERCISES
Write the following questions and their answers in your
notebook. Express your results in decimal fractions.
Turn the barrel until the entire linear scale is shown,
1. How many divisions are marked on the linear scale ?
2. Determine the unit of the linear scale, whether it is a
centimeter or an inch. This can be done by comparison with
the English and the metric scales marked on your ruler.
3. How long in inches or centimeters is the linear scale ?
4. What is the length of each division of the linear scale ?
Turn the barrel until the screw comes in contact with
the stop.
5. Into how many divisions is the circular scale along the
edge of the barrel graduated ?
6. (a) Does the zero line of the circular scale coincide with
the line of reference of the linear scale ?
(ft) How far are they apart ? Count the number of divisions
of the circular scale between them. This is known as taking
the zero reading.
Turn the barrel until the zero line coincides with the line
of reference. From this position turn the barrel until two
divisions of the linear scale have been passed over.
7. How many complete turns were made ?
Bring the zero line of the barrel back to the line of reference
of ths linear scale. Give the barrel several complete turns and
count the number of divisions passed over on the linear scale.
The relation between the number of turns and the number of
divisions should be carefully noted.
8. How many divisions are passed over in (a) two turns ?
(ft) four turns ? (c) six turns ? (d) one turn ?
9. How far in centimeters or inches does the barrel move in
one complete turn ?
14 APPLIED MATHEMATICS
Bring the zero line opposite the line of reference. Now
move the barrel until the line 5 of the circular scale is oppo
site the line of reference.
10. (a) What part of a turn has the barrel made ?
(i) How far in centimeters or inches did the barrel move ?
(c) How far will the barrel move in passing over one division
of the circular scale ?
Turn the barrel until its edge coincides with the fifth division
of the linear scale, and the zero line of the circular scale coin
cides with the line of reference.
11. What is the length of the opening at the end of the
screw ? Record the distance, and then as a rough check
verify by measuring with a ruler.
With the barrel in the same position as before (at the fifth
division) continue to turn so as to increase the opening at the
end of the screw. Turn the barrel until the seventeenth division
of the circular scale is opposite the line of reference.
12. How much is the opening at the end of the screw ?
The following will illustrate : Suppose the divisions of the
linear scale are ^ (.025) of an inch, and there are 25 divisions
on the circular scale. The value of one division of the circular
scale will be 7^ X fa = .001 in. Each division of the circular
scale, therefore, measures .001 in. In Exercise 12 the distance
for 5 linear divisions would be .025 x 5 = .125. This added to
the value of the 17 circular divisions gives .125 + .017 = .142 in,
for the reading.
EXERCISES
Record the readings in your notebooks and give the work
of the computations in full.
1. Measure the thickness of a coin. Hold the barrel lightly
so that it will slip in the fingers as contact occurs. There is
danger of straining the screw if it is turned up hard. Take
four readings at different places on the coin and average the
results.
VERNIER AND MICROMETER CALIPERS 15
2. Get a metal solid from your instructor and measure its
dimensions. Take the average of three readings. Compute the
volume. Check by using an overflow can.
3. Measure the diameter of a wire. After taking a reading
release the wire and turn it about its axis through 90 ; take
a second reading. If the two readings do not agree, the wire is
slightly flattened in section. Take several readings at different
places on the wire, and the average of the readings will proba
bly be very close to the standard diameter of the wire.
4. Find the volume of a shot. Using the specific gravity of
lead, find the number of shot to the pound.
5. Find the thickness of one of the pages of your textbook.
Compute its volume.
6. Devise other exercises and record them in your notebooks.
CHAPTER III
WORK AND POWER
13. Work. When a man lifts a bar of iron or pulls it along
the floor, he is said to do work upon it. Evidently it takes
twice as much effort to lift 50 Ib. as it does to lift 25 lb., and
five times as much effort to lift a box 5 ft. as it does to lift it
1 ft. That is, work depends upon two things, distance and
pressure.
Hence a foot pound is taken as the unit of work. It is the
work done in raising 1 lb. vertically 1 ft., or it is the pressure
of 1 lb. exerted over a distance of 1 ft. in any direction. If a
nian exerts a pressure of 25 lb. in pushing a wagon 20 ft., he
has done 500 ft. lb. of work.
Foot pounds = feet x pounds.
14. Illustrations. Tie a string to a 1lb. weight, attach a
spring balance and lift it 1, 2, 3 ft. How many foot pounds of
work ? Lower it 1, 2, 3 ft. How much work ? Pull the string
horizontally over the edge of a ruler to raise the weight 1 ft.
How much work ? Is the amount of pressure given by the
spring balance or by the pound weight ? Pull the weight along
the top of the desk 1 ft. How much work ? Hook the spring
balance under the edge of the desk and pull 2 lb. How much
work ? Drop the weight 1 ft. How much work ought the
weight to do when it strikes the floor ?
A boy weighing 60 lb. climbs up a ladder 10 ft. vertically.
How much work ? How much work is done when he comes
down the ladder ?
16
WORK AND POWER 17
A boy weighing 60 Ib. walks up a flight of stairs. How much
work has he done when he has risen 10 ft. ? Why should the
answer be the same as in the preceding problem ?
A stone weighing 50 Ib. is on the roof of a shed 10 ft. from
the ground. How much work was done to get it in that posi
tion ? If it is pushed off, how much work ought it to do when
it strikes the ground ? Why ought the two answers to be the
same number of foot pounds ?
15. Power. Time is not involved in work. A man may take
4 hr. or 10 hr. to raise a ton of coal 15 ft. ; in either case he
has done 30,000 ft. Ib. of work. But in the first case he is
doing work at the rate of 125 ft. Ib. per minute, while in the
second case he is working at the rate of 50 ft. Ib. per minute.
To compare the work of men or machines, or to determine the
usefulness of a machine, it is necessary to take into considera
tion the time required for the work.
Power is the rate of doing work. Thus if an electric crane
raises a steel beam, weighing 500 Ib., 80 ft. in 2 min., its rate
of work is = 20,000 ft. Ib. per minute.
2i
The unit of power, the horse power, is the power required to
do work at the rate of 33,000 ft. Ib. per minute. If a steam
crane lifts 90 T. of coal 11 ft. in 20 min., neglecting friction,
the horse power of the engine is
2000 x 90 x 11 _ o
P ' ~ 33,000 x 20
When we speak of the horse power of an engine we usually
mean the indicated horse power (i. h. p.), which is calculated
from the dimensions of the cylinder and the mean effective
steam pressure obtained from the indicator card. The horse
power actually available for work is called the brake horse
power (b. h. p.), and is determined by the Prony brake or a
similar device.
18 APPLIED MATHEMATICS
The horse power of a steam engine is given by the equation
, p I a n
= "~
where p = mean effective pressure in pounds per square inch,
I = length of stroke in feet,
a = area of piston in square inches,
n = number of strokes per minute, or twice the number
of revolutions per minute.
PROBLEMS
In these problems no account is taken of friction and other
losses.
1. If a man exerts a pressure of 56 Ib. while wheeling a
barrow load of earth 25 ft., find the number of foot pounds of
work he does.
2. How much work is done by a steam crane in lifting a
block of stone weighing 1.2 T. 30 ft. ?
3. A hole is punched through an iron plate \ in. thick. If
the punch exerts a uniform pressure of 40 T., find the work
done.
4. A horse hauling a wagon exerts a constant pull of 75 Ib.
and travels at the rate of 4 mi. per hour. How much work will
the horse do in 3 hr. ? If the driver rides on the wagon, how
much work does he do ?
5. A man weighing 150 Ib. carries 50 Ib. of brick to the
top of a building 40 ft. high. How much work has he done
(a) in getting himself to the top ? (ft) in carrying the brick ?
How much work is done on his return trip down the ladder ?
6. If a pump is raising 2000 gal. of water per hour from
the bottom of a mine 400 ft. deep, how many foot pounds of
work are done in 2 hr. ? (A gallon of water weighs 8.3 Ib.)
7. How many gallons of water would be raised per minute
from a mine 600 ft. deep by an engine of 180 h. p. ?
WORK AND POWER 19
8. The plunger of a force pump is 4 in. in diameter, the
length of the stroke is 3 ft., and the pressure of the water is
40 Ib. per square inch. Find the work done in one stroke.
9. A well 6 ft. in diameter is dug 30 ft. deep. If the earth
weighs 125 Ib. per cubic foot, find the work done in raising the
material.
10. A basement 20 ft. by 15 ft. is filled with water to a
depth of 4 ft. How much work is done in pumping the water
to the street level, 6 ft. above the basement floor ? (The aver
age distance which the water is lifted is 4 ft.)
11. A chain 40 ft. long weighing 10 Ib. per foot is hanging
vertically in a shaft. Construct a curve to show the work done
on each foot in lifting the chain to the surface. (Assume that
the first foot is lifted ft., the second l ft., and so on.) What
is the total work done in lifting the chain ?
12. How much work is done in rolling a 200lb. barrel of
flour up a plank to a platform 6 ft. high ?
13. A boy who can push with a force of 40 Ib. wants to roll
a barrel weighing 120 Ib. into a wagon 3 ft. high. How long a
plank must he use ? (Length of plank x 40 = 3 x 120. Why ?)
14. A man can just lift a barrel weighing 200 Ib. into a wagon
3 ft. high. How much work does he do ? How long a plank
would he need to roll up a barrel weighing 400 Ib. ? 600 Ib. ?
15. A horse drawing a cart along a level road at the rate of
3 mi. per hour performs 42,000 ft. Ib. of work in 5 min. Find
the pull in pounds that the horse exerts in drawing the cart.
16. A horse attached to a capstan bar 12 ft. long exerts a pull
of 120 Ib. How much work is done in going around the circle
100 times ?
17. How long will it take a man to pump 800 cu. ft. of
water from a depth of 16 ft. if he can do 2000 ft. Ib. of work
per minute ?
18. How much work can a 2 h. p. electric motor do in
10 min. ? in 15 sec. ?
20 APPLIED MATHEMATICS
19. What is the horse power of an electric crane that lifts
4 T. of coal 30 ft. per minute ? If 40 per cent of the power is
lost in friction and other ways, what horse power would be
required ?
20. Find the horse power of an engine that would pump
40 cu. ft. of water per minute from a depth of 420 ft., if 20
per cent of the power is lost.
21. A locomotive exerts a pull of 2 T. and draws a train at
a speed of 20 mi. per hour. Find the horse power.
22. The weight of a train is 120 T. and the drawbar pull is
7 Ib. per ton of load. Find the horse power required to keep
the train running at the rate of 30 mi. per hour.
23. The drawbar pull of a locomotive pulling a passenger
train at a speed of 60 mi. per hour is 5500 Ib. At what horse
power is the engine working ?
24. What is the horse power of Niagara Falls if 700,000 T.
of water pass over every minute and fall 160 ft. ?
25. If a 10 h. p. pump delivers 100 gal. of water per minute,
to what height can the water be pumped ?
26. A derrick used in the construction of a building lifts an
Ibeam weighing 2 T. 50 ft. per minute. What is the horse
power of the engine, if 20 per cent of the power is lost ?
27. In a certain mine 400 T. of ore are raised from a depth
of 1000 ft. during a day shift of 10 hr. Neglecting losses, what
horse power is required to raise the ore ?
28. In supplying a town with water 8,000,000 gal. are raised
daily to an average height of 120 ft. What is the horse power
of the engine ?
29. A belt passing around two pulleys moves with a velocity
of 15 ft. per second. Find the horse power transmitted if the
difference in tension of the two sides of the belt is 1100 Ib.
30. What is the difference in tension of the two sides of a
belt that is running 3600 ft. per minute and is transmitting
280 h. p. ?
WORK AND POWER
21
31. Find the number of revolutions per minute which a
driving pulley 2 ft. in diameter must make to transmit 12 h. p.,
if the driving force of the belt is 250 Ib.
32. A belt transmits 60 h. p. to a pulley 20 in. in diameter,
running at 250 r. p. m. What is the difference in pounds of
the tension on the tight and slack sides ?
33. In a power test of an elec
tric motor a friction brake con
sisting of a strap, a weight, and
a spring balance was used. The
radius 01 the pulley was 2 in.,
the pull was 7 Ib., and the speed
of the shaft was 1800 r. p. m.
What horse power did the test
give?
, 2 x 22 x 2 x 1800 x 7
SOLUTION. h. p. = = A.
1 7 x 12 x 38,000
34. In a power test of a small dynamo the pull was 6 Ib.
and the speed was 1500 r. p. m. If the radius of the driving
pulley was 3 in., find the horse power.
35. In testing a motor witli a Prony brake the pull was 12 Ib.,
length of brake arm was 18 in., and the speed was 500 r. p. m.
Find the horse power.
FIG. 4. FRICTION BRAKE
FIG. 6. PRONY BRAKE
36. In testing a Corliss engine running at 100 r. p. m. a
Prony brake was used. The lever arm was 10.5 ft. and the
22 APPLIED MATHEMATICS
pressure exerted at the end of the arm was 2000 Ib. What was
the horse power ? In a second test with a pressure of 2200 Ib.
the speed was 90 r. p. m. Find the horse power.
37. Calculate the horse power of a steam engine from the
following data: stroke, 2ft.; diameter of cylinder, 16 in. ;
r. p. 111., 100 ; mean effective pressure, 60 Ib. per square inch.
38. The diameter of the cylinder of an engine is 20 in. and
the length of stroke is 4 ft. Find the horse power if the engine
is making 60 r. p. m. with a mean effective pressure of 60 Ib.
per square inch.
39. Find the horse power of a locomotive engine if the mean
effective pressure is 90 Ib. per square inch, each of the two
cylinders is 16 in. in diameter and 24 in. long, and the driv
ing wheels make 120 r. p. m.
40. On a sidewheel steamer the engine has a 6ft. stroke,
the shaft makes 35 r. p. m., the mean effective pressure is 30 Ib.
per square inch, and the diameter of the cylinder is 4 ft. Find
the horse power of the engine.
41. Find the horse power of a marine engine, the diameter of
the cylinder being 5 ft. 8 in., length of stroke 5 ft., r. p. in. 15,
and mean effective pressure 30 Ib. per square inch.
42. The diameter of the cylinder of a 514 h. p. marine engine
is 5 ft., length of stroke 6 ft., r. p. m. 20. Find the mean effec
tive pressure.
43. Find the diameter of the cylinder of a 525 h. p. steam
engine : stroke, 6 ft. ; r. p. m., 15 ; mean effective pressure, 25 Ib.
per square inch.
44. What diameter of cylinder will develop 10.3 h. p. with a
6in. stroke, 300 r. p. in., and a mean effective pressure of 90 Ib.
per square inch ?
45. The cylinder of a 55 h. p. engine is 12 in. in diameter
and 28 in. long. If the mean effective pressure is 60 Ib. per
square inch, find the number of revolutions per minute.
WORK AND POWER 23
16. Mechanical efficiency of machines. The useful work
given out by a machine is always less than the work put into
it because of the losses due to the weight of its parts, friction,
and so on. If there were no losses, the efficiency would be 100
per cent.
The efficiency of a machine is the quotient obtained by divid
ing the useful work of the machine by the work put into it.
T,^ . Output
Efficiency = ,
Input
^M .  . Brake horse power
J^mciencu of a steam engine = =
Indicated horsepower
In general the efficiency of a machine increases with the load
up to a certain point, and then falls off. Small engines are
often run at an efficiency of less than 80 per cent ; large
engines usually have an efficiency of 85 to 90 per cent.
PROBLEMS
1. A steam crane working at 3 h. p. raises a block of granite
weighing 8 T., 50 ft, in 12 min. Find the efficiency of the crane.
c ^ . , 2000 x 8 x 50 . u . ,
SOLUTION. Output = ft. Ib. per minute.
U
Input = 3 x 33,000 ft. Ib. per minute.
_ . 2000 x 8 x 50
EffiC16nCy = 12 x a x 33,000 = 67 Per ^
2. A 6 h. p. electric crane lifts a machine weighing 15 T.
at the rate of 5 ft. per minute. What is the efficiency ?
3. An engine of 150 h. p. is raising 1000 gal. of water per
minute from a mine 500 ft. deep. Find the efficiency of the
pumping system.
4. An elevator motor of 50 h. p. raises the car and its load,
2800 Ib. in all, 120 ft. in 15 sec. Find the efficiency.
5. How long will it take a 20 h. p. engine to raise 2 T. of
coal from a mine 300 ft. deep, if the efficiency is 80 per cent ?
24 APPLIED MATHEMATICS
6. What is the efficiency of an engine if the indicated horse
power is 250 and the brake horse power is 225 ?
7. In lifting a weight of 256 Ib. 20 ft. by means of a tackle
a man hauls in 64 ft. of rope with an average pull of 110 Ib.
Find the efficiency of the tackle.
8. The efficiency of a set of pulleys is 75 per cent. How
many pounds must be the pull, acting through 88 ft., to raise
a load of 525 Ib. a distance of 20 ft. ?
9. A pump of 10 h. p. raises 54 cu. ft. of water per minute
to a height of 80 ft. What is its efficiency ?
10. A steam crane unloads coal from a vessel at the rate of
20 T. in 8 min., and lifts it a total distance of 24 ft. If the
combined efficiency of the engine and crane is 70 per cent,
what is the horse power of the engine ?
11. Eind the power required to raise 4800 gal. of water 60 ft.
in 2 hr. if the efficiency of the pump is 60 per cent.
12. A centrifugal pump whose efficiency when lifting water
12 ft. is 62 per cent, is required to lift 18 cu. ft. per second to
a height of 12 ft. What must be its horse power ?
13. A dock 200 ft. long and 50 ft. wide is filled with water
to a depth of 30 ft. It is emptied in 40 min. by a centrifugal
pump which delivers the water 40 ft. above the bottom of the
dock. If the combined efficiency of the engine and pump is
70 per cent, what is the horse power of the engine ? (A cubic
foot of sea water weighs 64 Ib. The average distance which
the water is lifted is 25 ft.)
14. A steam engine having a cylinder 10 in. in diameter and
a stroke of 24 in. makes 80 r. p. m. and gives a brake horse
power of 34 h. p. If the mean effective pressure is 50 Ib. per
square inch, find the efficiency.
15. In testing a Corliss engine running at 80 r. p. m. a Prony
brake was used. The lever arm was 10.5 ft. and the pressure
at the end of the arm was 1600 Ib. The indicated horse power
was 290. Find the efficiency of the engine.
WORK AND POWER
25
16. The efficiency of a boiler is 70 per cent and of the en
gine 80 per cent. What is the combined efficiency ?
SOLUTION. .80 x .70 = 56 per cent.
17. Power is obtained from a motor. If the efficiency of the
motor is 88 per cent, of the dynamo 85 per cent, and of the
engine 86 per cent, what is the combined efficiency ?
18. The engine which furnishes power for a centrifugal
pump has an indicated horse power of 14 and an efficiency of
88 per cent. What is the efficiency of the pump if it is raising
3000 gal. of clear water 12 ft. high per minute ?
19. In a test to find the efficiency of a set of pulleys the fol
lowing results were obtained. Construct the efficiency curve.
Weight lifted (pounds)
5
10
15
20
25
30
35
Distance (feet) . .
1
1
1
1
1
1
1
Pull in pounds . .
3
6
6.5
8
9.5
11
12.8
Distance (feet) . .
3
3
3
3
3
3
3
20. In a test to determine the relative efficiency of centrif
ugal and reciprocatipg pumps the following results were ob
tained. Construct the efficiency curves.
Lift in feet
<?,
5
10
ir>
90
95
30
35
40
r >0
60
80
100
T>0
160
00
40
>RO
Efficiency of reciprocat
ing pump (per cent) .
30
45
55
61
66
68
71
75
77
82
85
87
90
89
88
85
Efficiency of centrifugal
pump (per cent) . . .
50
56
64
68
69
68
66
62
58
50
40
21. In a laboratory experiment to determine the efficiency of
a set of pulleys the following results were obtained. Construct
the efficiency curve.
Load in grains .
Efficiency . .
40
13.2
00
20.5
140
36.0
100
43.2
240
49.0
290
53.6
340
57.4
390
00.6
440
63.1
490
65.5
26
APPLIED MATHEMATICS
22. The following results were obtained in an experiment to
find the efficiency of a set of differential chain pulley blocks.
Find the efficiency in each test and construct the efficiency
curve.
Load in pounds .
7
21
36
49
70
98
112
126
140
Distance (feet) . .
1
1
1
1
1
1
1
1
1
Pull in pounds
3.22
6.73
8.40
11.03
15.13
20.17
23.17
26.00
29.06
Distance (feet) . .
16
16
16
16
16
16
16
16
16
23. Find the efficiency of the following engines :
No.
Type
Pressure
in Ib.
per sq.in.
Stroke in
inches
Diameter of
cylinder in
inches
Revolutions
per minute
Brake
horse
power
1
Marine . . .
37
168
110
17
4440
2
Marine . . .
26
72
70
15
441
3
Corliss . . .
90
48
30
86
1180
4
6
Gas engine * .
Locomotive .
62
80
16
24
12
17
160
260
18
504
6
7
Highspeed .
Mediumspeed
60
75
16
36
12
24
246
100
100
633
Explosion every two revolutions.
CHAPTER IV
LEVERS AND BEAMS
17. Law of the lever. A rigid rod movable about a fixed
point may be held in equilibrium by two or more forces. To
find the relation between these forces when the lever is in
a state of balance, we will make a few experiments.
EXERCISES
1. Balance a meter stick at its center ; suspend on it two
unequal weights so that they balance. Which weight is nearer
the center? Multiply each
weight by its distance from
the center and compare the i i
products. Do this with 
several pairs of weights. LJ
What seems to be true ?
2. Balance a meter stick
as before, and put a 500g.
weight 12 cm. f roiii the cen
ter ; then in turn put on the
following weights so Law O f the iever . p . PF ^ w . WF .
that each balances p
the 500g. weight. / """
In each case record  Fl  7
the distance from the weight to the center.
PIG. 6
Grams
450
400
350
300
250
200
150
120
Measured distance .
Computed distance .
27
28 APPLIED MATHEMATICS
Locate a point on squared paper for each weight. Units:
horizontal, 1 large square = 5 cm. ; vertical, 1 large square =
50 g. Draw a smooth curve through the points. Take some
intermediate points on the curve and test the readings by put
ting the weights on the meter stick. On the same sheet of
squared paper draw the curve, using the computed distances.
3. Suspend two unequal weights on one side of the center
and balance them with one weight. What is the law of the
lever for this case ?
Definitions. The point of support is called the fulcrum. The
product of a weight and its distance from the fulcrum is called
the leverage of the weight. The quotient of the length of one
arm of the lever divided by the length of the other is called
the mechanical advantage of the lever.
The force which causes a lever to turn about the fulcrum
may be called the power (/>), and the body which is moved may
be called the weight (?#).
18. Three classes of levers. Levers are divided into three
classes, according to the position of the power, fulcrum, and
R r w
FIG. 8
Class I. When the fulcrum is between the power and the
weight. Name some levers of this class.
Class II. When the weight is between the fulcrum and the
power. Name some levers of this class.
Class III. When the power is between the fulcrum and
the weight. Name some levers of this class.
LEVERS AND BEAMS 29
19. Levers of the first class. With a lever of this class a
large weight may be lifted by a small power; time is lost
while mechanical advantage is gained.
PROBLEMS
In these problems on levers of the first class either the lever
is " weightless/' that is, it is supposed to balance at the ful
crum, or else the weight of the lever is neglected. Draw a
diagram for each problem.
1. What weight 12 in. from the fulcrum will balance a 6lb.
weight 14 in. from the fulcrum ?
SOLUTION. Let w = the weight.
12 w = 6 x 14.
w = l.
Check. 12 x 7 = 6 x 14.
84 = 84.
2. How far from the fulcrum must a 74b. weight be placed
to balance a 4lb. weight 35 cm. from the fulcrum ?
3. What is the weight of an object 10 in. from the fulcrum,
if it balances a weight of 3 Ib. 14.4 in. from the fulcrum ?
4. A meter stick is balanced at the center. On one side are
two weights of 10 Ib. and 4 Ib., 4 in. and 1\ in. from the ful
crum respectively. How far from the fulcrum must a 7lb.
weight be placed to balance?
5. Two books weighing 250 g. and 625 g. are suspended
from a meter stick to balance. The heavier book is 12 cm. from
the center. How far is the other book from the center ?
6. A 5g. and a 50g. weight are placed to balance on a
meter stick suspended at its center. If the leverage is 100,
how far is each weight from the center?
7. An iron casting weighing 6 Ib. is broken into two pieces
which balance on a meter stick when the mechanical advantage
is 4. Find the weight of each piece.
30 APPLIED MATHEMATICS
8. Two boys weighing 96 and 125 Ib. play at teeter. If the
smaller boy is 8 ft. from the fulcrum, how far is the other boy
from that point ?
9. Two boys playing at teeter weigh 67 Ib. and 120 Ib. and
are 7 ft. and 6 ft. respectively from the fulcrum. Where must
a boy weighing 63 Ib. sit to balance them ?
10. Two bolts weighing together 392 g. balance when placed
50 cm. and 30 cm. respectively from the fulcrum. Find the
weight of each.
11. A boy weighing 95 Ib. has a crowbar 6 ft. long. How
can he arrange things to raise a block of granite weighing
280 Ib. ?
12. A lever 15 ft. long balances when weights of 72 Ib. and
108 Ib. are hung at its ends. Find the position of the fulcrum.
PROBLEMS IN WHICH THE WEIGHT OF THE LEVER
is INCLUDED
Exercise 1. Test a meter stick to see if it balances at the
center. If it does not, add a small weight to make it balance.
Weigh the meter stick. It is found to weigh 162 g.
1.6.2. = 1.02 g. per centimeter.
Attach a 200g. weight to one
end and balance as in Fig. 9. l
The length of FW = 22.4 cm.
The length of PF = 77.6 cm.
The weight of
JFTr = 22.4x1.62 = 36.2 g.
The weight of
PF = 77.6 x 1.62 = 125.7 g.
161.9 g. Check. ElG ' 9
The 200g. weight and the short length of the meter stick balance
the long part. Let us suppose that the weight of each part is con :
centrated at the center of the part, and apply the law of the lever.
LEVERS AND BEAMS 31
77 fi 99 4
125.7 x l~i = 36.2 x f~ + 200 x 22.4.
125.7 x 38.8= 36.2 x 11.2 + 4480.
4877 = 4885.
This checks as near as can be expected in experimental work.
The measurements were made to three figures and the results differ
only by one in the third place.
Hence when a uniform bar is used as a lever we may assume
that the weight of each part is concentrated at the midpoint of
fche part.
A shorter method of solution is to consider the weight of the
lever as concentrated at its center. Thus, in the preceding exercise :
200 x FW = 162 (50  FW).
Solving, F W =22. 4.
Exercise 2. Make a similar test with a metal bar.
PROBLEMS
1. One end of a stick of timber weighing 10 Ib. per linear
foot, and 14 ft. long, is placed under a loaded wagon. If the
fulcrum is 2 ft. from the end, how many pounds does the tim
ber lift when it is horizontal ?
SOLUTION. Let x number of pounds lifted.
2 x + 20 x 1 = 120 x 6.
x = 350 Ib.
Check. 2 x 350 + 20 = 120 X 6.
720 = 720.
2. A lever 20 ft. long and weighing 12 Ib. per linear foot is
used to lift a block of granite. The fulcrum is 4 ft. from one
end and a man weighing 180 Ib. puts his weight on the other
end. How many pounds are lifted on the stone ?
3. A uniform lever 12 ft. long and weighing 36 Ib. balances
upon a fulcrum 4 ft. from one end when a load of x Ib. is
hung from that end. Find the value of x.
32 APPLIED MATHEMATICS
4. A uniform lever 10 ft. long balances about a point 1 ft.
from one end when loaded at that end with 50 Ib. What is the
weight of the lever ?
FIG. 10
SOLUTION. Let x weight of a linear foot.
50xl + xxJ^9xx4^.
x = 1 J. Ib.
10 x = 12$ Ib.
50 = 50f.
SECOND SOLUTION. Let x = weight of the lever.
x x 4 = 50 x 1.
4 x = 50.
Check. 12$ x 4 = 50.
5. A man weighing 180 Ib. stands on one end of a steel
rail 30 ft. long and finds that it balances over a fulcrum at
a point 2 ft. from its center. What is the weight of the rail
per yard ?
6. A teeter board 16 ft. long and weighing 32 Ib. balances at
a point 7 ft. from one end when a boy weighing 80 Ib. is seated
1 ft. from this end and a second boy 1 ft. from the other end.
How much does the second boy weigh ?
7. A uniform lever 12 ft. long balances at a point 4 ft. from
one end when 30 Ib.' are hung from this end and an unknown
weight from the other. If the lever weighs 24 Ib., find the
unknown weight.
20. Levers of the second class. With a lever of the first
class the weight moves in a direction opposite to that in which
the power is applied. How is it with a lever of the second class ?
LEVERS AND BEAMS
EXERCISES
33
1. Place a meter stick as shown in Fig. 11, and put weights
in the other pan to balance. This arrangement makes a weight
less lever.
(a) Put 100 g. 18 in. from F. How many grams are required
to balance it ?
FIG. 11
SOLUTION.
p.PF=?o. WF.
36;; = 100 x 18.
p = 50 g.
Check by putting a 50g. weight in the other pan.
(6) Put 100 g. 9 in. from F and find p. Check.
(c) Put 100 g. 27 in. from F and find p. Check.
(d) Put 200 g. 12 in. from F and find p. Check.
2. Lay a uniform metal bar 2 or 3 ft. long on the desk and
lift one end witli a spring balance. Compare the reading with
the weight of the bar. Make two or three similar tests. What
seems to be true ? Where is the fulcrum ? Where is the
power? Where is the weight with reference to the fulcrum
and power ? Suppose the weight of the bar to be concentrated
at the center and see if the law of the lever p  PF = w WF
holds true.
3. Place a 2lb. weight on a meter stick lying on the desk
at distances of (a) 40 cm., (i) 50 cm., (V) 60 cm., and (d) 80 cm.
from one end. In each case compute the pull required to lift
the other end of the meter stick. Check by lifting with a spring
balance.
4. Construct a graph to show the results obtained in Exer
cise 3. Why should it be a straight line ?
34 APPLIED MATHEMATICS
PROBLEMS
1. A lever 6 ft. long has the fulcrum at one end. A weight
of 120 Ib. is placed on the lever 2 ft. from the fulcrum. How
many pounds pressure are required at the other end to keep the
lever horizontal, (r/) neglecting the weight of the lever ? (i) if
the lever is uniform and weighs 20 Ib. ?
2. A man uses an 8ft. crowbar to lift a stone weighing
800 Ib. If he thrusts the lever 1 ft. under the stone, with what
force must he lift to raise the stone ?
3. A man is using a lever with a mechanical advantage of 6.
If the load is 1 J ft. from the fulcrum, how long is the lever ?
4. A boy is wheeling a loaded wheelbarrow. The center of the
total weight of 100 Ib. is 2 ft. from the axle and the boy's hands
are 5 ft. from the axle. What lifting force does he exert ?
5. A uniform yellowpine beam 10 ft. long weighs 38 Ib. per
linear foot. When it is lying horizontal a man picks up one
end of the beam. How many pounds does he lift ?
6. To lift a machine weighing 3000 Ib. a man has a jack
screw which will lift 800 Ib. and a beam 12 ft. long. If the
jackscrew is placed at one end of the beam and the other end
is made the fulcrum, how far from the fulcrum must he attach
the machine in order to lift it ?
21. Levers of the third class. In all levers of this class
the power acts at a mechanical disadvantage since it must be
greater than the weight. Therefore this form of the lever is
used when it is desired to gain speed rather than mechanical
advantage.
EXERCISES
Attach a meter stick to the base of the balance, as shown in
Fig. 12, and let the meter stick rest on a triangular block placed
in one pan of the balance. Put weights in the other pan to
balance. This makes a weightless lever.
Let PF = 9 cm. *
LEVERS AND BEAMS 35
1. Put 100 g. 18 em. from F. How many grams arc required
to balance it ?
SOLUTION.
p.pF^ w . WF.
9;> = 100 x 18.
p = 200.
Check by placing 200 g.
in the pan. FIG. 12
2. Put 100 g. 27 cm. from F and find p. Check.
3. Put 50 g. 36 cm. from F and find p. ('heck.
4. Put 50 g. 45 cm. from F and find p. Check.
5. Put one end of a meter stick just under the edge of the
desk. Hold the stick horizontal with a spring balance. Where
are the fulcrum, weight, and power ? Where may we consider
the weight of each part of the meter stick to be concentrated ?
Weigh the meter stick and compute the pull required to hold
it horizontal. Check by reading the spring balance.
6. Make the same experiment with a uniform metal bar.
PROBLEMS
1. A lever 12 ft. long has the fulcrum at one end. A pull of
80 Ib. 3 ft. from the fulcrum will lift how many pounds at the
other end ? Neglect weight of lever.
2. The arms of a lever of the third class are 2 ft. and 6 ft.
respectively. How many pounds will a pull of 60 Ib. lift ?
3. With a lever of the third class a pull of 65 Ib. applied
6 in. from the fulcrum lifts a weight of 5 Ib. at the other end
of the lever. How long is the lever ? Neglect its weight.
4. If the mechanical advantage of a lever is ^, a pull of
how many pounds will be required to lift 40 Ib. ?
5. Construct a curve to show the mechanical advantage of a
lever 12 ft. long, as the power is applied 1 ft., 2 ft., 3 ft. ...
from the weight, the whole length of the lever being used.
36
APPLIED MATHEMATICS
22. Beams. The following exercises will show that a straight
beam resting in a horizontal position on supports at its ends
may be considered a lever of the second class.
EXERCISES
1. Test two spring balances to see if they are correct. Weigh
a meter stick. Suspend it on two spring balances, as shown in
Fig. 13. Read each balance. Note that each should indicate
one half the weight of the meter stick. Place a 200g, weight
at the center. Read each balance.
FIG. 13
2, With the meter stick as in Exercise 1, place a 200g.
weight 10, 20, 30, ... 90 cm. from one end, and record the
reading of each balance after the meter stick has been made
horizontal. Construct a curve for the readings of each balance
on the same sheet of squared paper.
far
40
FIG. 14
To compute the reading of the balance we need only think of the
beam as a lever of the second class.
Thus, when the weight is 40 cm. from one end,
p x 100 = 200 x 60,
77 = 120;
and q x 100 = 200 x 40,
7 = 80.
Check. 80 + 120 = 200.
LEVERS AND BEAMS
37
3. Suspend a 500g. weight 20 cm. from one end of the meter
stick. Read the balances after the stick has been made horizontal.
Correct for its weight. Compare with the computed readings.
4. Make similar experiments with metal bars and with two
)r three weights placed on the bar at the same time.
PROBLEMS
1. A man and a boy are carrying a box weighing 120 Ib. on
i stick 8 ft. long. If the box is 3 ft. from the man, what weight
is each carrying ?
SOLUTION.
Arithmetic.
Algebra. Let
Solving,
FIG. 15
3 + 5 = 8.
 of 120 = 45 Ib., weight boy carries.
I of 120 = 75 Ib., weight man carries.
120. Check.
x = number of pounds man carries.
y = number of pounds boy carries.
x + y = 120
3 x = 5 y.
x = 75.
y = 45.
M
A general solution.
FIG. 16
x(m + n) = n W.
=
m + n
y(m + n) = m W.
y =
m + n
88 APPLIED MATHEMATICS
2. Two men, A and B, carry a load of 400 Ib. on a pole be
tween them. The men are 15 ft. apart and the load is 7 ft.
from A. How many pounds does each man carry ?
3. A man and a boy are to carry 300 Ib. on a pole 9 ft. long.
How far from the boy must the load be placed so that he shall
carry 100 Ib. ?
4. A beam 20 ft. long and weighing 18 Ib. per linear foot
rests on a support at each end. A load of 1 T. is placed 6 ft.
from one end. Find the load on each support.
. 5. A locomotive weighing 56 T. stands on a bridge with its
center of gravity 30 ft. from one end. The bridge is 80 ft. long
and weighs 100 T. ; it is supported by stone abutments at the
ends. Find the total weight supported by each abutment.
6. A man weighing 192 Ib. walks on a plank which rests on
two posts 16 ft. apart. Construct curves to show the pressure
on each of the posts as he walks from one to the other.
MISCELLANEOUS PROBLEMS
1. One end of a crowbar 6 ft. long is put under a rock, and
a block of wood is put under the bar 4 in. from the rock. A
man weighing 200 Ib. puts his weight on the other end. How
many pounds does he lift on the rock, and what is the pressure
on the block of wood ?
2. A nutcracker 6 in. long has a nut in it 1 in. from the
hinge. The hand exerts a pressure of 4 Ib. What is the pres
sure on the nut ?
3. What pressure does a nut in a nutcracker withstand if
it is 2.8 cm. from the hinge, and the hand exerts a pressure of
1.5 kg. 12 cm. from the hinge ?
4. Two weights, P and Q, hang at the ends of a weightless
lever 80 cip. long. P = 1.2 kg. and Q = 3 kg. Where is the
fulcrum if the weights balance?
LEVERS AND BEAMS 39
5. A man uses a crowbar 7 ft. long to lift a stone weighing
600 Ib. If he thrusts the bar 1 ft. under the stone, with what
force must he lift on the other end of the bar ?
6. A safety valve is 2 in. in diameter and the lever is
18 in. long. The distance from the fulcrum to the center of
the valve is 3 in. What weight must
be hung at the end of the lever so
that steam may blow off at 100 Ib.
per square inch, neglecting weight
of valve and lever ?
7. What must be the length of "F I?"
the lever of a safety valve whose
area is 10 sq. in., if the weight is 180 Ib., steam pressure 120 Ib.
per square inch, and the distance from the center of the valve
to the fulcrum is 3 in. ?
8. Find the length of lever required for a safety valve 3 in.
in diameter to blow off at 60 Ib. per square inch, if the weight
at the end of the lever is 75 Ib. and the distance from the
center of the valve to the fulcrum is 2 in.
9. In a safety valve of 3^ in. diameter the length of the
lever from fulcrum to end is 24 in., the weight is 100 Ib., and
the distance from fulcrum to center of valve is 3 in. Find
the lowest steam pressure that will open the valve.
10. A bar 4 m. long is used by two men to carry 160 kg. If
the load is 1.2 m. from one man, what weight does each carry ?
11. A bar 12 ft. long and weighing 40 Ib. is used by two
men to carry 240 Ib. How many pounds does each man carry
if the load is 6 ft. from one man ?
12. A man and a boy have to carry a load slung on a light
pole 12 ft. long. If their carrying powers are in the ratio 8 : 5,
where should the load be placed on the pole ?
13. A wooden beam 15 ft. long and weighing 400 Ib. carries
a load of 2 T. 5 ft. from one end. Find the pressure on the
support at each end of the beam.
40
APPLIED MATHEMATICS
14. A beam carrying a load of 5 T. 3 ft. from one end rests
with its ends upon two supports 20ft. apart. If the beam
is uniform and weighs 2 T., calculate the pressure on each
support.
15. The horizontal roadway of a bridge is 30 ft. long and its
weight is 6 T. What pressure is borne by each support at the
ends when a wagon weighing 2 T. is one third the way across ?
16. An iron girder 20 ft. long and weighing 60 Ib. per foot
carries a distributed load of 1800 lb. ? and two concentrated
loads of 1500 Ib. each 6 ft. and 12 ft. respectively from one
support. Calculate the pressure on each support.
17. One end of a beam 8 ft. long is set solidly in the wall,
as in Fig. 18. If the beam weighs 40 Ib.
per linear foot, what is the bending moment
at the wall ?
SOLUTION. The bending moment at any t
point A is equal to the weight multiplied by
its distance from A . We may assume that the
weight of the beam is concentrated at its center
4 ft. from the wall. Hence the bending moment
= 320 x 4 = 1280 Ib. ft.
FIG. 18
18. In Fig. 18 a weight of 800 Ib. is placed at the end of
the beam away from the wall. What will be the total bending
moment ?
19. A steel beam weighing 100 Ib. per linear foot projects
20 ft. from a solid wall. What is the bending moment at the
wall ? What weight must be placed at the outer end to make
the bending moment five times as great ?
20. A stiff pole 15 ft. long sticks out horizontally from a
vertical wall. It would break if a weight of 30 Ib. were hung
at the end. How far out on the pole may a boy weighing 80 Ib.
go with safety ?
21. A steel beam 15 ft. long projects horizontally from a
vertical wall. At the end is a weight of 400 Ib. Construct a
LEVERS AND BEAMS 41
curve to show the bending moments of this weight at various
points on the beam from the wall to the outer end.
Suggestion. The bending moment at the wall is 400 x 15 = 6000
Ib. ft. ; 1 ft. from the wall it is 400 x 14 = 5600 Ib. ft., and so on.
22. A beam projects horizontally 15 ft. from a vertical wall.
Construct a curve to show the relation between the distance
and the weight if the bending moment at the wall is kept at
1200 Ib. ft.
CHAPTER V
SPECIFIC GRAVITY
23. Mass. The mass of a body is the quantity of matter
(material) contained in it. The English unit of mass is a cer
tain piece of platinum kept in the Exchequer Office in London.
This lump of platinum is kept as a standard and is called a
pound. The metric unit of mass is a gram; it is the mass of
a cubic centimeter of distilled water at 4 C. (39.2 F.).
24. Weight. The weight of a body is the force with which
the earth attracts it. The mass of a pound weight would not
change if it were taken to different places on the surface of the
earth, but its weight would change. A piece of brass which
weighs a pound in Chicago would weigh a little more than a
pound at the north pole and a little less than a pound at the
equator. Why ? The masses of two bodies are usually com
pared by comparing their weights.
25. Density. The density of a body is the quantity of mat
ter in a unit volume. Thus with the foot and pound as units
the density of water at 60 F. is about 62.4, since 1 cu. ft. of
water at 60 F. weighs about 62.4 Ib. In metric units the
density of water at 4 C. is 1, since 1 ccm. of water at 4 C.
weighs 1 g. The density of lead in English units is 707 ; that
is, 1 cu. ft. of lead weighs 707 Ib. In metric units the density
of lead is 11.33, since 1 ccm. of lead weighs 11.33 g.
26. Specific gravity. The specific gravity or relative density
of a substance is the ratio of the weight of a given volume of
the substance to the weight of an equal volume of water at
4 C. (39.2 F.). Thus if a cubic inch of copper weighs .321 Ib.
42
SPECIFIC GRAVITY
43
and a cubic inch of water weighs .0361 Ib., the specific gravity
of this piece of copper is .321 ~ .0361 = 8.88. If we are told
that the specific gravity of silver is 10.47, it means that a cubic
foot of silver weighs 10.47 times as much as a cubic foot of
water.
APPROXIMATE SPECIFIC GRAVITIES
Aluminum . .
2.67
Ice ....
.917
Oak, white
.77
Brass . . .
7.82
Iron, cast . .
7.21
Pine, white .
.55
Copper . . .
8.79
Iron, wrought .
7.78
Pine, yellow .
.66
Cork . . .
.24
Lead ....
11.3
Silver . . .
10.47
Glass, white .
2.9
Marble .
2.7
Steel . . .
7.92
Granite . .
2.6
Mercury, at 60
13.6
Tin ....
7.29
Gold ....
19.26
Nickel . . .
8.8
Zinc
7.19
Exercise. Find the specific gravity of several blocks of wood
and pieces of metal.
Problem. The dimensions of a block of cast iron are 3^ in.
by 2f in. by 1 in., and it weighs 37.5 oz. Find its specific gravity.
3 X 2 x 1 = 8.94 cu. in.
1 cu. in. of water = .0361 Ib.
8.94 cu. in. of water = .0361 x 16 x 8.94 oz.
= 5.15 oz.
Q Weight of block of metal
Weight of equal volume of water
^37.5
""5.15
= 7.28.
PROBLEMS
1. What is the weight of 1 cu. in. of copper ? p jp
SOLUTION. 1 cu in. of water = .0361 Ib. .0361
Specific gravity of copper is 8.79 ; that is, copper is 8.79 264
times as heavy as water. 52
.. 1 cu. in. of copper = .0361 x 8.79 Ib. 1
= .317 Ib. .317
44 APPLIED MATHEMATICS
2. What is the weight of 1 cu. ft. of cast iron ? p$.
SOLUTION. 1 cu. ft. of water = 62.4 Ib. 7  21
Specific gravity of cast iron is 7.21. 437
.. 1 cu. ft. of cast iron = 62.4 x 7.21 Ib. 12
= 450 Ib. 1
450
3. Find the weight of 1 cu. in. of (a) aluminum ; (ft) cork ;
(c) lead ; (d) gold ; (e) silver ; (/) zinc.
4. Find the weight of 1 cu. ft. of (a) granite ; (ft) ice ;
(c) marble ; (d) white oak ; (e) yellow pine.
5. What is the weight of a yellowpine beam 20 ft. long,
8 in. wide, and 10 in. deep ?
6. The ice box in a refrigerator is 24 in. by 16 in. 'by 10 in.
How many pounds of ice will it hold ?
7. A piece of copper in the form of an ordinary brick is
8 in. by 4 in. by 2 in. What is its weight ? How much would
a gold brick of the same size weigh ?
8. A flask contains 12 cu. in. of mercury. Find the weight
of the mercury.
9. Find the weight of a gallon of water.
10. What is the weight of a quart of milk if its specific
gravity is 1.03 ?
11. How many cubic inches are there in a pound of water ?
SOLUTION. 1 cu. in. = .0361 Ib.
.. 1 Ib. = cu. in.
.0361
= 27.7 cu. in.
12. An iron casting weighs 50 Ib. Find its volume.
SOLUTION. Let x = number of cubic inches, in the casting.
.0361 x = weight of x cu. in. of water.
7.21 x .0361 x = weight of x cu. in. of cast iron.
50
x =
7.21 x .0361
=r 192 cu. in.
SPECIFIC GRAVITY 45
13. What is the volume of 60 Ib. of aluminum ?
14. How many cubic feet are there in 50 Ib. of cork ?
15. How many cubic inches are there in a flask which just
holds 6 Ib. of mercury ?
16. A cubic foot of bronze weighs 552 Ib. What is its spe
cific gravity ?
17. Find the specific gravity of a block of limestone if a
cubic foot weighs 182 Ib.
18. A cubic inch of platinum weighs .776 Ib. What is its
specific gravity ?
19. A cedar block is 5 in. by 3 in. by 2 in. and weighs 10.5 oz.
Find its specific gravity.
20. .0928 cu. ft. of metal weighs 112 Ib. Find its specific
gravity.
21. Each edge of a cubical block of metal is 2 ft. If it weighs
4450 Ib., what is its specific gravity ?
22. A metal cylinder is 15.3 in. long and the radius of a
cross section is 3 in. If it weighs 176.6 Ib., what is its specific
gravity ?
23. The specific gravity of petroleum is about .8. How many
gallons of petroleum can be carried in a tank car whose capacity
is 45,000 Ib. ?
27. Advantage of the metric system. So far we have been
using the English system, and we have had to remember that
1 cu. in. of water weighs .0361 Ib. But in the metric system
the weight of 1 ccm. of water is taken as the unit of weight and
is called a gram. Thus 8 ccm. of water weighs 8 g. If a cubic
centimeter of lead weighs 11.33 g., it is 11.33 times as heavy
as water; hence its specific gravity is 11.33. The weight in
grams of a cubic centimeter of any substance is its specific gravity.
Exercise. To show that 1 ccm. of water weighs 1 g.
Balance a glass graduate on the scales. Pour into it 10, 20, 30 ccm.
of water, and it will be found that the weight is 10, 20, 30 g.
46 APPLIED MATHEMATICS
What is the weight of 80 ccm. of water ? A dish 8 cm. by
5 cm. by 2 cm. is full of water ; how many grams does the
water weigh ? A block of wood is 12 cm. by 10 cm. by 5 cm. ;
what is the weight of an equal volume of water ? A brass
cylinder contains 125 ccm. ; what is the weight of an equal vol
ume of water ? Hence the volume of a body in cubic centi
meters is equal to the weight in grains of an equal volume
of water.
28. First method of finding specific gravity,
1. Weigh the solid in grams.
2. Find the volume of the solid in cubic centimeters.
1 ccm. of water == 1 g.
/. the volume in cubic centimeters equals the weight of an
equal volume of water.
Weight in grams
Weight of an equal volume of water
__ Weight in grams __
Volume in cubic centimeters *
Exercise. Find the specific gravity of (a) a brass cylinder;
(&) a brass prism ; (c) a steel ball ; (dT) a copper wire ; (e) an
iron wire; (/) a pine block; (g") a piece of oak. Can you
expect to obtain the specific gravities given in the table ?
Why not ? *
PROBLEMS
1. A block of metal 13.8 cm. by 14.2 cm. by 27.0 cm. weighs
60 kg. Find its specific gravity.
2. A cylinder is 84.3 mm. long and the radius of its base is
15.4 mm. If it weighs 157 g., what is its specific gravity ?
3. A metal ball of radius 21.5 mm. weighs 292.6 g. Find its
specific gravity.
4. The altitude of a cone is 42.1 mm. and the radius of the
base is 14.6 mm. Find its specific gravity if it weighs 22.3 g.
SPECIFIC GRAVITY
47
5. How many times heavier is (a) gold than silver? (#)
gold than aluminum? (c) mercury than copper? (rf) steel
than aluminum ? (e) platinum than gold ? (/) cork than lead ?
6. The pine pattern from which an iron casting is made
weighs 15 Ib. About how much will the casting weigh ? (The
usual foundry practice is to call the ratio 16 : 1.)
29. The principle of Archimedes. This principle furnishes a
convenient method of finding the specific gravity of substances.
Exercise. Weigh a brass cylinder ; weigh it when suspended
in water and find the difference of the weights. Lower the
cylinder into an overflow can filled with
water and catch the water in a beaker
as it flows out. Compare the weight
of this water with the difference in the
weights. Do this with several pieces
of metal. What seems to be true?
Imagine a steel ball submerged in
water resting on a shelf. If the shelf
were taken away, the ball would sink to the bottom of the tank.
Now suppose the surface of the ball contained water instead
of steel, and suppose the inclosed water weighed 5 oz. If the
shelf were removed, the water ball would be held in its posi
tion by the surrounding water ; that is, when the steel ball is
suspended in water, the water holds up 5 oz. of the total weight
of the ball.
PRINCIPLE OF ARCHIMEDES. Any body when suspended in
water loses in weight an amount equal to the weight of its own
volume of water.
30. Second method of finding specific gravity.
1. Weigh a piece of cast iron, 156.3 g.
2. Weigh it when suspended in water, 134.3 g.
3. 156.3 134.3 = 22.0 g. This is the weight of an equal
volume of water.
FIG. 19
48 APPLIED MATHEMATICS
. 156.3 _ 1A
4. Sp. gr = 2^0 = 7.10.
Let W = the weight of the substance in air.
w = the weight of the substance suspended in water.
W
Then = the specific gravity of the substance.
W w
Exercise. Find by this method the specific gravity, of (a) brass ;
(V) copper ; (<?) cast iron ; (d) glass ; (e) lead ; (/) porcelain ;
(<7) an arclight carbon.
PROBLEMS
1. How much will a brass 50g. weight weigh in water ?
SOLUTION. Let x = the weight in water.
* = 7 . 82 .
50 x
Solving, x = 43.6 g. Check by experiment.
2. Compute the weight in water of (a) 100 g. of copper ;
(b) 500 g. of zinc ; (c) 1 kg. of silver ; (d) 200 g. of pine ;
(V) 100 g. of cork.
3. Find the weight in water of (a) 1 Ib. of cast iron ; () 1 Ib.
of lead ; (c) 5 Ib. of aluminum ; (d) 1 T. of granite ; (e) 10 Ib.
of cork.
4. If a boy can lift 150 Ib., how many pounds of the follow
ing substances can he lift under water: (a) platinum ? (6) lead ?
(c) cast iron ? (d) aluminum ? (e) granite ?
SOLUTION, (a) The problem is to find the weight in air of a
mass of platinum which weighs 150 Ib. in water.
Let w = the weight in air.
w
= 22 (specific gravity of platinum).
Solving, w = 157 Ib.
5. Construct a curve to show the weight in air of masses
which weigh 1 Ib. in water, the specific gravity varying from
1 to 20.
SPECIFIC GRAVITY 49
6. A coppei* cylinder weighs 80 Ib. under water. How much
does it weigh in air ?
7. A cake of ice just floats a boy weighing 96 Ib. How many
cubic feet are there in it ?
Suggestion. 1 cu. ft. of water weighs 62.4 Ib. How much does
1 cu. ft. of ice weigh? How many pounds will 1 cu. ft. of ice float?
How many cubic feet of ice are required to float 96 Ib. ?
8. A pine beam 1 ft. square is floating in water. If its spe
cific gravity is .55, how long must it be to support a man
weighing 180 Ib. ?
9. Construct a graph to show the weight in water of masses
of cast iron weighing from 1 to 100 Ib. in air, given that the
specific gravity of cast iron is 7.2. Why should the graph be
a straight line ?
MISCELLANEOUS PROBLEMS
1. Find the weight of 50 ccin. of copper.
SOLUTION. 1 ccm. of water = 1 g.
Specific gravity of copper = 8.79.
.. Weight of 50 ccm. of copper = 50 x 8.79 g.
= 440g.
2. Find the weight of (a) 100 ccm. of mercury ; (&) 150 ccm.
of zinc ; (c) 300 ccm. of aluminum.
3. Find the volume of 300 g. of zinc.
SOLUTION. 1 g. of water has a volume of 1 ccm.
Specific gravity of zinc = 7.19.
. 7.19 g. of zinc has a volume of 1 ccm.
300 ., _
= 41.7 ccm.
4. Find the volume of (a) 50 g. of brass ; () 100 g. of cork ;
(c) 100 g. of gold ; (d) 150 g. of marble ; (e) 1 kg. of silver.
5. The dimensions of a rectangular maple block are 8.1 cm.,
5.2 cm., and 3,5 cm. If it weighs 100 g., find its specific gravity.
50 APPLIED MATHEMATICS
6. 109 com. of copper and 34 ccm. of zinc are melted to
gether to form brass. Find its specific gravity.
SOLUTION. Let s = the specific gravity of the brass.
109 + 34 = 143 ccm., volume of the brass.
143 s = weight of the brass.
109 x 8.79 = weight of the copper.
34 x 7.19 = weight of the zinc.
1436109 x 8.79 + 34 x 7.19.
Solve for s and check.
7. 58.8 g. of copper and 25.2 g. of zinc are combined to form
brass. What is its specific gravity ?
SOLUTION. Let s specific gravity of the brass.
58.8 + 25.2 = 84 g., weight of the brass.
= volume of the brass.
5
F\ Q Q
~
o. y
25 2
l
7 .19
= 6.69 = volume of the copper.
= 3.50 = volume of the zinc.
= 6.69 + 3.50.
Solve for s and check.
8. The specific gravity of a piece of brass weighing 123 g. is
8.22. How many grams of copper and of zinc are there in it ?
SOLUTION. Let c = number of grams of copper.
z = number of grams of zinc.
**
= volume of the copper.
8.79
z
719
123
8.22
c + z = 123.
c z 123
8.79 7.19 8.22
Solve and check.
= volume of the zinc.
= volume of the brass.
SPECIFIC GRAVITY 51
9. An alloy was formed of 79.7 ccm. of copper and 51.4 ccm.
of tin. Find its specific gravity.
10. 475.2 kg. of hard gun metal was made by combining
411 kg. of copper and 64.2 kg. of tin. What was its specific
gravity ?
11. 336 Ib. of copper and 63 Ib. of zinc were combined to
make brazing metal. Find its specific gravity.
Suggestion. To reduce pounds to grams multiply by 453.6. Since
this factor occurs in each term of the equation, it may be divided out.
12. Nickelaluminum consists of 20 parts of nickel and 80
parts of aluminum. Find its specific gravity.
13. What is the specific gravity of bell metal consisting of
80 per cent copper and 20 per cent tin ?
14. Find the specific gravity of Tobin bronze, which consists
of 58.22 per cent copper, 2.30 per cent tin, and 39.48 per cent
zinc.
15. 516 g. of copper, 258 g. of nickel, and 226 g. of tin are
combined to form German silver. Find its specific gravity.
16. How much copper and how much aluminum must be
taken to make 200 kg. of aluminum bronze having a specific
gravity of 7.69 ?
17. A mass of gold and quartz weighs 500 g. The specific
gravity of the mass is 6.51 and of the quartz is 2.15. How
many grains of gold are there in the mass ?
CHAPTER VI
GEOMETRICAL CONSTRUCTIONS WITH ALGEBRAIC
APPLICATIONS
NOTE. Make all drawings and constructions in a notebook.
Record all the work in full, having it arranged neatly on the page.
Make the constructions as accurately as possible.
31. Drawing straight lines. Keep the pencil sharp, and
make the lines heavy enough to be clearly seen.
Exercise 1. Draw a line 2 in. long.
,4 f i
FIG. 20
To do so most accurately, draw an indefinite line AB. Then put
your compasses on the scale of the ruler so that the points are 2 in.
apart. With A as a center strike an arc at C. A C is the required line.
Exercise 2. Using this method, draw lines as follows :
(a) If in. ; (b) 1 dm. ; (?) 1 cm. ; (d) 83 mm. ; (e) 3.5 cm. ;
(/) 136mm.
32. Drawing to scale. Choose a scale that will give a good
sized figure, and below every figure record the scale used.
Exercise 3. The distance between two towns A and B is 30
mi. How could a line 6 cm. long represent that distance ?
Draw such a line and explain the relation that exists between
the distance and the line.
Exercise 4. Draw a line 3 in. long and let it represent a dis
tance of 36 mi. What distance is represented by 1 in. ? by
2 in. ? by l in. ? by 2 in. ? In this exercise the distance is
said to be represented on a scale of 1 in. to 12 mi.
52
GEOMETRICAL CONSTRUCTIONS 53
Exercise 5. With a scale of 1 in. to 16 ft. (1 in. = 16 ft.)
draw lines to represent the distances (a) 8 ft. ; (b) 12 ft. ;
(c) 24 ft. ; (d) 36 ft. ; (e) 18 ft.
33. Measuring straight lines. With an unmarked ruler or
with the edge of your book draw a line AB. To locate the ends
of the line as accurately as possible, make small marks in the
paper at A and B with the point of the compasses. Care should
be taken that the marks do not penetrate to the surface below.
Place one point of the compasses at A and let the other fall at
B. With this opening of the compasses place the points against
the scale of a ruler, one point on the division marked 1 cm.,
and count the number of centimeters and tenths of a centi
meter between the points of the compasses. On the line AB
write its length as you have found it. (The end divisions
of a ruler are not usually so accurate as the middle divisions ;
hence in making a measurement it is best not to start at the
zero of the scale.)
Exercise 6. Make two crosses in your notebook and call the
points of intersection M and N. Using the compasses, measure
MN in inches and centimeters and record the result.
Exercise 7. Draw an indefinite line AX and mark off on it
,4 5 = 2.8 cm., BC = 1.7 cm., and CD = 3.4 cm. Then with
your compasses measure AD. Kecord the length and compare
it with the sum of the numbers.
Exercise 8. (a) Measure the lines Afi, CD, and EF. Record
the measurements and add them.
A BC f> f f
FIG. 21
(ft) Draw an indefinite line AX and mark off on it AB, CD,
and EF, the point C falling on B and the point E on D.
Measure AF and record the result. Compare with that ob
tained in (a).
54 APPLIED MATHEMATICS
34. Angles. An angle is formed by two lines that meet.
Thus the lines EC and BA meet at the vertex B, forming the
angle ABC, JB, or ra. When three letters
are used to denote an angle the letter
at the vertex is read between the other
two. The single small letter should be FIG. 22
used to denote an angle when convenient
The size of an angle depends on the amount of opening be
tween the lines.
A right angle is an angle
of 90.
An acute angle is less than
90.
An obtuse angle is greater than 90 and less than
Thus a is an acute angle and b is an obtuse angle.
35. The protractor. To measure an angle place the pro
tractor so that the center of the graduated circle is ,at the ver
tex of the angle and its straight side lies along one arm of
the angle. Note the graduation under which the other arm of
the angle passes.
Exercise 9. Take a piece of paper and fold it twice so that the
creases will form four right angles at a point. Test one of the
angles with the protractor.
Exercise 10. About a point construct angles of 42, 85, and
53. What is the test of accuracy of construction ?
Exercise 11. At each end of a line AB, 7 cm. long, coiistrtict
an angle of 60 so that AB is one arm of each angle and the
other arms intersect at C. Measure angle A CB, and write the
number of degrees in each angle. Measure AC and BC. What
is the test of accuracy of construction ? Bisect angle A CB by
the line CD, D being on A B. How much longer is A C than AD?
Exercise 12. Draw a large triangle. Measure each angle and
write the results in the angles. What ought to b6 the sum ?
GEOMETRICAL CONSTRUCTIONS
55
Exercise 13. Make an angle A = 37. On the horizontal arm
take J.C = 6cm. and on the other arm take A B = 7.6 cm.
Draw BC. Guess the number of degrees in angle ACB. Meas
ure it.
Exercise 14. To find the distance across a lake from A to ,
a surveyor selected a point C from which he could see both A
and B. He measured the angle A CB, 72, with a transit and
found the distances CA and CB to be 450 ft. and 400 ft. re
spectively. From these measurements draw the figure to scale ;
measure A Bund determine what distance it represents.
Exercise 15. To find the height of a building AB across a
river DB measurements were made as follows : angle A CB =
16, angle A DB = 37, A
and CD = 100 ft. Draw
to scale, and find the
height of the building
and the width of the
river.
Exercise 16. A man wishing to find the distance between two
buoys, .4 and , measured a base line CD 1500 ft. in length
along the shore. At its extremities, C and D, he measured the
following angles : angle DCB = 36 15', angle BCA = 50 45',
angle CD A = 43 30', and angle ADB = 72. Draw to scale,
and find the distance between the buoys.
36. From a point in a line to draw a line at right angles
(perpendicular) to it.
CONSTRUCTION. Let C be the point in AB from which the line is
FIG. 24
to be drawn. Place one point of the com
passes at Cand mark off on AB the equal
distances CD and CE. With D and E
as centers and a convenient radius de
scribe arcs intersecting at F. Draw CF.
FCB is a right angle, and CF is said to
be perpendicular to AB.
F
/K
FIG. 25
TB
56
APPLIED MATHEMATICS
Example. To construct a right triangle whose legs are 6 cm.
and 8 cm. respectively.
CONSTRUCTION. Draw an indefinite
line A X and mark off A C = 8 cm. At
the point C construct the perpendicular
CY and take CD = 6 cm. Draw AB,
and ABC is the required triangle.
Measure c = 9.05 cm.
Check your construction by the
formula
where a and 6 are the legs of a right triangle and c is the
hypotenuse.
a a + ft 8 = 6 a + 8 2
= 36 + 64.
c 2 = 100.
c 2 = 9.95 2 = 99.0.
9.95
896
89
5
99.0
Exercise 17. Construct to scale if necessary and check as in
the preceding exercise, given a and b. (a) 3.5 cm. and 6.8 cm. ;
(V) 4.3 cm. and 9.6 cm. ; (c) 84 mm. and 64 mm. ; (rf) 42 in. and
18 in. ; (e) 28 ft. and 16 ft. ; (/) 120 mi. and 200 ml
Exercise 18. Construct a square
whose side is 4 cm. & c
CONSTRUCTION. Make AB 4 cm.
At B draw BX perpendicular to AB.
Cut off BC = 4 cm. With A and C
as centers and a radius of 4 cm. draw
arcs intersecting at D. Draw A D and
CD. A BCD is the required square.
Measure the diagonal and record the
result on the figure. Check by apply
ing the formula of the right triangle.
/\
B
FIG. 27
Exercise 19. Construct to scale squares whose sides are
(a) 12 in. ; (&) 1.8 m. ; (c) 540 mm. Check by formula.
GEOMETRICAL CONSTRUCTIONS 57
Exercise 20. Construct to scale and check, rectangles whose
sides are (a) 78 and 48 cm. ; (ft) 32 and 54 in. ; (e) 482 and
615 ft.
37. To construct a perpendicular to a line from a point
outside the line.
Let AB be the line and C the point.
With C as a center describe an arc
cutting AB at D and E. With D and . v
E as centers and a convenient radius \/
describe arcs intersecting at F. Draw
CF, the required perpendicular. FIG. 28
Exercise 21. Construct right triangles whose legs are (a) 6 and
12 cm. ; (K) 5 and 9 cm. Draw perpendiculars from the vertex
of the right angle to the hypotenuse. Measure and check.
Exercise 22. Draw a large triangle and construct a perpen
dicular from the vertex to the base. Measure the sides of the
two right triangles formed and check by the formula.
38. To construct a triangle whose sides are given.
Exercise 23. Construct a triangle whose sides are 7, 8, and
10 cm. respectively.
CONSTRUCTION. Draw a line A B 10 cm. long. With A as a center
and a radius of 7 cm. describe an arc. With B as a center and a
radius of 8 cm. describe an arc cutting the first arc at C. Draw A C
and BC, and ABC is the required triangle.
Exercise 24. From C in the figure of Exercise 23 draw a per
pendicular to AB. Measure the sides of the right triangles and
check by the formula.
Exercise 25. Construct a triangle whose sides are 7.5, 8.5, and
11 cm. respectively. Draw a perpendicular from the vertex to
the base and find the area of the triangle. Check by drawing a
perpendicular to another side and use its length to find the area.
The perpendicular from the vertex to the base is called the
altitude of the triangle.
58
APPLIED MATHEMATICS
39. To bisect a given line.
Exercise 26. Bisect a given line AB.
CONSTRUCTION. With A and B as cen
ters and a convenient radius describe arcs
intersecting at C and D. Draw CD, inter
secting AB at E. Then AE = EB. Check
by measuring.
Exercise 27. Draw an indefinite line
'\
FIG. 29
AB and divide it into four equal parts, using the method of
arcs. Check.
Exercise 28. Construct an equilat
eral triangle ABC whose sides are
each 9 cm. Divide the base into four
equal parts. Draw CD and CF and
measure their lengths. Measure the
angle ADC. Applying the formula
of the right triangle, compute CD
and CF.
40. To bisect an angle.
Exercise 29. Make an angle BA C and bisect it.
CONSTRUCTION. With A as a center and with a rather large radius
mark two points D and E on AC and
AB respectively. With D and E as cen
ters and the same radius describe arcs
intersecting at F. Draw AF, and angle
BAF= angle FAC. Check with the
protractor.
Exercise 30. Draw an obtuse angle
and bisect it. Check.
FIG. 31
Exercise 31. Construct a triangle ABC with ^45 == 7.6 cm.,
AC = 6.5 cm., and angle A = 45. Construct the altitude CD
and measure its length. Check by computing the length of CD,
using the formula of the right triangle.
GEOMETRICAL CONSTRUCTIONS 59
41. Parallel lines. Lines that lie in the same plane and do
not meet however far produced are called parallel lines.
Exercise 32. Construct a rectangle whose dimensions are 4.35
and 7.85 cm. respectively. Find the area to three significant
figures. The opposite sides of a rectangle are parallel. Write
in your notebook the sides that are parallel.
42. Parallelograms. If the opposite sides of a foursided
figure are parallel, the figure is called
a parallelogram. A BCD is a paral
lelogram.
Exercise 33. Construct a paral /
lelogram with AB = 8 cm., AD = FIG. 32
5 cm., and angle A = 65. The point
C can be obtained with arcs, as in Exercise 18. Name the par
allel sides. Measure all the angles.
Exercise 34. Construct a parallelogram with AB = 9.45 cm.,
BC = 4.15 cm., and angle B = 115. From D construct DE
perpendicular to AB, E being on AB. The line DE is the alti
tude of the parallelogram. Measure DE and find the area of
the parallelogram.
43. To draw a line parallel to a given line.
Exercise 35. Construct a triangle with AB = 8 cm., BC = 9 cm.,
and A C = 6 cm. Take CD = 4 cm.
Through D draw DE parallel to
AB. (Construct the parallelogram
ADFG.) Measure CE, or y, and
record its length. The equation
4 ?/
 = * will give the length of
Z \) y
CE. Solve the equation and com
pare with the measured length.
Exercise 36. Construct a triangle ABC whose sides are:
AB = 1 cm., BC = 9 cm., and CA = 11 cm. On BC take
60 APPLIED MATHEMATICS
BD = 3 cm., and through D draw a parallel to AB. Measure
the lengths of the two parts of A C and check by an equation
like that in Exercise 35.
44. To construct an angle equal to a given angle.
Exercise 37. At the point D on DE to construct an angle
equal to angle A.
CONSTRUCTION. With A as a center and a rather large radius de
scribe the arc EC cutting A X at B and AY at <?. With D as a cen
ter and the same radius describe an arc FG cutting DE at F. Takt
off with the compasses the distance JSC; then with F as a center
and BC as a radius. describe an arc cutting FG at H. Draw /)//.
Angle D is the required angle equal to A. Check with the protractor.
Exercise 38. Make angles of (a) 40, (ft) 58, (c) 140, and
construct angles equal to them.
Exercise 39. Construct a triangle ABC, making AB = 8.4 cm.,
BC = 6.8 cm., and AC = 7.2 cm. Draw a line DE = 4.2 cm.
At D make an angle EDF equal to angle BA C, and at J make
an angle DEF equal to angle ABC. Produce the two lines till
they meet at F. Measure the sides and angles of the triangle
DEF and compare them with the corresponding parts of the
triangle ABC.
Triangles which have their corresponding angles equal and
their corresponding sides proportional are called similar tri
angles.
Exercise 40. The angle of elevation of a church steeple at a
point 300 ft. from its base was found to be 16. Construct a
GEOMETRICAL CONSTRUCTIONS 61
similar triangle, that is, draw to scale and find the height of
the steeple.
Exercise 41. At a distance of 500 ft. the angle of elevation of
the top of one of the " big trees " of California is 31. How tall
is the tree ?
Exercise 42. Make some practical problems and solve them.
PROBLEMS
Record all measurements and give the work in full in your
notebooks.
1. The two legs of a right triangle are 15 and 36 ft. respec
tively. Construct the triangle to scale, stating scale used.
Measure the hypotenuse. Check by applying the formula of
the right triangle.
2. Construct a rectangle 4 cm. by 7 cm. Measure the diag
onal. Check.
3. A right angle may be constructed
as shown in Fig. 35. ABC is an equi
lateral triangle. CD = BC. AD is drawn
audBAD is a right angle. Construct a
right angle DAB. On AB take A E = 8.4
cm., and on A D take A F = 3.5 cm. Meas
ure EF. Check. 
* & /*
4. The hypotenuse of a right triangle ^ IG 35
is 19.4 ft. and one leg is 14.2 ft. Com
pute the length of the other leg. Check by constructing the
triangle to scale and measuring the required leg.
5. The base of. a right triangle is a*, the altitude is x + 1,
and the hypotenuse is x + 2. Find x by applying the formula
of the right triangle. Check by constructing a right triangle
with the legs x and x + 1. Measure the hypotenuse and com
pare with the value of x + 2.
6. The following sets of expressions represent the sides of
a right triangle. Solve and check as in Problem 5.
62 APPLIED MATHEMATICS
LEGS HYPOTENUSH
(a) x and x + 3 x + 6
(6) x and a: 4 7 a? + 8
(c) and z 2 a: + 2
(6/) a: and x + 4 a; f 8,
(e) and a; 7 x + 1
(/) a: and 2 a;  4 2 .r  2
(</) a; and a; + 1 2 a:  11
(ft) a: and x f 5 2 x 5
7. The altitude of a rectangle is 1 ft. less than the base, and
the area is 20 sq. ft. Find the dimensions. Cheek by drawing
on squared paper and counting the squares.
8. The following sets of expressions represent the sides
and the area of a rectangle. Find the dimensions and check
as in Problem 7.
SIDES AREA
(a) x and x 10 24
(b) x and a; 7 30
(c) x and x + 12 85
(d) x and x f 9 90
(e) x and 2 x + 5 18
(/) x and 2 x + 1 36
(</) a: and 3 x 7 40
(h) x and 4 x 10 24
9. Construct a right triangle 4J5C, denoting the base by x
and the altitude by y. Complete the rectangle xy. How is the
area of the rectangle found ? What algebraic expression repre
sents it in this case ? What part of the rectangle is the triangle
ABC? What algebraic expression represents the area of the
triangle ? What reason can you give for the correctness of the
expression for the area of the triangle ?
10. The legs of a right triangle are x and x + 6. Its area
is 20. Find the sides of the triangle. To check, draw on
squared paper a right triangle whose legs are x and x + 6.
Find the area by counting the large squares inside the triangle.
GEOMETRICAL CONSTRUCTIONS 63
When a part of a square looks less than a half, it is not
counted; but if it looks greater than a half, it is counted as a
whole square.
11. The following sets of expressions represent the legs and
area of a right triangle. Find the length of the legs in each
case, and check on squared paper as in Problem 10.
LEGS AREA
(a) x and x  11 30
(6) x and x 12 14
(c) x and x + 10 28
(rf) x and x  15 27
(e) x and 2 x 7 15
(/) x and 5 x 9 40
(g) x and 3 x  1 35
(A) x and 4 x 9 45
12. Construct a parallelogram A BCD. Bisect the angles A
and J5, and let the bisectors meet at F. Measure the angle
APR. Measure AB, BF, and FA. Apply the formula of the
right triangle. Make the test in several parallelograms and
state what seems to be true of the bisectors of two consecutive
angles of a parallelogram.
13. Construct a triangle ABC with CB = 8 cm., AB = 10.5 cm.,
and A C = 5.5 cm. On CA take CD = 2 cm., and from D draw
a line parallel to CB intersecting AB at E. From the formula
 = = find AE. Check by measuring AE.
14. In the figure of Piobleml31et^Z)=a:,JDC=3,^JS ===
and EB = 5. Use the formula to find x. Find the sides A C
and AB. Check by construction, taking the base any conven
ient length.
15. The following sets of values are the segments of the
sides of a triangle formed by a line parallel to the base. Find
the length of each segment and check by constructing the tri
angle and the parallel as in Problem 14.
64 APPLIED MATHEMATICS
AD DC AE EB
(a) x 3 2* 2 + *
(6) x 2 * + 5 x  1
(c) * 4 x + l x + 7
(d) x 5 4  * 3 + a:
(6)2: 3 a; + 2 z + 5
(/) a: x + 2 x + 4 2x5
(g) x xf 3 2ar 1 a
x rr + 4 3*2 a: + 5
16. The legs of a right triangle are # and y, and their sum
is 15. If the area of the triangle is 27, find x and y. To check
the result, construct on squared paper a right triangle whose
legs are x and y. Count the large squares and compare with
the given area.
17. The sum of the legs and the area of a right triangle are
given by the following sets of numbers. Find x and y, and
check.
LEGS SUM OF THE LEGS AREA
(a) x and y 16 14
(6) x and y 25 42
(c) x and y 15 28
(d) x and y 19 36
18. The difference of the legs of a right triangle and the
area are given by the following sets of numbers. Find x and y,
and check.
LEGS DIFFERENCE OF THE LEGS AREA
(a) x and y 12 32
(6) x and y 10 48
(c) x and y 8 64
(d) x and y 9 35
(e) x and y 5 102
CHAPTER VII
THE USE OF SQUARED PAPER
I. GRAPHICAL REPRESENTATION OF TABLES OF VALUES
45. The results of experiments and observations, statistical
tables, and tabulated numerical data of all kinds can be repre
sented by lines and curves. The graph shows at a glance rela
tions which are not so evident in a table of values ; and it also
enables one to find readily values which lie between those
given in the table.
Exercise. Construct a graph to represent this record of tem
perature, given in The Chicago Daily News.
SP.M 78
4 P.M 77
5 P.M 76
6 P.M 75
7 P.M 76
SP.M 75
9 P.M 73
10 P.M 74
11 P.M 75
12 midnight 73
1 A.M 73
2A.M 73
3 A.M 73
4 A.M 73
5 A.M 72
6 A.M 72
7 A.M 72
8 A.M 71
9 A.M 71
10 A.M 76
11 A.M 74
12 noon 75
1 P.M 76
We have here two quantities, hours and degrees, so related
that to a change in one there is a corresponding change in the
other.
65
66
APPLIED MATHEMATICS
The sheets of squared paper we use have seventeen large
squares each way ; the side of a large square is a centimeter,
and of a small square a millimeter.
The units for representing an hour and a degree should be
chosen so that the picture may be of good size arid still allow
the whole table to be represented. Let the horizontal lines
represent time and the vertical lines represent temperature.
77
76
75
D74
U73
\
Y
II 1AM ,3
Time
FIG. 36
ii
IPM
The horizontal and vertical lines from which we count degrees
and hours are called axes. We will always mark them OX
and OY respectively, and call them the #axis and the yaxis.
The point is called the origin.
Since the number of degrees is always greater than 70, we
may call the aaxis 70 to save space. At 3 P.M. the tempera
ture is 78 ; hence on the 3 o'clock line we put a point at 78, and
so on for the other hours, as shown in Fig. 36. A smooth curve
is then drawn through the points, and we have a curve which
shows at a glance the change in temperature during the day.
THE USE OF SQUARED PAPER 67
The curve does not, of course, show the exact reading of the
thermometer between the hours. However, it shows when the
temperature was falling and when rising, whether the change
was rapid or gradual, and in general gives a fairly correct
representation of the temperature for the day.
46. Hints on the use of squared paper. All graphical work
should be done in a book of squared paper where it can be re
ferred to from time to time. Much can be learned by looking
back over the curves and noting the relations between the
various problems and curves. Frequently it will be found that
a curve of the same shape is constructed in solving several
different problems.
Each graphical solution ought to be complete in itself. The
table of values or other data should be written on the sheet
with the curve, or on the blank page at the left of the graph in
the notebook. The axes should be lettered OX and OF, and
the units written on them. It is not necessary that the units
should be the same for both axes, but they should be chosen
so that the whole range of values may be plotted in a figure
which extends well over the sheet of squared paper.
When a curve is constructed for the sole purpose of reading
off intermediate values, a large square should represent 1, 5,
10, 20, 50, 100 , numbers which give easy readings.
If the curve is made simply to show general changes or to
solve a problem, the unit may be chosen to locate the points
with the least work.
If two or more curves are constructed on the same axes, they
should be numbered to correspond with the tables or data, and
they can be more readily distinguished if a different kind of
line is used for each curve, for example, thick and thin con
tinuous lines, dotted lines, and so on. When convenient the
various curves may be drawn in different colors ; in this case
the table of values should be written in the same color that is
used for the curve which represents it.
68
APPLIED MATHEMATICS
EXERCISES
1. Construct a curve from the record of temperature given
above with the same time unit, but let 5 mm. = 1. From which
curve can the changes be read most easily ?
2. Construct several temperature curves from the weather
reports in the daily papers.
3. On the same axes construct temperature curves for a day
in summer and a day in winter.
4. Construct on the same axes temperature curves for several
cities, e.g. Boston, Chicago, and San Francisco.
5. Place a thermometer outside the classroom window and
take readings at the beginning and end of the recitation hour
for two or three weeks. Construct the curve.
6. Construct several curves from tables found in newspapers,
magazines, The Daily News Almanac, The World Almanac,
Kent's " Mechanical Engineers' PocketBook," city, state, and
government reports, price lists, and so on. Try to find reasons
for any marked peculiarities in the curves.
7. Construct curves to show the number of hours of daylight
per day for the year. (Let a heavy horizontal line near the
center represent noon. From an almanac make a table of the
time of sunrise and sunset on the first day of each month;
locate the points and draw the two curves.) On the same
sheet of squared paper make curves for different latitudes,
e.g. Chicago and Dawson, Alaska, and compare the amounts
of daylight.
8. A price list of the Western Electric Company gives the
following price of bells. Construct the curve.
Size of gong in inches .
Price in dollars . . .
2*
1.68
3
1.74
Si
1.85
4
1.96
6
2.84
6
3.20
7
4.66
8
5.00
10
8.00
12
10.00
What is the probable price of a 9in. gong ? of anllin. jong?
THE USE OF SQUARED PAPER
69
9. The water in a glass is at a temperature of 60 F. Heat
is applied to the glass, and the temperature, T, at the end of
t minutes is as follows :
Minutes . .
Degrees . .
60
5
68
10
76
15
83.2
20
89.6
25
95.5
30
101
35
106
40
110
Construct the curve. What temperature would you expect
at the end of 7 min. ? of 32 min. ?
10. A boat is rowed straight across a river and soundings
are taken at various distances from the bank. From the table
draw a section of the river bed.
Distance from bank in feet
Depth in feet ....
5
1
10
3
15
5
20
8
25
15
30
14
35
16
40
18
45
18
50
12
52
8
54
5
56
4
58
3
11. From the top of a cliff 1500 ft. high a bullet was shot
horizontally with a velocity of 100 ft. per second. Construct
a curve to show its path, if at the end of each second it has
fallen the following number of feet :
Number of seconds .
Distance fallen
1
16
2
64
3
144
4
256
5
400
6
576
7
784
8
1024
9
1296
10
1600
Take the xaxis at the top of the sheet. On the xaxis let
1 cm. = 1 sec., or 100 ft. ; on the ?/axis, 1 cm. = 100 ft. In how
many seconds will the bullet reach the ground if it is level ?
How far from the foot of the cliff will it fall ? In how many
seconds will it fall 600 ft. ? How far will it fall in 5 sec. ?
II. THE GRAPH AS A " HEADY RECKONER"
47. Straightline graphs* In the following exercises the
graph is a straight line. Choose convenient units and let the
graph extend well over the sheet of squared paper.
70
APPLIED MATHEMATICS
EXERCISES
1. Construct a graph to change inches into centimeters and
centimeters into inches, given (
1 in. = 2.5 cm.
CONSTRUCTION.
in. = cm.
4 in. = 10 cm.
*
Locate these two points, __
and P, and draw a straight line B
through them. Test a few
points on the graph to see if the
results are approximately cor
rect. Thus at M 2 in. = 5 cm.
Hm
A
Inches
FIG. 37
2. Construct a graph to
change pints to liters, given that 11. = 2.1 pt.
3 Construct a graph to find the circumferences of circles of
diameter from to 18 in., given that the circumference equals
TT times the diameter.
4. Construct a graph to find the velocity of a falling body,
given that the velocity at any second equals 32 times the
number of seconds.
5. Construct a graph to change miles per hour to feet per
second, given that 30 mi. per hour equals 44 ft. per second.
6. Construct a graph to change cents to marks, given that
1 mark equals 24 cents.
7. Construct a graph to change cubic inches to gallons, given
that 1 gal. equals 231 cu. in.
8. Construct on the same axes graphs to find the simple in
terest of $100 at 4 per cent, 5 per cent, and 6 per cent.
9. Construct a graph to find the number of amperes in a
circuit of 10 ohms resistance as the voltage increases from 10
to 100 volts, given that the number of volts divided by the
number of ohms equals the number of amperes.
THE USE OF SQUARED PAPER 71
10. The formula for the number of revolutions per minute
o o
of cutting tools in lathes is n = ' ? where n = revolutions
a
per minute, s = the speed in feet per minute, and d = the
diameter of the rotating tool in inches. Construct a graph for
a tool 6 in. in diameter, with speeds from 5 to 50 ft. per
minute.
11. The resistance r of a train in pounds per ton, due to
o
speed, is given by the formula r = 3 +  Construct a graph
for speeds from 5 to 60 mi. per hour.
12. The pressure of the atmosphere in pounds per square
inch for readings of the barometer is given by the formula
p = .491 />, where p = the pressure in pounds per square inch,
and b = the reading of the barometer. Construct a graph for
barometer readings from 28 in. to 31 in. Use the given formula
to find the pressure for the readings 28.75 in., 29.50 in., and
30 in., and compare with the pressures read from the graph.
13. Write the equations which express the relation between
the two quantities in each of the preceding exercises.
Thus in Exorcise 1 to change inches to centimeters we mul
tiply the number of indies by 2.5. Therefore, representing
centimeters by c and inches by /, c 2.5 i is the equation
which expresses the relation between centimeters and inches.
48. Equations expressing the relation between two quan
tities. In the first list of exercises the curves were constructed
from tables of values determined by observation or experiment.
In many cases there is no known relation between the sets of
corresponding numbers. Thus in the table of temperatures the
thermometer was read at intervals of one hour, and we do not
know any law which will tell what the reading will be. But in
the second list there is in each case a known law or relation
which may be written in the form of an equation. Thus
1 in. = 2.5 cm. ; hence the number of centimeters equals the
72 APPLIED MATHEMATICS
number of inches multiplied by 2.5, or c = 2.5 i. From this
equation we can make a table of values, and from the table
locate points and construct the graph. If we know that the
graph is a straight line, it is necessary to determine only two
points and draw a straight line through them.
All the equations in this exercise are of the first degree
and all the graphs are straight lines. We may assume that
when the relation between two quantities is expressed by an
equation of the first degree the graph is a straight line
(see sect. 52 for proof).
49. Curves. When the equation is not of the first degree
the graph will be a curve which must be constructed by locat
ing a number of points sufficient for the problem in hand.
EXERCISES
1. Construct a curve to find the area of squares whose sides
are from to 10 in. (Let 1 cm. horizontally = 1 in., and 1 cm.
vertically = 10 sq. in.) If a = the area and s = a side of the
square, what is the equation that connects the area and side ?
Find from the equation and from the graph the area of a
square whose side is (a) 3.5 in. ; (ft) 7.5 in. ; (c) 9.25 in.
2. Construct a graph to find the surface of cubes whose
edges are from to 10 in.
3. Construct a graph to find the area of circles of radii from
to 10 in., given area = Trr*.
4. Construct a graph to find the volume of cubes whose
edges are from to 10 in. What is the equation connecting v
and e?
5. Construct a graph to find the space passed over by a
falling body, given s = 16 2 , t = number of seconds.
6. The power of doing work possessed by a body in motion
1/911
(kinetic energy) is given by K = 5 > where w = the weight
THE USE OF SQUARED PAPER 73
in pounds, v = the velocity of the body in feet per second, and
g = 32. Construct a graph to show the kinetic energy of a
24lb. shot as its velocity changes from 1600 to 600 ft. per
second.
7. The volume of a gas diminishes in the same ratio as the
pressure on it is increased, or pv = a constant. Given pv = 120,
make a table of values and construct a curve to show the
volume as the pressure increases from 1 Ib. to 60 Ib. per
square inch.
8. The centrifugal force of the whole rim of a flywheel
wv*
equals ? where w = weight of the rim in pounds, r = mean
radius of the rim in feet, v = velocity of the rim in feet per
second, and g = 32.2. Given w = 3220 Ib. and r = 5 ft., con
struct a curve for velocities from 10 to 100 ft. per second.
9. The safe load in tons, uniformly distributed, on horizontal
yellowpine beams is w = ? where b = breadth of beam
lo I
in inches, d = depth of beam in inches, and I = distance between
the supports in inches. Construct a curve to show the safe load
on yellowpine beams 4 in. in breadth, 12 ft. between supports,
and depths from 8 to 18 in.
10. The resistance of a copper wire at 68 F. to the passage
of an electric current is given by R = ~ > where I = length
d
of wire in feet and d = diameter of wire in mils (.001 in.).
Construct a curve for the resistance of 1000 ft. of copper wire
of diameter from 5 to 100 mils.
11. The volume of air transmitted in cubic feet per minute
in pipes of various diameters is given by Q = .327 vd?, where
v = velocity of flow in feet per second and d = diameter of
the pipe in inches. Construct a curve to show the volume of
air transmitted in pipes of diameters from 2 to 10 in. with a
flow of 12 ft. per second. Without further computation con
struct a curve for a velocity of 24 ft. per second.
74
APPLIED MATHEMATICS
III. THE SOLUTION OF PROBLEMS
50. In a graphical solution do not make a table of values
unless it is necessary.
PROBLEMS
1. A travels 6 mi. per hour and B 10 mi. per hour. If B
starts 2 hr. after A, when and where will they meet?
SOLUTION. Choose units and axes as in Fig. 38. A travels 24 mi. in
4 hr. Locate this point M, and draw OA through the points and M.
B starts 2 hr. after A ; hence
the graph of his journey begins
at C. He travels 20 mi. in 2 hr.
Locate this point N and draw
CB through the points C and
N. P 9 the intersection of OA
and CB, shows when and where
they meet, 5 hr. after A starts
and 30 rni. from the starting
point.
The figure also shows how
far they are apart at any time.
Thus at the end of 3 hr. they
are 8 mi. apart ; this number of miles is given by the part of the 3hr,
line included between the lines OA and CB.
Solve this problem and some of the others in this list algebraically
and compare the results with the graphical solution.
2. A travels 7 mi. per hour and B 5 mi. per hour. They
start at the same time and travel east, A from a town M and
B from a town N 15 mi. east of M. When and where will
they meet?
3. Two trains start at the same time from Chicago and
St. Louis respectively, 286 mi. apart ; the one from Chicago
travels 50 mi. per hour and the other 40 mi. per hour. When
and where will they meet ?
On the xaxis let a large square = 20 mi. Let St. Louis be at
the lower lefthand corner, and Chicago 14.3 squares to the right.
X
A
B
X
30
mile
p
>^
^
i^
^?
S>
^
X
M
fl
^x
^
/
^
8
/
'
j:
10
XI
Miles
FIG. 38
THE USE OF SQUARED PAPER
75
)5ueors
Draw the line to represent the journey of the St. Louis train to the
right, and the Chicago train to the left.
4. A cyclist starts at the rate of 300 yd. per minute, and
5 min. later another cyclist sets off after him at the rate of
500 yd. per minute. When and where will they meet ? When
are they 700 yd. apart ?
5. A, traveling 20 mi. per day, has 80 mi. start of B, who
travels 25 mi. per day. When will B overtake A ?
6. A invests $500 at 6 per cent and B invests $1000 at
5 per cent. In how many years will A's interest differ from
B's by $300 ?
SOLUTION. Choose axes and
units as in Fig. 39. Interest of
$500 for 10 yr. is $300 ; locate
point P, and draw OA through
P to represent A's interest. In
a similar manner draw OB to
represent IVs interest. Three
squares vertically represent
$300. Mark off three squares
on the edge of a piece of paper
and with it find on what verti
cal line the distance between
OA and OB is three squares;
result, 15 yr.
7. In how many years will the interest on $1500 at 5 per
cent be $240 greater than the interest on $1000 at 6 per cent ?
When will it be $120 greater ?
8. A invests $1000 at 5 per cent and B invests $5000 at
4 per cent. In how many years will the amount of A's invest
ment equal the interest of B's ?
9. A invested $2000 at 4^ per cent, and two years later B
invested $2400 at 5 per cent. How many years elapsed before
they received the same amount of interest ? When was the
difference of the interest $120 ?
I
B
700
A
600
/
500
v
/
4OO
/
^
^300
/
/
/
r
1
3poo
/
/
y
P
too
/
/
/
S
C
i
\ *
\ <
5 I
Yec
& \
jrs
o \
Z. I
4 \e
FIG. 39
76
APPLIED MATHEMATICS
FIG. 40
10. A man walks a certain distance and rides back in 8 hr. ;
he could walk both ways in 10 hr. How long would it take
him to ride both ways ?
SOLUTION. Let OA =
10 hr. (Fig. 40). Mis the
midpoint of OA. MP is
any convenient length.
OP A represents the jour
ney when the man walks
both ways, and OPB
when he walks and rides
back. It is two squares
from A to B ; take C two
squares from and join CP. Then CPB represents the journey when
he rides both ways. CB = 6 hr., the time it takes him to ride both ways.
11. A man can walk to Lincoln Park in 3 hr. If he walks
to the park and rides back in 5% hr., how long would it take
him to ride both ways ?
12. A man walks to town at the rate of 4 mi. per hour and
rides back at the rate of
10 mi. per hour after re
maining in town 1 hr.
He was absent 8 hr.
How far did he walk?
y
v;
\
f
* N
/
V>
\
/
/
\
/
'
po
c
A
D
X
rest
\
10
J
jfr
/
\
D
,S
<
'
\
X
\
p
SOLUTION. Choose axes
and units as in Fig. 41.
OP = 8 hr. OA is the
graph of the walk and PB
is the graph of the ride. If
he had not remained in
town, the distance of the
point of intersection of the
two lines from the araxis
would give the distance he walked. Since he remained in town 1 hr.
we find where the horizontal distance from OA to P.B equals 1 hr.
This is CD on the 20mL line ; hence the man walks 20 mi.
2 3
a e
9
TIG. 41
THE USE OF SQUARED PAPER
77
13. A man rides to a city at the rate of 10 mi. per hour,
remains in the city 2 hr., and returns in an automobile at the
rate of 15 mi. per hour. If he was absent 10 hr., how far was
it to the city ?
14. A boy starts out on his bicycle at the rate of 6 mi. per
hour. His wheel breaks down and he walks home at the rate
of 2J mi. per hour. How far did he ride if he reached home
8 J hr. after starting ?
Construct the graphs for the walk and ride, as in Problem 10.
The intersection of the lines gives the distance.
15. A man rows at the rate of 6 mi. per hour to a town down
a river and 2 mi. per hour returning. How many miles distant
was the town if he was absent 12 hr. and remained in town 6 hr. ?
16. If A and B can build
a sidewalk in 6 and 4 da.
respectively, in what time
can they build it working
together ?
SOLUTION. Take OX in
Fig. 42 any convenient
length, and let OA = 6 da.
and XB = 4 da. Draw XA
and OB] P is the point of
intersection. PM = 2.4 da., JP IG 42
the required time.
17. A can do some work in 30 da. and B can do it in 20 da.
How long will it take them working together ?
18. If A can do some work in 12 hr. that he and B can do
together in 4 hr., in what time can B do it ?
As in Problem 14, draw XA for A's work. On XA take P 4 units
above OX ; draw OP and produce it to meet X B at B. XB = 6 da.,
the required time.
19. Two men can dig a ditch in 8 da. If one alone can dig
it in 40 da., how long will it take the other man to dig it ?
t
6
5
A
3
Z
o
A
X
x
^v
N
X
x
^
"""
x
_^
*
^
^
^
1
X
x
^s
^
X,
X
rt
78
APPLIED MATHEMATICS
14 i
I.
10 20
00 40 50
Pounds
eo TO 60 go loo x
20. A man bought 100 Ib. of brass for $13.60, paying for the
copper in it 16 cents per pound and for the zinc 10 cents per
pound. How many pounds of each metal are there in the brass ?
y
SOLUTION. In work
ing problems take the
units as large as possi
ble ; they are taken
small here to save space.
In Fig. 43 OS = $13.60.
OC, OZ, and OB are
the graphs for the cop
per, zinc, and brass re
spectively. Draw BM
parallel to OC, intersect
ing OZ at P. Draw
PN JL OX. ON = 40 Ib., FlG ' 4S
the number of pounds of zinc ; and NT = 60 Ib., the number of
pounds of copper.
Check. 40 + 60 = 100.
40 x .10 + 60 x .16 = 13.60.
Show that the same results are obtained by drawing BM' parallel
to OZ, intersecting OC in P'. (A geometrical proof of the construc
tion may be made by advanced students.)
21. An aluminumzinc alloy weighing 300 Ib. was sold for
$60, the cost of the material. If the aluminum cost 25 cents
per pound and the zinc 10 cents per pound, how many pounds
of each metal were in the alloy ?
22. A man bought 100 A. of land for $3250. If part of it
cost him $40 an acre and part of it $15 an acre, how many acres
of each kind were there ?
23. A man starts off rowing at the rate of 6 mi. per hour,
and half an hour later a second man sets out after him at the
rate of 8 mi. per hour, (a) When is the first man overtaken ?
(ft) How far has he rowed when overtaken ? (c) How far apart
are they when the first man has rowed 1 hr. ?
THE USE OF SQUARED PAPER
79
24. The distance from Chicago to Milwaukee is 85 mi. An
automobile leaves Chicago at 1.00 P.M. at the rate of 15 mi.
per hour and another leaves Milwaukee at 1.30 P.M. at the
rate of 18 mi. per hour. When and where will they meet ?
25. A man walked to the top of a mountain at the rate of
2J mi. per hour, and down the same way at the rate of 3^ mi.
per hour. If he was out 5 hr., how far did he walk ?
26. From the same place on a circular mile track two boys,
A and B, start at the same moment to walk in the same direc
tion, A 4 mi. per hour and B 3 mi. per hour. How often and
at what times will they meet if they walk 1 hr. ?
27. If the two boys in Problem 26 walk in opposite directions
around the track, how often and at what times will they meet ?
28. A with an old automobile travels 15 mi. an hour, and
stops 5 min. at the end of each hour to make repairs. B on a
new car travels 25 mi. per hour. If B starts 3 hr. after A,
when and where will he overtake A ?
IV. THE GRAPHICAL REPRESENTATION AND SOLUTION
OF EQUATIONS
51. Equations of the first degree. We have graphed equa
tions which arose in concrete problems, and we will now apply
the same methods to abstract equations containing the two
unknowns x and y.
Exercise. Construct the graph of x + y = 5.
Transposing, y = 5 x.
By giving values, to x we have the following table :
X
y
8
3
7
2
6
1
5
4
1
3
2
2
3
1
4
5
1
6
2
7
3
8
For the first time in our graphical work we have to deal with
negative numbers. This will cause no trouble, however, for we
80
APPLIED MATHEMATICS
FIG. 44
will simply count off the positive values of x to the right of
the origin, and the negative values to the left. For positive
values of y count up from the
&axis, and for negative values
count down.
Taking heavy horizontal and
vertical lines near the center of
the page for the ,raxis and yaxis
respectively, locate the points
from the table and draw a line
through them. The axes should
always be lettered as in Fig. 44,
and the units indicated on the
axes or on the sides of the
diagram.
It is not worth while to plot many equations of the first
degree by locating points, since it will be proved in the next
paragraph that such a graph is always a straight line. Hence
in plotting equations of the first degree it is necessary to locate
only two points. These points should be some distance apart
in order that the graph may be fairly accurate.
52. Theorem. The graph of an equation of the first degree
is a straight line.
Proof. Any equation of the first degree can be reduced to the form
y = mx + b (1)
by transposing, uniting, and divid
ing by the coefficient of y. Let P be
a point on the graph of y = mx f b.
Draw PM OX. Then, for the
point P, OM = x and PM = y. In
equation (1) put x = ; then y = b,
that is, the graph of (1) cuts the
yaxis at the point (0, 6), Let OA = b.
Through P and A draw the straight
line AC. Through A draw AF parallel to OX, cutting PM at N.
.
/ /
N F
X' >" O
fl *
V
FIG. 45
THE USE OF SQUARED PAPER
81
From (1),
From the figure,
and
Therefore
m =
 b

x
Why?
Why?
PN yb

AN x
That is, for any point P on the graph of y = mx + b the ratio
PN/AN is constant, since m is some fixed number. Hence, by the
properties of similar triangles (what are they ?), any point whose x
and y satisfy equation (1) lies on the straight line AC.
EXERCISES
Plot the following equations :
1. x + y = 6. 3. x + y = 6.
2. x y = 6. 4. x f y = 6.
53. Equations of degree higher than
the first. The graph of an equation of
degree higher than the first is a curve,
which can be drawn with sufficient accu
racy by locating a number of points.
Exercise. Plot y = x 1 6 x + 5.
If we wish to take the side of a large
square = 1 on both axes, it is necessary
to begin the table of values with some
value of x that will bring the point on
the paper. If we start with x = 8, then
y = 21, and the point (8, 21) is off the
paper ; hence we begin with x = 7.
5. 2*
6. 5x
FIG. 46
X
y
7
12
6
5
5
4
3
3
4
2
3
1
5
1
12
82
APPLIED MATHEMATICS
Usually it is necessary to locate points close together to
determine the true shape of the curve at some particular point.
Thus from the given equation :
X
y
3.5
3.75
3.2
3.96
3.1
3.99
2.9
3.99
2.8
3.96
2.5
3.75
These additional points show that the curve is rounded at
(3, 4). This point is called the turning point of the curve.
54. The purpose of graphical representation. From this
curve we may learn two things : (1) the x of the points where
it intersects the xaxis, 1 and 5, are the roots of the equation
x 2 6 x + 5 = ; (2) the y of the turning point, 4, gives the
least value of the expression # 2 Qx + 5 (see Chapter VIII).
EXERCISES
Plot these equations. In the first four find the least value of
the expression and the roots of the equation when y = :
1. y = x* 4 x 5. 5. x 2 + if = 25 (circle).
2. y = x 2  6x + 9. 6. y 2 = Sx (parabola).
3. y = ^2 _ x _ 6 7. 9 a 2 + 25 y 2 = 225 (ellipse).
4. y = x 2 + x  2. 8. 4 *  9 if = 36 (hyperbola).
55. A short method of computing the table of values for
equations of degree higher than the second. This method can
be used also in checking the roots of equations.
Exercise 1. Plot y = a 8  5z a 2x + 24.
Let x = 6. x* = xx 2 = 6 x 2 .
.. * 8  5 x 2  2 x + 24 = 6 x 2  5 x 2  2 * + 24
= * 2  2 x + 24.
a: 2 = # = 6 x.
.. z 2  2 x + 24 = 6 x  2 a? + 24 = 4 x + 24.
4^ = 4x6.
.. 4 x + 24 = 24 + 24 = 48.
.*. y = 48 when x = 6.
THE USE OF SQUARED PAPER
83
The coefficients only need be written and the work can be
put in the following form :
152 + 24 [6
6 + 6 + 24
1+4 + 48
After the coefficients are written we multiply the first one
at the left by 6 and add the product to the second, obtaining 1.
This sum is multiplied by 6 and added to the third coefficient,
and so on.
If any power of x is lacking, write for the coefficient of
the missing term. Thus, if y = x 4 + 3 x 2 + 2 x + 5, write the
coefficients 1+0+3+2+5.
TABLE OF VALUES FOR y = x 3 5 x 2 2x + 24
X
y
6
48
5
14
4
TIP* n\00
CO 11
\
3
2
8
1
18
24
1
20
2
c
42
Locate axes and c
ient units, as in Fig. 4
for x 4 and x 3,
Choose
:7. Si
it is ]
points
get t
3 root
5* + 5
14.
= .r 8 
TAB
5 con
nee i/
necesf
* beto
lie ci
;s of
M r
ven
0
So
Y
sary
.reen
irve
the
I are
2.
25
/^
f\
to locate 01
x = 4 and
fairly acci
equation x
seen to be
Exercise
Exercise
ie or more
x = 3 to
irate. Tin
20
/
\
1
IS
/
\
10
/
o .
3, am
oty =
ot
13:
/ ;ti
LE OF
s
1
v
/
X
?
2. PI
J. PI
o
V
4 ,
Y
2
4
fe
VALUES
FIG. 47
X
y
4
120
3.5
44.2
3
2
18
1
12
1
2
30
3
48
4
4.6
72
84
APPLIED MATHEMATICS
Find the table of values by the short method. The choice of
units in Fig. 48 makes the curve of good form for a study of
its properties. The roots
of the equation x 4 + # 8
13z a a + 12 = are
seen to be 4, 1, 1, and
3. How can the position
of the three turning points
be found ?
56. Helpful principles
in plotting curves. For
.equations in the form y
equal an expression con
taining x, with no root
signs and no term in the
denominator containing x,
the following principles are useful in plotting the curves :
1. The number of turning points cannot be greater than the
degree of the equation less one. Thus an equation of the
fourth degree cannot have more than three turning points.
2. A line parallel to the ?/axis can cut the curve only once.
3. If the equation is of odd degree, the ends of the curve
are on the opposite sides of the xaxis.
4. If the equation is of even degree, both ends of the curve
are on the same side of the o>axis.
5. The number of times the curve cuts the oraxis cannot be
greater than the degree of the equation.
EXERCISES
Construct curves to represent the following equations :
1. y = x 8 + 2x*  x  2. 4. y = x*  4^ 2 .
2. y = x 8 + x 2  x  1. 5. y = x 4  lOx* + 8.
3. y = x* + 3x 2 6xS. 6. y = x 4  4;r 2 + 4z  4.
THE USE OF SQUARED PAPER 85
57. Solution of simultaneous equations. Equations like
x + y = 8 , a; 2 + z/ 2 = 25
. ~ and
can be solved by plotting the curves on the same axes and not
ing where they intersect. The x and the y of each point of in
tersection gives a pair of values which satisfies each equation.
The graphical solution shows clearly how many pairs of values
there are, and why a certain value of x must be taken with a
certain value of y. In many cases, however, the algebraic solu
tion can be made more quickly. But squared paper is of real
service in solving equations of degree higher than the second
containing one unknown.
58. Solution of equations of any degree ; real roots. The
principle involved in graphical solution is readily seen by look
ing at the curves already plotted. Suppose we wish to solve
the equation x 2 6x + 5==0; that is, we want to find values
of x which make the expression x 2 6 x  5 zero. Put y =
or* 6 x + 5 and we obtain the curve in Fig. 46. At the point
where the curve cuts the #axis y is 0. Since the curve cuts
the ojaxis at x = 1 and x = 5, the solutions of # 2 6# + 5 =
are 1 and 5. Look over the curves you have plotted and de
termine the solutions when possible. If the roots of an equa
tion are small whole numbers, they can easily be found by
factoring the given expression. If the given expression cannot
be factored, the roots can be found to as many decimal places
as are needed by graphical methods.
Exercise. Solve x*  5z 2  2x + 20 = 0.
Put y = x 9 5 a: 2 2 x + 20 and compute the following table of
values :
X
y
5
10
4.5
.875
4
4
3.5
 5.375
3
4
2.5
 .625
2
4
1
14
20
1
16
1.5
8.375
2
4
86
APPLIED MATHEMATICS
Time is saved by plotting the curve rather accurately where
it cuts the jraxis.
Fig. 49 shows that the roots of the equation lie between 4
and 5, 2 and 3, and 1 and 2. We will find the first root
to two decimal places. Since the curve seems to cut the #axis
between x = 4.4 and x = 4.5, we substitute these two values in
3
6
\
3
FIG. 49
the equation, obtaining for x = 4.4, y = .416 ; and fora? = 4.5,
y = .875. The change in sign shows that the curve does cut
the #axis between these two points, and the root to two figures
is 4.4.
The next thing is to draw the part of the curve between x = 4.4
and x = 4.5 to a larger scale, as in Fig. 49. The two points P
and P' may be joined by a straight line which, in general, will
lie close to the curve. The curve seems to cross the rraxis be
tween x = 4.43 and x = 4.44. For x = 4.43, y = .0462 ; and
for x = 4.44, y = .0803. The change of sign shows that the
curve does cross the zaxis between these two values of x.
Hence the root to two decimal places is 4.43. In a similar
manner the root could be found to any desired number of
decimal places.
Find the other two roots to two decimal places.
THE USE OF SQUARED PAPER
87
PROBLEMS
Find the roots of these equations to three decimal places :
1. a; 8 3 3 a 2o: + 5 = (root between 1 and 2).
2. x 8 4 or 2 6x + 8 = (root between 4 and 5).
3. x 8 + 2 x 2 4 x 43 = (positive roots).
4. x 4 12 x + 7 = (positive roots).
5. z 8 5z 2 + 8x 1 = (root between and 1).
6. z 8 + 2z 2 3z9 = (root between 1 and 2).
7. z 8 7 x + 7 = (root between 3 and 4).
8. x 3  2x 2  x + 1 = (3 roots).
9. z 8 3# + l = (3 roots).
V. DETERMINATION OF LAWS FROM DATA OBTAINED BY
OBSERVATION OR EXPERIMENT
59. Exercise. Find the law of a helical spring.
In the physics laboratory a helical spring was loaded with
weights of 100 g., 200 g., , and the elongation for each load
was recorded in the following table :
x (grams) . . .
y (centimeters) .
100
.9
200
3
300
G.4
400
10.4
500
14.6
600
18.6
700
22.6
800
26.8
900
30.9
Plot these points care
fully, choosing the units
to get as large a figure as
possible. Stretch . a fine
thread along the points
and it will be found that
it can be placed so that
most of the points will lie
close to it or on it, and
that they will be rather
35
3o
^
/
"1 . 8 <S
GenHmehers
/
f
l/j
J
Y*
10
/*
/
5
./
200
Qrx
400
>ms
60O
BOO
FIG. 60
88 APPLIED MATHEMATICS
evenly distributed above and below. Hence it is evident that an
equation of the first degree connects the grams and centimeters.
In this statement the first two loads are omitted, and no load
greater than 900 g. is considered, since at that load the spring
showed signs of breaking. Draw a straight line in the position
of the thread.
Let us suppose that the law or equation is in the form
y = mx + b. (1)
The values of m and b must be found that will best fit the data.
Take two points which lie close to the straight line and some
distance apart, and substitute the x and y of these points in (1).
Taking the fourth and ninth points, we have
10.4 = 400 m + b. (2)
30.9 = 900 m + b. . (3)
(3)  (2), 20.5 = 500 m. (4)
m = .061. (5)
Substituting (5) in (2), b = 6. (6)
Therefore y .041 x 6 is the required equation or law.
Check. Substitute the x and y of sixth point.
18.6 = 600 x .041  6
= 18.6.
Substitute the x and y of the seventh point, we obtain 22.6 = 22.7.
60. Straightline laws. When the results of experimental
work are plotted it frequently happens that the points lie nearly
in a straight line. In such cases it is not difficult to find the
law or equation by the method used in the preceding exercise.
Since there are always errors in experimental work the points
will not, of course, lie exactly in a straight line. If some of
the points lie at a rather large distance from the straight line
through several of them, it may be that the equation is not
of the first degree. In the following exercises the graphs are
straight lines.
THE USE OF SQUARED PAPER
89
EXERCISES
1. Make a helical spring by coiling a wire around a small
cylinder. Arrange the spring to carry a load ; take readings of
the elongation for several loads and find the law of the spring.
2. Put a Fahrenheit and a Centigrade thermometer in a
dish of water and take the reading of each. Vary the tempera
ture of the water by adding hot water or ice and take sev
eral readings. Find the law connecting the readings of the
two thermometers.
3. Load a thin strip of pine supported at points two feet apart
and note the deflection. Vary the load and find that for loads
under a certain weight the deflection is proportional to the load.
For what weight does the law begin to fail ?
4. Find the laws of the following helical springs :
x (ounces)
4
5
6
7
8
9
10
1
y (inches)
5.2
5.5
5.8
6.1
6.4
6.7
7
2
y (inches)
13.2
14.0
14.8
15.6
16.4
17.2
18
3
y (inches)
3.8
5.0
6.2
7.4
8.6
9.8
10
5. I is the latent heat of steam in British thermal units
(B. t. u.) at f F. Find an equation giving I in terms of t.
t
I
170.1
995.2
193.2
979.0
212.0
965.7
240.0
945.8
254.0
935.9
6. V is the volume of a certain gas in cubic centimeters at
the temperature t C. If the pressure is constant, find the law
connecting V and t.
t
V
27
110
33
112
40
115
55
120
68
125
90
APPLIED MATHEMATICS
7. A steel bar 107 cm. long was supported at the ends and
loaded at the center with the following results. Find the equa
tion connecting the load and deflection.
Grams ....
Deflection . .
500
1.18
1000
2.35
2000
4.72
3000
7.15
4000
9.42
8. In an arclight dynamo test the voltage for the revolutions
per minute was recorded. Find the laws connecting the volts
and revolutions per minute.
.Revolutions per minute . .
Volts
200
165
300
253
400
337
500
421
600
507
700
590
9. P is the pull in pounds required to lift a weight W
by means of a differential pulley. Find the law connecting
P and W.
W
P
50
8.0
100
13.4
150
19.0
200
24.4
250
30.1
300
35.6
10. When the weight W was lifted by a laboratory crane
the force applied to the handle was P pounds. Find the law
connecting P and W.
W
P
50
7.4
100
8.3
150
9.5
200
10.3
250
11.6
300
12.4
350
13.6
400
14.5
CHAPTER VIII
FUNCTIONALITY; MAXIMUM AND MINIMUM VALUES
61. Number scale. Real numbers are represented graphi
cally by a straightline scale. Zero is the dividing point between
the positive and the negative field, and may be considered either
positive or negative.
In going down the negative scale further and further from zero
the numbers are getting smaller ; that is, 10 is less than 3.
The actual magnitude of a number, without regard to its sign
or quality or position in the scale, is called its absolute value.
<0 4 3 2 I Q H ^+? 43 +.4 +00
FIG. 61
Beginning at the extreme left and passing constantly to
the right, numbers may be said to increase continuously from
oc through to + GO. Beginning at the extreme right and
passing constantly to the left, numbers may be said to decrease
continuously from + <*> through to GO. Beginning at any
point and passing to the right gives increasing numbers, while
passing to the left gives decreasing numbers.
62. Variables. A variable is a number which changes and
passes through a series of successive values. It may pass
through the whole scale of values from oo to + oo> or it
may pass through a certain portion of the scale only. If the
variable is confined to a portion of the number system, as
from the position 15 in the scale to the position + 6, it is
said to have the interval 15 to + 6.
A number is said to vary continuously in a given interval,
a to b y if it starts with the value a and increases (or decreases)
91
92 APPLIED MATHEMATICS
to the value ft in such a way as to assume all values betwee
a and ft (integral, fractional, and irrational) in the order o
their magnitude.
63. Inequality of numbers. One number is greater than
second if a positive number must be added to the second t
produce the first. Thus 3 is greater than 8, since + 5 mus
be added to 8 to obtain 3.
One number is less than a second if a positive number mug
be subtracted from the second to obtain the first. Thus 1
is less than 12, since + 5 must be subtracted from 12 t
obtain 17.
The relation of inequality is usually expressed by a symbo
Thus  3 >  8, 10 > 4,  17 <  12, 2 < 7.
64. Function of a variable. The value of an expression ii
volving a variable depends upon the value of the variabL
The expression is called a function of the variable. Thu
x 2 1 is a function of x (written f(x) = x 2 1, and rea
" function of x equals x 2 1 "), for when x has the value
2, 1, 0, +1, +2 respectively, or* 1 has the values {
0,  1, 0, 3.
The variable to which we may give values at will is calle
the independent variable ; but the expression or variable whic
depends upon it for its value is called the dependent variabl
or function. The volume of a cube is a function of the edg<
v = f(e) = e 8 . The area of a circle is a function of the radiui
a = f(r) = Trr 2 . The distance through which a body falls is
function of the time, s = /():= gfi. Name the independer
and dependent variables in the preceding illustrations.
Exercise. Plot the graph of the function 2x 8 3 or 2 12 x H
Give x integral values from 3 to 4 and obtain the following table
X
2x 8 3x 2  12x + 4
3
41
2
 1
11
4
1
9
2
16
3
6
4
36
FUNCTIONALITY
93
40
You have been constructing curves by locating points from
a table and drawing a smooth curve through them ; you should
now see that this method of plotting a function is based on the
assumption that the given expression is a continuous function of
x. In this case a small change in x makes a small change in the
given function ; hence if all values of x were taken, there would
be a continuous succession of points forming a smooth curve.
In Fig. 52 imagine a perpendicular to the xaxis drawn to
the curve from x = 3. The length of this perpendicular is
the value of the function
for x = 3. Now imagine
the perpendicular to move
to the right to x = + 4, and
you have a mental picture
of the function varying con
tinuously in value from 41
to + 11, then to 16, and
finally to + 36.
For certain intervals of
values of x the function is
greater than zero, and for
certain intervals it is less
than zero. For certain defi
nite values of x the function
has the value zero. The value of the function is greater than
zero in the intervals from x = 2 to x = .4 (about), and from
x = 2.9 (about) to x = + oo. The function is less than zero from
x = oo to x = 2, and from x = .4 (about) to x = 2.9 (about).
The function has the value zero for x = 2 and x = .4 (about).
65. Maximum and minimum values. As x increases from
 3 to 1, 2x*  3x 2  I2x + 4 increases from  41 to + 11.
As x increases from 1 to + 2, the function decreases from
+ 11 to 16. As x increases from + 2 to + 4, the function
increases from 16 to + 36. We observe that as the variable
x increases continuously, the value of the function may either
zo
FIG. 62
D4 APPLIED MATHEMATICS
increase or decrease. At any point where the function stops
increasing and begins to decrease, it is said to have a maximum,
value or to be a maximum. In this case it occurs when x = 1,
3r when the function has the value + 11.
When the function stops decreasing and begins to increase,
.t is said to have a minimum value or to be a minimum. Here
it occurs when x = 2, or when the function has the value 16.
In other words, a function is a maximum when its value is
greater than the values immediately preceding and following.
En the same way a function is a minimum when its value is
less than the values immediately preceding and following.
The point on the curve at which there is a maximum or
ninimum value of the function is called a turning point.
66. To investigate functional variation and get an idea of
regional increase and decrease, and maximum and minimum
values. Plot enough points to give the shape of the curve. The
regions of increase and decrease are then readily noted. To check
%n apparent maximum or minimum value of the function, cal
culate values of the function for points close together in the im
mediate neighborhood and on both sides of the apparent value.
That value of the function which is either greater or less than
M those which immediately precede or follow is the value desired.
PROBLEMS
1. A line 10 in. long is divided into two segments which
ire taken as the base and altitude of a rectangle, (a) Express
the area of the rectangle as a function of one of the segments.
(7v) Plot this function, (c) Discuss the increase and decrease of
area as the length of one segment changes from to 10 in.
(d) What length of segment gives a maximum area ? (e) What
is the maximum area ? (/) Is there a minimum area ?
Suggestion. Let x = one segment.
10 x = other segment.
x (10 x) = area.
FUNCTIONALITY 95
2. Express the sum of a variable number and its reciprocal
as a function of the number. Plot the function and investigate
for regional changes. What is the minimum value of the sum
of a number and its reciprocal ?
3. An opentop tank with a square base is to be built to
contain 32 cu. ft. What should be the dimensions in order to
require the smallest amount of steel plate for construction ?
Suggestion. Let x = a side of the base.
Q9
Then f = depth of the tank.
x*
128
x * ^ = surface of the tank.
x
128
Plot the function x 2 H and determine x for the minimum value.
x
4. Express the area of a variable rectangle inscribed in a
circle whose radius is 4 in., as a function of the base. What
are the dimensions of the rectangle of greatest possible area ?
Suggestion. Make a drawing of the circle and rectangle and
note how the area changes as the base of the rectangle increases
from to 8 in. A diagonal of the rectangle is a diameter of the
circle. Why ?
Let x = base of the rectangle.
Then V64 x 2 = altitude of the rectangle.
x V64 X* = area of the rectangle.
Plot this function and determine the value of x that makes it a
maximum.
5. Show that the largest rectangle having a perimeter of
24 in. is a square..
6. What are the dimensions of the greatest rectangle in
scribed in a right triangle whose base is 12 in. and altitude
8 in.?
7. From the cube of a variable number six times the num
ber is subtracted. What value of the variable would make this
function a minimum ? Discuss the functional variation in full,
96
APPLIED MATHEMATICS
8. From a variable number its logarithm is subtracted.
What value of the variable number would make this difference
a minimum ?
9. Two towns A and B (Fig. 53)
are 3 and 4 mi. respectively from
the shore of a lake CD. If CD is
a straight line 7 mi. long, where
must a pumping station P be built
to supply the towns with water with
the least amount of pipe ?
10. If t represents the number of tons of coal used by a
steamer on a trip, and v represents the speed of the boat per
hour, the following relation holds : t = .3 + .001 v*. Other ex
penses are represented by one ton of coal per hour. What speed
would make the cost of a 1000mi. trip a minimum ?
11. The cost of an article is 35 cents. If the number sold at
different prices is given by the following table, find the selling
price which would probably give the greatest profit.
7X P
FIG. 53
Selling price in dollars . .
Number articles sold . . .
.50
3600
.60
3100
.75
2640
.90
2080
1.00
1300
1.10
700
Suggestion. First from the given table plot a curve to show the
probable number sold at prices from 50 cents to $1.10. Then on
the same axes with different vertical units plot the curve to show
the profits at the various prices. Profit = (selling price cost) x
number sold.
To determine the turning point of the second curve somewhat
closely it will be necessary to locate intermediate points; e.g. for
the selling price at 80 cents and 85 cents. The number probably
sold at these prices may be found from the first curve.
12. Devise other problems in maxima and minima and
solve them.
CHAPTER IX
EXERCISES FOR ALGEBRAIC SOLUTION IN PLANE GEOMETRY
67. During the year given, to plane geometry these exercises
not only serve as a review of algebra, but they should also
develop in the pupils an ability to attack successfully many
geometrical problems from the algebraic side. The figures for
the first exercises should be carefully drawn with ruler, com
passes, and protractor, and the drawing should check the
algebraic work. Later the figures may be sketched. The num
bers and letters should be put on the given and required parts
in the drawing, and the equations set up from the figures.
Represent lines, angles, and areas by a single small letter.
Check all results.
COMPLEMENTARY AND SUPPLEMENTARY ANGLES
1. Find two complementary angles whose difference is (a)
20; (6) 52; (r) 5 8' 10"; (d) x.
2. x/2 and x/3 (x + 40 and x 30) are complementary
angles. Find x and the angles.
3. Find the angle that is the complement of (a) 8 times
itself ; (&) 7 times itself ; (c) 3 times itself ; (d) n times itself.
4. How many degrees are there in the complementary angles
which are in the ratio (a) 1 : 2 ? (ft) 4 : 5 ? (c) 3.5 : 6.5 ? (d) m : n ?
5. Find the value of two supplementary angles if one is 9
(15) times as large as the other.
6. How many degrees are there in an angle that is the sup
plement of (a) 4 times itself ? (&) 7 times itself ? (c) of itself ?
(d) n times itself ?
97
98 APPLIED MATHEMATICS
7. Of two supplementary adjacent angles, one lacks 7 of
being 10 times as large as the other. How many degrees in
each?
8. If 10 (7) be added to one of two supplementary angles
and 20 (8) to the other, the resulting angles will be in the
ratio 2 : 5 (3 : 4). Find the angles.
9. If 6 (5) be taken from one of two supplementary angles
and added to the other, the ratio of the two angles thus found
is 2 : 7(13 : 5). What are the angles ?
10. To one of two supplementary angles add 11 (9) and
from the other subtract 16 (5). The two angles thus obtained
will be to each other as 3 : 4(5 : 12). Find the angles.
11. How many degroes are there in an angle whose supple
ment is (a) 5 times its complement ? (6) f of its complement ?
(c) n times its complement ?
12. Find the angle whose supplement and complement added
together make 112 (208).
13. If 3(8) times the complement of an angle be taken from
its supplement, the remainder is 10 (76). Find the angle.
14. If 3 times an angle added to 5 times its supplement
equals 20 times its complement (supplement), what is the
angle ?
15. The angles formed by one line meeting another are in
the ratio 7 : 11 (3 : 8). How many degrees in each ?
16. Construct a graph to show the complement of any angle.
(Take a large square each way equal 10. Locate a few points :
x = 10, y = 80 ; x = 40, y = 50 ; x = 90, y = ; and draw a
straight line through them.) What is the equation of this line ?
17. On the same sheet of squared paper construct a graph to
show the supplement of any angle. What is the equation of
the straight line ?
18. On the same sheet of squared paper as in the last two
problems draw a straight line from (x = 0, y = 0) to (x = 80,
EXERCISES FOR ALGEBRAIC SOLUTION 99
y = 160). Read off a few pairs of angles given by points on
this line. What is the equation of this line ? On this line,
mark the points that answer the question, If one of two comple
mentary (supplementary) angles is twice the other, how many
degrees in each ?
19. Find two complementary angles such that the sum of
twice one and 3 times the other is 210. Solve graphically.
20. Two complementary angles are in the ratio 2 : 3(7 : 8).
Find the number of degrees in each. Solve graphically.
21. Three angles make up all the angular magnitude about a
point. The difference of the first and second is 10 (20), and
of the second and third is 100 (2). How many degrees in each
angle ?
22. The sum of four angles about a point is 360. The
second is 3 times the first, the third is 10 greater than the sum
of the first and second, and the fourth is twice the first. Find
the angles.
23. Of the angles formed by two intersecting lines, one is
5(3^) times another. What are the angles ?
PARALLEL LINES
24. Two parallels are cut by a transversal making one ex
terior angle 3 (5f ) times the other exterior angle on the same
side of the transversal. Find all the angles.
, 25. If two parallels are cut by a transversal making two ad
jacent angles differ by 20 (36 20'), find all the angles.
26. If a transversal of two parallels makes the sum of 5 (4)
times one interior angle and 2 (3) times the other interior
angle on the same side of the transversal equal to 420 (625),
find all the angles.
27. The sum of one pair of alternateinterior angles formed
by a transversal of two parallels is 8 (6 ) times the sum of the
other pair. Find all the angles.
100 APPLIED MATHEMATICS
TRIANGLES
28. Of the angles of a triangle the second is twice the first,
and the third is 3 times the second. How many degrees in
each angle ?
29. Find the angles of a triangle ABC, given:
(a) A 3 times and B 4 times as large as C.
() A 3 times as large as C and B of C.
(r) A 44 and B 25 smaller than C.
(</) A :B: C = 2 : 3 : 4(3 : 5 : 10).
30. In a triangle ABC angle A is G times angle B, and angle
C is J of angle A. Find the three angles.
31. Find the angles of the triangle ABC when A is 43 more
than of B, and B is 18 less than 4 times C.
32. The sum of the first and second angles of a triangle is
twice the third angle, and the third angle added to 3 times the
second equals 140 less the third angle. Find the three angles.
33. In a triangle the sum of twice the first angle, 3 times the
second, and the third is 320 (400); and the sum of the first,
twice the second, and 3 times the third is 440 (310). Find the
angles.
34. In a triangle ABC, A lacks 106 of being equal to the
sum of B and C, and C lacks 10 of being equal to the sum of
A and B. Find the angles.
35. The vertical angle of an isosceles triangle is 68. Find
the base angles.
36. One base angle of an isosceles triangle is 25 (47). Find
the vertical angle.
37. Find the angles of an isosceles triangle if a base angle is
4(5) times the vertical angle.
38. In an isosceles triangle the vertical angle is 36 (75)
larger than a base angle. Find the angles.
39. In an isosceles triangle 5 times a base angle added to 3
times the vertical angle equals 490 (530). Find the angles.
EXERCISES FOR ALGEBRAIC SOLUTION 101
40. Find the angles of an isosceles triangle in which the ex
terior angle at the base is 95 (140).
41. The angle at the vertex of an isosceles triangle is (^) of
the exterior angle at the vertex. Find the angles of the triangle.
42. A base angle of an isosceles triangle is 12 (n) times the
vertical angle. Find the angles of the triangle.
43. What are the angles of an isosceles triangle in which the
vertical angle is 12 more than ^ Q) of the sum of the base
angles ?
44. Construct a graph to show the change in the vertical
angle y of an isosceles triangle as a base angle x increases from
to 90.
45. The vertical angle of an isosceles triangle lacks 8 (20)
of being ^ (.9) of a right angle. Find all the angles.
46. The acute angles of a right triangle are x and 2 x(3y
and 5 y). Find them.
47. The difference of the acute angles of a right triangle is
18 (37). Find them.
48. If the acute angles of a right triangle are in the ratio
(a) 2 : 3, (/>) 7 : 8, (c) m : n, find the angles.
49. In a right triangle the sum of twice one acute angle and
3 times the other is 211 (192). Find the angles.
POLYGONS
50. How many sides has a polygon the sum of whose inte
rior angles is 720 (2340)?
51. An interior angle of a regular polygon is 165 (160).
How many sides has the polygon ?
52. How many sides has a polygon the sum of whose inte
rior angles equals 2 (12) times the sum of the exterior angles ?
53. How many sides has a polygon the sum of whose interior
angles exceeds the sum of the exterior angles by 1080 (2700) ?
102 APPLIED MATHEMATICS
54. Construct a graph to show the sum of the angles of a
polygon as the number of sides increases from 3 to 12.
55. Construct a graph to show the number of degrees in
each angle of a regular polygon of n sides for values of n from
3 to 36.
56. If the number of sides of a regular polygon be increased
by 2(3), each of its interior angles is increased by 15 (10).
How many sides has the polygon ?
57. By how many must the number of sides of a regular
polygon of 12(15) sides be increased in order that each inte
rior angle may be increased 18 (6)?
58. By how many must the number of sides of a regular
polygon of 8(20) sides be increased if each exterior angle is
diminished 5 (6)?
59. Construct a curve to show the number of degrees in an
exterior angle of a regular polygon as the number of sides
increases from 3 to 18.
60. The perimeter of a triangle is 176 (50.4) ft. in length
and the sides are as 1 : 3 : 4(2 : 5 : 7). Find the sides.
61. The perimeter of a triangle bears to one side the ratio
3 : 1 (15 : 4) and to another side the ratio 4 : 1 (5 : 2). What
part of the perimeter is the third side ?
62. The sum of the three sides, a, b, and c, of a triangle is
35 ft. ; twice a is 5 ft. less than the sum of b and c, and twice
c is 4 ft. more than the sum of a and b. Find each side.
63. If the perimeter and base of an isosceles triangle are in
the ratio 4 : 1 (5 : 2), what part of the perimeter is one of the
equal sides ?
64. Find the perimeter of an isosceles triangle if it is 4 (8^)
times the base, and one of the equal sides is 4 (55) ft. longer
than the base.
65. In an isosceles right triangle the perpendicular from
the vertex to the hypotenuse is 12 (30) cm. long. How long is
the hypotenuse ?
EXERCISES FOR ALGEBRAIC SOLUTION 103
66. If the hypotenuse of an isosceles right triangle is 26 (8) in.
long, what is the length of the perpendicular from the vertex to
the hypotenuse ?
PARALLELOGRAMS
67. One angle of a parallelogram is 4 (9) times its consecu
tive angle. Find all the angles.
68. An angle of a parallelogram is 3(2) times one of the
other angles. Find all the angles.
69. Find the angles of a parallelogram if the difference of
two consecutive angles is 20 (90).
70. If two consecutive angles of a parallelogram are in the
ratio 17 : 1 (4 : 5), how many degrees in each angle ?
71. How many degrees in each angle of a parallelogram when
an angle exceeds () of its consecutive angle by 30 (56)?
72. The number of degrees in one angle of a parallelogram
equals J of the square of the number of degrees in the con
secutive angle. Find all the angles.
73. Prove algebraically that if two angles x and y of a quad
rilateral are supplementary, the other two angles a and b are
also supplementary.
74. Find the sides of a parallelogram if one side is () of
another side and the perimeter is 28 (84) cm.
75. One side of a parallelogram is 4(5) in. longer than an
other side and the perimeter is 36 (58) in. Find the sides.
76. The sum of two adjacent sides of a rhomboid is () of
the difference of those sides. Find the sides if the perimeter
is 18.3 (82) cm. 
77. One angle of a rhombus is 60. If 5 (2) times the perim
eter exceeds the square of the shorter diagonal by 19(13f),
find a side of the rhombus.
78. In a rhomboid two of whose sides are a and i, 3 times
a exceeds twice b by 11, and the sum of twice a and 5 times b
is 20. Find the perimeter.
104 APPLIED MATHEMATICS
79. In one of the triangles formed by the diagonals of a
rhombus and one of the sides of the rhombus the two smaller
angles are in the ratio 2 : 3(1 : 3). Find all the angles of the
rhombus.
80. The perimeter of a parallelografri is 16(9.6), and the
square of one side added to 4 (2) times an adjacent side equals
37(8.6). Find the sides of the parallelogram.
81. In a rhombus one of whose angles is 60 the shorter
diagonal is 10 in. (5 ft. 6 in.). Find the perimeter.
82. Two sides of a rectangle are x and x* (3 x and 7 x) and
the perimeter is 60(40). Find the sides.
CIRCLES
83. The circumference of a circle is divided into three parts.
Find the number of degrees in each part if the second contains
3(6) times as many as the first part, and the third part con
tains 5 (7) times as many as the first part.
84. In a circle a diameter and a chord are drawn. The
diameter is 4(5) in. longer than the chord and the diameter
and chord together are 18 (20) in. long. How long is each ?
85. There are 100 (aj) in one of the arcs subtended by a
chord. How many degrees are there in the other arc ?
86. In one of the arcs subtended by a chord there are
50 (120) more than in the other arc. How many degrees
in each arc ?
87. Find the side of a square inscribed in a circle whose
radius is 30 (42.5) mm.
88. A triangle whose perimeter is 36 (72) mm. is inscribed
in a circle. The first side is % of the second and of the third.
Find the three sides.
89. In a circle of radius 8 (12) in. a chord is drawn equal
in length to the radius. How far is it from the center ?
EXERCISES FOR ALGEBRAIC SOLUTION 105
90. A circle containing 280(308) sq. ft. is divided into three
parts by radii. The third part equals twice the second, and
the second part is 20 sq. ft. larger than the first. Find the
area of each part.
91. A line 1 (3.6) ft. long intersects a circumference in two
points. If the part inside the circumference is twice the length
of the part outside, how long is the part which forms the
chord ?
92. A number of coins are placed in a row touching one
another, and the length of the row is measured. 3 quarters,
2 nickels, and 5 dimes measure 204 mm. ; 1 quarter, 3 nickels,
and 2 dimes measure 123 mm. ; and 1 quarter, 1 nickel, and
1 dime measure 63 mm. Find the diameter of each coin. Check.
93. A boy has 20 copper disks ; part of them are 20 mm. in
diameter and the rest are 30 mm. The sum of their diameters
is 520 mm. How many of each kind has he ?
94. Two diameters are drawn in a circle, making at the
center one of the supplementary adjacent angles 3 times the
other. How many degrees in each angle ?
95. A chord 6 (4) in. long is 4 (6) in. from the center of a
circle. Find the radius of the circle.
96. A chord 16(4) in. long is at a distance of 6(8) in. from
the center of a circle. What is the length of a chord which is
3 (1) in. from the center ?
97. A chord 8(12) in. long bisects at right angles a radius.
How long is the radius ?
98. The radius of a circle is 5 (3) in. How far from the
center is a chord 8(4) in. long ?
99. The radius of a circle is r. What is the length of a
chord whose distance from the center is ( J) r ?
100. Find the length of the longest and shortest chords that
can be drawn through a point 9 (6) in. from the center of a
circle whose radius is 15(8) in.
106 APPLIED MATHEMATICS
101. The sum of the longest and the shortest chords through
a point 3 (8) in. from the center of a circle is 18 (64) in. Find
the radius and the two chords.
102. Construct a curve to show the length of a chord in a
circle of radius 8 in. as the distance of the chord from the
center increases from to 8 in.
103. A circle is circumscribed about a right triangle whose
legs are 6 and 8 (5 and 12) in. Find the radius of the circle.
104. The legs of a right triangle inscribed in a circle are
5 x and 12 x (x and 3 x) and the radius of the circle is 13 (5) in.
Find the sides of the triangle.
105. From the point of tangency P, a distance PA equal to
twice the radius is measured off on the tangent. If the distance
from A to the center of the circle is 10(6) in., find the radius.
106. In a circle of radius 8 (5) in. two parallel chords lie on
opposite sides of the center. One is twice as far from the center
as the other. If the sum of the squares of the half chords is
123 (10) in., find the distance each chord is from the center.
107. The perimeter of an inscribed isosceles trapezoid is
38(88) in. One of the parallel sides is (.7) f the other and
one of the nonparallel sides is 9 (30) in. shorter than the
longest side of the trapezoid. Find each side.
108. Two circles touch each other and their centers are
8 (a) in. apart. The diameter of one is 10 (d) in. What is the
diameter of the other ?
109. Two circles are tangent externally. The difference of
their radii is 8 (a) in. and the distance between their centers is
12 (6) in. Find the radii.
110. The distance between the centers of two circles is
18 (a) in., which is one half the sum of their radii. Find the
radii.
111. One angle of an inscribed triangle is 35 (50) and one
of its sides subtends an arc of 113 (150). Find the other
angles of the triangle.
EXERCISES FOR ALGEBRAIC SOLUTION 107
112. The circumference of a circle is divided into three arcs
in the ratio 1 : 2 : 3(2 : 3 : 5). Find the angles of the triangle
formed by the chords of the arc.
113. A triangle is inscribed in a circle. The sum of the first
and third angles is twice the second angle, and the difference
of the first and second is 20. How many degrees in each of
the three arcs ?
114. Construct a graph to show the change in an inscribed
angle y y as the arc intercepted by its sides increases from
to 180.
115. An isosceles triangle is inscribed in a circle. The number
of degrees in the arc upon which the vertical angle stands is
8(3) times the number of degrees in a base angle of the
triangle. Find the angles of the triangle.
116. Consecutive sides of an inscribed quadrilateral subtend
arcs of 82, 99, 67, and x respectively. Find each angle of
the quadrilateral ; also each of the eight angles formed by a
side and a diagonal.
117. How many degrees in each angle of a quadrilateral
inscribed in a circle, if the sides subtend arcs which are in
the ratio 1:2:3:4(2:3:5:6)?
118. A right triangle is inscribed in a circle. If one acute
angle of the triangle is (f ) of the other, how many degrees in
each of the three arcs ?
119. ADCD is an inscribed trapezoid. If the angle A is twice
angle C, find each angle.
120. Two chords AB and CD intersect within a circle at P.
The angle ^PC'is 50, arc DB is 40, and arc AD is 160.
Find the other arcs and angles.
121. Two chords AB and CD intersect within a circle at P.
Arc BD is twice arc AC, and arc CB is twice arc DA. Angle
DP A is twice angle APC. Find the arcs and angles.
122. The angle y is formed by two chords AB and CD inter
secting in a circle, and the two intercepted arcs A C and DB are
108 APPLIED MATHEMATICS
90 and x respectively. What is the equation connecting y
and x ? Construct a graph to show the change in y as x
increases from to 90.
123. From a point without a circle two secants are drawn,
making one of the intercepted arcs 3(5) times the other. If
the sum of the other two arcs is 200 (300), what is the angle
formed by the secants ?
124. The angle y is formed by two secants intersecting with
out a circle. The intercepted arcs are 90 and #(x<90),
What is the equation connecting y and x ? Construct a graph
to show the change in y as x increases from to 90.
125. Two tangents drawn from an exterior point to a circle
make an angle of 60 (80). Find the two arcs. Join the points
of tangency and find the other two angles in the triangle thus
formed.
126. Through the ends of an arc of 45 (100) tangents to
the circle are drawn. Find the angle formed by the tangents.
Find the other two angles in the triangle formed by joining
the points of tangency.
127. Find the angle formed by two tangents to a circle drawn
from a point at a distance from the center of the circle equal
to the diameter.
128. From P, a point without a circle, two tangents PA and
PB, and a secant PC are drawn. The arc AB equals 160 (100).
If the difference of the angles BPC and CPA is 10 (25), find
the angles.
129. From a point without a circle of radius 4 (8) in. a
secant through the center and a tangent are drawn. If the
angle formed by the secant and tangent is 30 (60), find
the distance from the point to the center of the circle, and
the length of the tangent.
130. In an equilateral triangle whose sides are 40 (60) mm.
a circle is inscribed. Find the radius of the circle. Find the
radius of the circumscribed circle.
EXERCISES FOR ALGEBRAIC SOLUTION 109
RATIO
131. Express the ratio of the following pairs of numbers in
the simplest form :
(a) 168 and 252. (A) 148 z 8 and 185 x 4 .
(ft) 387 and 602. (i) x 2 + 5x + 6 and x + 3.
(c)  and f. (j) x * + 2x 15 and * + 5 
(e)
(/) .125 and 3.75. ( . x + 2 x*+6x
a*x and 30a*x. V ' x + 3 x* + 7x
132. Squares are constructed on the lines a and ft. Find the
ratio of the areas :
(a) a = 5 in., 6 = 10 in. (c) a = 4 cm., b = 12 cm.
(ft) a = 3^ in., ft = 7 in. (d) a = 14 mm., ft = 35 cm.
133. On a sheet of squared paper let the bottom line be the
xaxis and the left border line be the yaxis, and the side of a
square each way = 1. Draw a straight line through the points
(0, 0) and (8, 16). Make a table of corresponding values of x
and ?/. What is the ratio of y to x ? What is the equation of
the fine ?
134. The width y of a field is to be made f of the length x.
What is the equation connecting y and x ? Construct a graph
to show the width of the field for a length from 10 to 100 rd.
135. If the ratio of y to x is 2 : 3, construct a graph to show
the relation. What is the equation of the straight line ?
136. If 14 x 9 y = 2 x y, find the ratio of x : y. Construct
the graph.
137. What is the ratio ofx:y, if. 7 x 6y = 3x + 4y?
138. If x : y = 4 : 5, find the value of the ratio 2x + y:7 x y.
Construct the graph.
139. Find the value of the ratio 3 x 2 + 2 y 2 : xy + y*, if x : y
= 1:2.
110 APPLIED MATHEMATICS
PROPORTION
140. Test the correctness of the following proportions \
84 = 42 1.25^120 '
^ 180 ""90" ( ' .26 " 24 "
48 _96 a 2 + 2 ab + b* _ a + b
225~45* (*' a?  6 s ~" a  ft "
87 111 , ^ x 2 + 7# + 10 x + 2
CO
259 w ' (x + 5) 2 x +
141. Find x in the following proportions :
, , 18 32
28^35 a? _ 1
^ 48 s3 ^24* ^ x = 9'
142. What number can be added to 7, 12, 1, and 3 (5, 19,
16, and 52) so that the resulting numbers will form a pro
portion ?
143. Find the numbers proportional to 1, 2, 3, 4 (2, 5, 1, 3)
that may be added regularly to 5, 10, 15, 40 (11, 20, 8, 14) so
as to form a proportion.
144. The line joining the midpoints of the nonparallel sides
of a trapezoid is 20 (42) in. long. Find the bases if one is
(.4) of the other.
145. In a triangle ABC the line PQ parallel to BC divides
the side AC in the ratio 3 : 4 (5 : 9). If AB = 20 (9.8) in., find
the two segments of AB.
146. The sum of the two sides of a triangle is 45 (63) in. A
line parallel to the third side cuts off from the vertex segments
10 and 8 (4 and 20) in. long. Find the two sides.
147. A line 100 (6) ft. long is divided into parts in the ratio
1 : 2 : 3 : 4 (2 : 3 : 7). Find each part.
EXERCISES FOR ALGEBRAIC SOLUTION 111
148. Three lines are in the ratio 2 : 3 : 4 (2 : 1 : 6) and their
fourth proportional is 30 (24). Find the length of each line.
149. The sum of two sides of a triangle is 20 (5) in. The
third side, 18 (4) in. long, is a third proportional to the other
two sides. Find them.
150. One side of a triangle is 2 in. longer than the first side,
and the third side is 5 in. longer than the first. If one side is
a mean proportional between the other two, find the three sides.
151. The three sides of a triangle are x, y, and 3. The cor
responding sides of a similar triangle are 10, 20, and 15. Fii\d
x and y.
152. The sum of the three sides of a triangle, x, y, and z, is
15, and the corresponding sides of a similar triangle are x f 3,
y + 7, and z + 5. Find the sides of each triangle.
153. The three sides of a triangle are 3.r, 6x, and Sx
(x, x + 1, x + 2), and the corresponding sides of a similar tri
angle are 3x 2 , 6x 2 , and 8x 2 (x 2 , x 2 + x, and x 2 + 2*). If the
sum of the perimeters of the two triangles is 102 (75), find the
sides of each triangle.
154. The sides of a triangle are 5, 8, 12 (12, 16, 20) in. Find the
segments of each side made by the bisector of the opposite angle.
155. The sum of two sides of a triangle is 24 in., and the
bisector of the included angle divides the third side into parts
4 and 8 in. long. Find the three sides.
156. In a triangle ABC, AB = 12 and BC = 36. From a
point on A B at a distance x from A a line y is drawn to AC
parallel to the base. Construct a graph to show the length of
y as x increases from to 12.
RIGHT TRIANGLES
157. The hypotenuse of a right triangle is 8 in. and one
angle is 30. Find (a) the other two sides ; (6) the perpendic
ular from the vertex of the right angle to the hypotenuse ; (c)
the segments of the liypotenuse.
112 APPLIED MATHEMATICS
158. One leg of a right triangle is 2 (3) ft. longer than the
other and the hypotenuse is 4 (7) ft. longer than the shorter
leg. Find the three sides.
159. The legs of a right triangle are 12 and 16 (5 and 12)
ft. Find (a) the hypotenuse; (&) the perpendicular from the
vertex of the right angle to the hypotenuse ; (c) the segments
of the hypotenuse.
160. The perpendicular from the vertex of the right angle
of a right triangle to the hypotenuse is 12 (3) in. long and the
hypotenuse is 26 (6.25) in. long. Find the other two sides.
161. If the legs of a right triangle are a and i, find the per
pendicular from the vertex of the right angle to the hypotenuse,
and the segments of the hypotenuse.
162. One side of a right triangle is 4. Construct a curve to show
the length of the hypotenuse as the other side increases from
to 16. (Let the bottom line be the ,raxis, the left border line be
the yaxis, and the side of a large square each way = 1. Take the
side 4 on the vertical axis and locate the points of the curve
with compasses. Check a few of the points by computation.)
CHORDS, TANGENTS, SECANTS
163. The segments of a chord made by another chord are
7 and 9(15 and 13) in., and one segment of the latter chord
is 3 (10) in. What is the other segment ?
164. Two chords intersect, making the segments of one chord
2 and 12(4 and 8) in., and one segment of the other chord
2(14) in. longer than the other segment. Find the two chords.
165. One of two intersecting chords is 14(17) in. long, and
the product of the segments of the other chord is 45 (60). Find
the segments of the first chord.
166. Two secants intersect without a circle. The external
segment of one is 20 (2) in. and the internal segment is 5 (4) in.
If the external segment of the other secant is 10 (3) in., find
the length of the internal segment.
EXERCISES FOR ALGEBRAIC SOLUTION 113
167. From a point without a circle two secants are drawn
whose external segments are 5 and 6(6 and 8) in. The internal
segment of the former is 13 (16) in. What is the internal seg
ment of the latter ? What is the length of the tangent from
the same point ?
168. Two secants from a point without a circle are 24 in.
and 22 in. long. If the external segment of the lesser is 5 in. ?
what is the external segment of the greater? What is the
length of the tangent from the same point?
169. A tangent and a secant are drawn to a circle from an
external point. The external and internal segments of the
secant are respectively 2(3) in. and 1(4) in. shorter than the
tangent. What is the length of the tangent ?
170. From a point on the tangent of a circle 6(15) in. from
the point of tangency a secant is drawn whose internal seg
ment is 2(3) times the external segment. Find the length of
the secant.
171. A tangent intersects a secant which is drawn through
the center of a circle. The length of the tangent is 4() in.,
and the length of the external segment of the secant is 2 (s)
in. Find the radius of the circle and the secant.
172. In a circle of radius 17 in. a point P is taken on the
diameter 15 in. from the center. What is the product of the
segments of chords through P? Denoting the segments by x
and y, what is the equation that connects x and y ? In this
equation give values to x and make a table of values of x and y.
Construct a curve to show the change of y as x increases from
2 to 32 in.
173. From a point on the circumference of a circle of 9 in.
diameter a tangent 6 in. long is drawn. From the end of the
tangent secants are drawn. If y is the external and x the in
ternal segment of the secant, what is the equation connecting
x and y ? Construct a curve to show the length of y as x in
creases from to 9 in, and then decreases to 0.
114 APPLIED MATHEMATICS
AREA OF POLYGONS
174. The base of a triangle is 5(3) times the altitude and
the area is 90 (75) sq. in. Find the base and altitude.
175. The area of a triangle is 130(42) sq. in. and the altitude
is 7 in. less (5 in. more) than the base. Find these dimensions.
176. The sum of the base and altitude of a triangle is
12 (23) in. and the area is 16 (45) sq. in. Find the base and
altitude.
177. Find the area of a right triangle whose base is 20(32)
and the sum of whose hypotenuse and other side is 40(50).
178. The altitude of an equilateral triangle is 12 (7^) ft. Find
its sides and area.
179. The altitude of a triangle is 16 in. less than the base.
If the altitude is increased 3 in. and the base 12 in., the area
is increased 52 sq. in. Find the base and altitude.
180. If the hypotenuse of a right triangle is 1 (8) in. longer
than one leg, and 8(9) in. longer than the other leg, what is
the area of the triangle ?
181. If the area of an equilateral triangle is 16 V3 (60) sq. in.,
find the altitude and a side.
182. If a denotes the area, s a side, and h the altitude of an
equilateral triangle, express each in terms of the others.
183. If a rectangle is 7 (8) ft. longer than it is wide and
contains 170 (209) sq. ft., find its dimensions.
184. The perimeter of a rectangle is 72 (132) ft. and its
length is 2(5) times its width. Find its area.
185. A rectangle whose length is 8 (5) ft. greater than 3 (4)
times its width contains 115(3750) sq. ft. Find its dimensions.
186. The area of a rectangle is 36 sq. ft. Construct a curve
to show the altitude as the base increases from 1 to 36 ft.
187. The side of one square is 3 (4) times as long as that of
another square, and its area is 72 (90) sq. yd. greater than that
of the second square. What is the side of each square ?
EXERCISES FOR ALGEBRAIC SOLUTION 115
188. One side of a square is 3 (6) yd. less than 2 (3) times
the side of a second square, and the difference in area of the
squares is 45 (756) sq. yd. Find the area of each square.
189. One side of a rectangle is 10 (6) ft. and the other side
is 2(1) ft. longer than the side of a given square. The area
of the rectangle exceeds that of the square by 80 (174) sq. ft.
Find the side and area of the square.
190. The floor of a rectangular room contains 180 (240) sq. ft.,
and the length of the molding around the room is 56(62) ft.
Find the length and width of the room.
191. A picture including the frame is 10(9) in. longer than
it is wide. The area of the frame, which is 3 (6) in. wide, is
192(480) sq. in. What are the dimensions of the picture ?
192. The dimensions of a picture inside the frame are 12 in.
by 16 in. (Sin. by 12 in.). What is the width of the frame if
its area is 288(138) sq. in. ?
193. Around a square garden a path 2 ft. wide is made. If
376 sq. ft. are taken for the path, find a side of the garden.
194. Around a garden 100 ft. by 120 ft. a man wishes to
make a path which shall occupy ^ () of the area. How wide
must the path be made ?
195. A rectangular building having a perimeter of 140 ft.
is inclosed by a fence whose distance from the building is J the
width of the building. If the area between the fence and build
ing is 1800 sq. ft., find how far the fence is from the building.
196. An opentop box is made from a square piece of tin by
cutting out a 5 (2)in. square from each corner and turning up
the sides. How large is the original square if the box contains
180 (242) cu. in.?
197. An opentop box is formed by cutting out a l(3)in.
square from each corner of a rectangular piece of tin 2 (3) times
as long as it is wide, and turning up the sides. If the total
surface of the box is 284(936) sq. in., find the dimensions of
the piece of tin.
116 APPLIED MATHEMATICS
198. It is desired to make an opentop box from a piece of
tin 30 (24) (15) in. sq., by cutting out equal squares from each
corner and turning up the strips. What should be the length
of a side of the squares cut out to give a box of the greatest
possible volume ?
Suggestion. If x = side of square cut out, volume of the box =
Make a table of values of y, giving x the values 1, 2, 3 .
Locate the points and draw a smooth curve through them. The
turning point of the curve will show the value of x for the
greatest volume.
199. From a rectangular piece of tin 12 in. by 24 in. (16 in.
by 36 in.) it is desired to make ah opentop box of the largest
possible volume, by cutting out equal squares from the corners
and turning up the strips. What should be the length of a side
of the squares ?
200. The altitude of a trapezoid is 5 (14) in., the area is
10(455) sq. in., and the difference of the bases is 2 (11) in.
Find the bases.
201. The area of a trapezoid is 90 (495) sq. ft., the line join
ing the midpoints of the nonparallel sides is 6 (45) ft., and the
difference of the bases is 6 (12) ft. Find the bases and altitude.
202. In a trapezoid b and V are the bases, h the altitude, and
a the area. Find each in terms of the other.
203. The base of a triangle is 12 in. and the altitude increases
from to 20 in. Construct a graph to show the increase in
area of the triangle.
204. The base and altitude of a triangle increase uniformly,
and the altitude is always twice the base. Construct a curve
to show the change in the area of the triangle as the base
increases from to 10 ft.
205. The base and altitude of a triangle are 24 in. and 9 in.
respectively. What is the area of the triangle formed by a line
parallel to the base and 6 (8) (x) in. from the vertex ?
EXERCISES FOR ALGEBRAIC SOLUTION 117
206. In a triangle whose base is 12 in. and altitude is 16 in.
a line is drawn parallel to the base and at a distance x from the
vertex. If y = the area of the triangle cut off from the vertex,
what is the equation connecting x and y ? Construct a curve
to show the area of the triangle cut off as x increases from
to 16 in.
207. The altitude of a triangle is 2 (3) times its base.
Through the midpoint of the altitude a line is drawn parallel
to the base. If the area of the triangle cut off is 36 (5) sq. in.,
find the base and altitude of the given triangle.
208. The sum of the areas of two similar triangles is
240(290) sq. in., and the sides of one are 2(2^) times the cor
responding sides of the other. Find the area of each triangle.
209. The difference of the areas of two squares is 39 (324)
sq. ft., and a side of one is 3 (14) ft. longer than a side of the
other. Find a side of each square.
210. The sum of the areas of two squares is 13(221) sq. ft.,
and a side of one square is 1 (9) ft. shorter than a side of the
other. Find a side of each square.
211. A side of one square is 5(2) in. longer than a side of
another square, and the areas of the squares are in the ratio
4 : 1 (16 : 9). What is a side of each square ?
212. Construct a curve to show the area of a square as its
sides increase from to 13 in.
CIRCLES AND INSCRIBED POLYGONS
213. Construct a curve to show the area of a circle as its
radius increases from to 16 in. (Locate points for r = 0, 2, 4,
...,16.)
214. The radius of a circle is 5(8) (r) ft. Find a side and
the area of the inscribed square.
215. What is the radius of the circle inscribed in a square
whose area is 1600(5000) (a) sq. ft. ?
118 APPLIED MATHEMATICS
216. An equilateral triangle is inscribed in a circle of radius
6(12) (r) in. Find a side, the altitude, and area of the triangle.
217. The side of an inscribed equilateral triangle is 9 (1.732)
(s) in. Find the radius of the circle.
218. The sum of the side of an inscribed equilateral triangle
and the radius of the circle is 5 + 5 V3 (10.928) (18) in. What
is the length of a side and the radius ?
219. The area of a regular inscribed hexagon is 24 V3 (17.32)
(a) sq. ft. Find the radius of the circle.
220. An equilateral triangle and a regular hexagon are in
scribed in a circle. Find the radius of the circle if the sum of
the areas of the triangle and hexagon is 9 V3(l8 V3) (389.7)
sq. in.
221. The sum of the perimeters of two regular pentagons is
100 (225) ft., and their areas are in the ratio 1:9(25:16).
Find a side of each pentagon.
222. The difference of the perimeters of two regular octagons
is 40(80) ft., and their areas are in the ratio 1 : 4(9 : 25). Find
a side of each octagon.
223. The sum of the circumferences of two circles is
207r(176)ft., and the difference of their radii is 2 (14) ft.
What are the radii?
224. The radius of one circle is 6 (1) ft. longer than the
radius of another circle, and the sum of their circumferences
is 113} (31.416) ft Find the radii.
225. What is the radius of a circle whose area equals the
area of two circles of radii (a) 3 and 4 in. ? (i) 3.3 and 5.6 cm. ?
(c) 6.5 and 7.2 cm. ? (d) r and nr ?
226. What is the radius of a circle whose area equals the
sum of (a) 3, (b) 6, (c) n equal circles ?
227. What is the radius of a circle that is doubled in area
by increasing its radius 1 (3) ft. ?
EXERCISES FOR ALGEBRAIC SOLUTION 119
228. A square and a circle have the same perimeter. Find
the ratio of their areas.
229. If a square and a circle have the same area, what is
the ratio of their perimeters ?
230. If a circle and an equilateral triangle have the same
perimeter, what is the ratio of their areas ?
231. Construct on the same axes curves to show the change in
area of a circle and the inscribed regular hexagon, square, and
equilateral triangle, as the radius increases from to 10 in.
232. The area between two concentric circles is 20?r(286)
sq. ft. and the difference of the radii is 2(7) ft. Find the radii.
233. If the area between two concentric circles is 96?r(50)
sq. ft, and the radius of the inner circle is 2 (5) ft., find the
radius of the larger circle.
234. In a circle of radius 12 (>) in. it is desired to draw a
concentric circle which shall bisect the area of the given circle.
Find its radius.
235. The area of a circle of radius 8(r) in. is to be divided
by a concentric circle so that the area of the ring shall be a
mean proportional between the area of the given circle and of
the inner circle. Find the radius.
CHAPTER X
COMMON LOGARITHMS
68. Definitions. Numbers have been reduced to powers of
10. Thus 2 = 10  8010 , 3 = 10 4771 , 125 = 10 2  0969 .
These exponents are called logarithms. The integral part of
a logarithm, called the characteristic, can be determined easily
and is not given in a table of logarithms ; the decimal part,
called the mantissa, is always taken from the table.
69. Approximate numbers. In ordinary shop practice and
in much engineering work measurements are made usually to
three or four figures. Thus in making a rough estimate the
sides of a building lot may be measured to the nearest foot;
the length of a belt may be measured to the nearest quarter
of an inch ; an angle may be measured to the nearest tenth of
a degree. If the diameter of a pulley is measured and said
to be 12.3 in., the meaning is that the diameter lies between
12.25 in. and 12.35 in., that is, the third figure is doubtful.
In ordinary computations, where numbers with only three or
four figures are involved, a fourplace table of logarithms is
used. The logarithms are not exact; they are approximate
numbers in which the fourth figure is doubtful. Hence the
results should not be carried beyond four figures.
70. The mantissa. To find the mantissa of the logarithm
of a number from 1 to 999, e.g. 352, we look in the first column
of the table at the left for the first two figures, 35, and in the
column headed 2 we find the mantissa of 352, namely .5465.
The mantissa of 745 is .8722.
(Let the class read the mantissas of numbers from the table
till all can find the mantissa of any number quickly.)
120
COMMON LOGARITHMS 121
71. The characteristic. The method of finding the charac
teristic is readily obtained from the following table :
10 8 =
1000,
'log
1000
= 3.
log 6214
=
3
+
a
decimal.
10 2 =
100,
'log
100
log 518
=
2
+
a
decimal.
10' =
10,
'log
10
4
log 83
=
1
+
a
decimal.
10 =
1,
'log
1
= 0.
log 6
=
+
a
decimal.
10 J =
.1
'log
.1
=
1.
log .3
=
1
+
a decimal.
10 a =
.01,
'log
.01
=
2.
log .04
==
2
+
a decimal.
10 8 =
.001,
'log
.001
=
3.
log .008
=
3
+
a decimal.
Since 518 lies between 100 and 1000 its logarithm lies be
tween 2 and 3 ; that is, it is 2 plus a decimal.
The above table shows that the characteristic of the logarithm
of an integer is one less than the number of integral jigures in
the number.
From the table it is also seen that the characteristic of a
decimal is a negative number. Since the mantissa is always
positive, it is convenient to make a little change so that the
characteristic may be considered positive ; this is done by
adding and subtracting 10.
Thus log .2 =  1 + .3010 = 9.3010  10.
log .02 =  2 + .3010 = 8.3010  10.
log .002 =  3 + .3010 = 7.3010  10.
To find the characteristic of the logarithm of a decimal, begin
at the decimal point and count the zeros, 9, 8, 7, till the first
significant figure is reached. The last count with 10 written
after the mantissa is the characteristic.
72. The logarithm of a number. Since 10 is the base of
our number system, 10 is taken as the base of logarithms for
use in ordinary computations. This makes the work much
easier, because the mantissa does not change as long as the
figures in a number remain in the same order. Thus 216, 21.6,
.216, and .0216 have the same mantissa.
122
APPLIED MATHEMATICS
log 216 = 2.3345, i.e.
Dividing both
sides of the equa
tion by 10,
log 2 = 0.3010, i.e.
Multiplying both
sides of the equa
tion by 100,
^Q2.8846
10
= 216.
= 10
/.log 21.6 = 1.3346.
..log 2.16 = 0.3345.
..log .216 = 9.3345. 10.
/.log 200 = 2.3010.
iQl.8846
10
= 21.6
= 10
100.8846
10
= 2.16
= 10
J 09.8345 10
^00.8010
10 2
= .216
= 2.
= 100
1Q2.8010
= 200
Hence it is seen that moving the decimal point any number
of places to the right or left is multiplying or dividing by some
integral power of 10, and this affects only the characteristic.
The mantissas of numbers having one, two, or three figures
are taken directly from the table. The mantissas of fourfigure
numbers are easily found.
Find the logarithm of 1836. The mantissa of 1836 is the
same as the mantissa of 183.6, since moving the decimal point
does not change the mantissa. The mantissa of 183.6 lies be
tween the mantissas of 183 and 184; and it is .6 of the way
from the mantissa of 183 to the mantissa of 184.
Mantissa of 184  mantissa of 183 = 2648  2625
= 23.
23 x .6 = 14.
2625 + 14 = 2639.
.'. log 1836 = 3.2639.
Find log 49.23.
Mantissa of 493  mantissa of 492 = 6928  6920
= 8
8 x .3 = 2.
6920 + 2 = 6922.
/. log 49.23 = 1.6922,
COMMON LOGARITHMS 123
To find the logarithm of a number. Place the decimal po int
(mentally) after the third figure. Subtract the next lower man
tissa from the next higher. Multijjly the difference by the fourth
figure of the number regarded as tenths, disregarding a fraction
less than one half and calling one half or more one ; add the prod
uct to the next lower mantissa. Write the proper characteristic.
(Let the class find the logarithms of many numbers. The
work should be done mentally ; it can be done easily and quickly
with practice.)
73. To find a number from its logarithm. Given logft =
1.5927, required to find b. Looking in the table of mantissas,
it is seen that 5927 lies between 5922 and 5933 ; the cor
responding numbers are 391 and 392. Hence the number cor
responding to 5927 lies between 391 and 392 ; that is, it is 391
plus a fraction. To find the fraction, add a zero to the differ
ence of the given mantissa and the smaller, and divide it by the
difference of the next larger and next smaller mantissas.
391 5922
391.5 5927
392 5933
* 11)50(5
Since a difference of 11 in the mantissas makes a difference of
1 in the numbers, a difference of 5 makes a difference of T \ * n
the numbers. Hence the mantissa 5927 gives the number
391^ = 391.5. But the characteristic 1 shows that there are
two integral figures in the number. Therefore b = 39.15.
Given log w = 0.9145, m = 8.213.
log n == 8.8132  10, n = .06504.
To find a number from its logarithm. When the given man
tissa lies between tiro mantissas in the table, divide the differ
ence of these mantissas into the difference of the smaller mantissa
and the given mantissa, to one decimal figure. Add this decimal
124 APPLIED MATHEMATICS
figure to the number corresponding to the smaller mantissa
and place the decimal point in the position indicated by the
characteristic.
(All the work in finding a number from its logarithm should be
done mentally ; with practice it can be done easily and quickly.)
74. The use of logarithms in computation. Since logarithms
are exponents it follows that :
I. log(2x 3)= log 2 + log 3.
2 = 10 8010 , 3 = 10 4771 
2x3 = 10 8010 x 10 4771 = 10 7781 = 6.
The logarithm of a product is equal to the sum of the logarithms
of the factors.
II. log $ = log 3 log 2.
3 * 2 = 10 4771 T 10 m( > = 10 1761 = 1.5.
The logarithm of a quotient is equal to the logarithm of the
dividend minus the logarithm of the divisor.
III. Iog2 8 = 31og2.
2 8 = (10' 8010 ) 8 = 10  9080 = 8.
The logarithm of a power of a number is equal to the loga 
rithm of the number multiplied by the exponent of the power.
IV. log V3 = log 3* = \ log 3.
A/3 = 3* = (10 4771 )* = 10 288tt = 1.732.
The logarithm of the root of a number is equal to the logarithm
of the number divided by the index of the root.
PROBLEMS
1. Multiply 28.34 by 3.376.
log 28.34 = 1.
log 3.376 = 0.
log product =
product =
COMMON LOGARITHMS 125
Before finding the mantissas from the table always make out an
outline as above. This saves time and prevents mistakes. Keep the
signs of equality and the figures exactly in columns.
SOLUTION. log 28.34 = 1.4524
log 3.376 = 0.5284
log product = 1.9808
product = 95.68.
As a rough check we have 28 J x 3J = 94.
2. Multiply 1.251 by .6453.
SOLUTION. log 1.251 = 0.0973
log .6453 = 9.8098  10
log product = 9.9071  10
product = .8074
Rough check. .65 x \\ = .81.
3. Divide 31.87 by 641.2.
SOLUTION. log 31.87 = 11.5034  10
log 641.2 = 2.8070
log quotient = 8.6964  10
quotient = .04970.
Rough check. 32 * 640 = .05.
Since the characteristic of the logarithm of the divisor is
larger than the characteristic of the logarithm of the dividend,
10 is added to and subtracted from the logarithm of the divi
dend. Note that the quotient has four significant figures
(see sect. 2). The zero must be written at the right to show
that the division has been carried out to four figures.
4. Divide .8354 by .04362.
SOLUTION. log .8354 = 9.9219 10
log .04362 = 8.6397  10
log quotient = 1.2822
quotient = 19.15.
Rough check. .84 * .044 = 19.
126 APPLIED MATHEMATICS
5. Find .6874 8 .
SOLUTION. log .6874 = 9.8372  10
3
29.5110  30
log .6874* = 9.511(3 * 10
.G874 3 = .3248.
Hough check. .7 3 = .34.
6. Find V.8231.
SOLUTION. log .8231 = 9.9155  10
= 19.9155  20
\ log .8231 = 9.9578  10
V.8231 = .9074.
Rough check. V^& = .9.
Before dividing log .8231 by 2, 10 was added and subtracted in
order that the resulting logarithm should have a 10. Similarly,
in extracting the cube root of a decimal add and subtract 20.
7. 8.114 x 56.83. 17. (1.237) 5 .
8. 5.161 x .0471. 18. (.8734) 8 .
9. 86.31 x .07832. 19. Vl983.
10. .0447 x .9142. 20. Vl835~.
11. 6.320 x 3.106 x 8.141. 2 1. ^U42.
12  mf 22. ^0687:
13 &W
'^
891 x 3.62 x .5162
68.14 x 2.657
12.73 x 9.684
67.83
.4971
2.056 x .8666
4 x 3.142 x (1.651)'
.5382
16. (4.931) 8 .
Oft
3
86.3 x 4.5 x 3.142 x 15 2 x 200
33000
27. Find the area of a rectangular lot 323.8 ft. long and
112.3 ft. wide.
COMMON LOGARITHMS 127
28. The base of a triangle is 72.14 ft. and its altitude is
8.482ft. Find its area.
29. Find the area of a square whose side is 71.18 yd.
30. The parallel sides of a trapezoid are 69.14 ft. and 38.15 ft.
If the altitude is 12.83 ft., find the area.
31. Find the surface and volume of brass cylinders and
prisms, wooden blocks, and so on.
32. Find the area of the blackboard in square meters.
33. Find the area of the athletic field.
34. Find the area of the ground covered by the school
buildings.
35. Find the area of the block in which the school building
stands.
36. Construct the logarithmic curve.
37. The area of a rectangle is 1689 sq. yd. and the length
is 58.12 yd. Find its width.
38. Find the side of a square whose area is 77.83 sq. ft.
39. The volume of a cube is 2861 cu. in. Find the length
of an edge.
40. What is the diameter of a piston which has an area of
172.8 sq. in. ?
41. Find the diameter of a circular plate of iron of the
same weight and thickness as a rectangular plate 3 ft. 4 in.
by 2 ft. 8 in.
42. A steel shaft is 3.5 in. in diameter and 12 ft. 9 in. long
Find its weight if 1 cu. in. of steel weighs .283 Ib.
CHAPTER
THE SLIDE RULE
75. Use of the slide rule.* In ordinary practical work it is
usual to make measurements and carry results in computations
only to three or four significant figures. With the slide rule
multiplications and divisions can be performed mechanically to
the degree of accuracy required in this work. The slide rule is
1 Z 345678 010 0,3040 5060705090100
A 1
111 III
\*
B 1
III III
JB
1 Z 345678 910 203040 30 60 70600) KX)
\ Z 3 A567O910
tl
1 1 1
r
SI
1 1 1
Ib
1 Z 3 A 5 6 7 9 10
FIG. 54
widely used in technical schools and in shops and laboratories
where there is a large amount of computation. It serves as a
check upon the numerical solution of problems, and should be
used by engineering students.
76. Description of the slide rule. The slide rule is simply
a table of logarithms arranged in such a way that they can be
dded and subtracted conveniently. The logarithms are not
anted on the slide rule, but each number on it stands in the
ution indicated by its logarithm. In Fig. 54 BC is the slide,
luated on the upper and on the lower edges. These gradu
were made in the following manner: CC was divided
^00 equal parts ; log 2 = .301, therefore 2 was placed at
(board slide rules ready for the student to cut and fit together may be
tf the Central Scientific Company, Chicago, at $1.10 per dozen.
128
THE SLIDE RULE 129
the 301st graduation ; log 3 = .477, therefore 3 was placed at
the 477th graduation; and so on for all the integers from 1
to 1000.
To read the numbers from 1 to 1000 we must go over the
rule from left to right three times. Thus we read first 1, 2, 3,
 , 10 ; then beginning at 1 again and calling it 10, we read 10,
20, 30, , 100 ; then beginning at 1 again and calling it 100,
we read 100, 200, 300,    , 1000.
77. Operations with the slide rule. It is not difficult to
learn to use the slide rule if at first the operations are per
formed with small numbers. Whenever in doubt about any
operation perform it first with small numbers.
I. Multiplication. Multiply 3 by 2. Move the slide so as to
set 1 C on 3 D ; then under 2 C read the product 6 on D. Note
that this is simply adding logarithms.
To find the product of two numbers, set 1 C on one of the num
bers on /), and under the other number on C read the product on D.
Sometimes in multiplying we must use the 1 at the right end
of scale C. Thus multiply 84 by 2. Set 1 at the right end of
scale C on 84 Z), under 2 C read 168 on D. We use the 1 at
the left end or the right end of scale C according as it brings
the second factor over scale D. In the example above, if we
had set 1 at the left end of scale C on 84, then 2 C would have
been off scale D.
The decimal point is placed by inspection. Thus, multiply
12.5 by 1.8. Set 1 C on 18 7), under 125 C read 225 on D. But
making an approximate multiplication mentally, 12 x 2 == 24 ;
hence we know that there are two integral figures in the prod
uct, giving 22.5 as the result. In all operations with the slide
rule the decimal point can be placed by making an approximate
mental computation.
II. Division. Divide 8 by 2. Set 2 C on 8 D, under 1 C read
the quotient 4 on D. Note that this is simply subtracting
logarithms.
130 APPLIED MATHEMATICS
To divide one number by another, set the divisor on scale C
on the dividend on scale D, under 1 C read the quotient on scale D.
The decimal point is placed by inspection. Thus divide
3.44 by 16. Set 16 C on 344 D, under 1 C read' the quotient
215 on D ; but we see that 3716 = 'about .2 ; hence the quo
tient is .215.
III. Combined multiplication and division. Find the value
24 x 9
of  Set 6 C on 24 Z>, under 9 C read the result 36 on
D. Study this operation till the separate parts are seen clearly
and understood. First the division of 24 by 6 is made by set
ting 6 C on 24 7), under 1 C we might read the quotient ; but
we want to multiply this quotient by 9. As 1 C is already
on this quotient we have only to read the product '36 on scale
D under 9 C.
An important problem under this case is to find the fourth
term of a proportion. Thus, in the proportion 6 : 24 = 9 : x,
24 x 9
* = __
Hence to find the fourth term of a proportion, set the first
term on the second, under the third read the fourth.
IV. Continued multiplication and division. Here for conven
ience we need the runner. This is a sliding frame carrying a
piece of glass which has a line on it perpendicular to the length
of the rule.
1. Find the value of 3 x 8 x 5.
Set 1 C at the right on 3 D, set runner on 8 C, set 1 C at the
right on the runner, under 5 C read 12 on D. Hence
3x8x5 = 120.
54
2. Find the value of
3x6
THE SLIDE RULE 131
Set 3 C on 54 D, set runner on 1 C, set 6 C on runner, under
1C read result 3 on D. Note that we have simply made two
divisions.
3. Find the value of
24 x 6
Set 24 C on 157), set runner on 48 C, set 6 C on runner, under
1 C read result 5 on D.
4. Find the value of
oZ
Set 32 C on 8 7), set runner on 1 r, set 1 C at right end of
slide on runner, set runner on 9 C, set 1 C on. runner, under 4 C
read result 9 on D.
In a similar manner any number of continued multiplications
and divisions may be performed.
V. Squares and square root. The graduations on scale A at
the top of the slide are arranged so that the square of every
number on scale C stands directly above it on scale A. Thus
above 2 is 4, above 3 is 9, and above 25 is 625. On scale A
the distances of the numbers from 1 at the left end of the scale
are proportional to the logarithms of the numbers as on scale C ;
but it is easier to learn to use scale A by noticing its relation
to scale C. We read from left to right 1, 2, 3, , 10, 20, 30,
, 100 ; then beginning at 1 again and calling it 100, we read
100, 200, 300,.., 1000, 2000, 3000, , 10,000. The first 4
is either 4 or 400, that is, either the square of 2 or 20 ; the
second 4 is either 40 or 4000, that is, either the square of 6.32
or of 63.2.
To square any number, find the number on scale C and read
its square directly above it on scale A.
To extract the square root of any number, find the number
on scale A and read its square root directly below it on scale C.
The upper scale is very convenient when multiplying or
dividing by square roots, finding the area of circles, and so on.
132 APPLIED MATHEMATICS
1. Find the value of 8 A/3.
Set 1 C at right end of scale on 34, under 8 C read result
13.85 on D.
8
2. Find the value of p '
V3
8 8V3
V3 3
Set 3 C on 3 A, under 8 C read result 4.61 on D.
~ , ., , f V8 xVl2
3. Find the value of 7=
V5
Set 5B on 84, under 12 B read result 4.38 on D.
4. Find the area of a circle whose radius is 4 ft.
Set 1 C on 4 Z>, above TT on B read the area, 50.3 sq. ft., on A.
PROBLEMS
1. Find the value of :
1. 78 x 5. 12.8 48.8 16.8 x 4.2
2. 38.4 x 25. 15 ' 2.93 ' 31.4
3. 8.63x424. 944 84x13 16V39
4.121x6.38. 16.3' 15 ' 33
2. Find the area of the rectangle whose dimensions are
3.26 in. by 4.21 in.
3. The area of a rectangle is 18.6 sq. cm. and its base i&
5.34 cm. Find its altitude.
4. Find the area of a circle whose radius is (a) 5 in. ; (U) l.
in. ; (c) 2.56 cm. ; (d) 3.22 ft.
5. Construct a curve to show the area of circles of radius
from 1 in. to 10 in.
6. Find the surfaces and volumes of brass cylinders, prisms
blocks of wood, and so on.
THE SLIDE RULE 133
7. To make 865 Ib. of admiralty metal, used for parts of
engines on naval vessels, 752.5 Ib. of copper, 43.3 Ib. of zinc,
and 69.2 Ib. of tin were melted together. Find the per cent
of each metal used.
8. 17 Ib. of copper, 85 Ib. of tin, 595 Ib. of lead, and 153 Ib.
of antimony were melted together to make 850 Ib. of type
metal. What per cent of each metal was used ?
9. If sea water contains 2.71 per cent of salt, how many
tons of sea water must be taken to give 100 Ib. of salt ?
10. The safe load in tons, uniformly distributed, for white
oak beams is given by the formula
where W = the safe load in tons, b = the breadth in inches,
d = the depth in inches, and I = the distance between the
supports in inches.
Construct a curve to show the safe load in tons for white
oak beams having a breadth of 3 in., distance between supports
13 ft., and depth from 3 in. to 15 in.
11. If w = the weight of 1 Ib. of any substance when sus
pended in water, and s its specific gravity, then
s = ~  , or w =
Lw
Construct a curve showing the weight of substances sus
pended in water, the specific gravity varying from .5 to 15.
CHAPTER XII
ANGLE FUNCTIONS
78. Angles. Let two lines AP and AM be coincident.
Suppose AP to revolve about the point A away from AM^
the amount of turn, indicated by the arrow, is called an angle.
The amount of turn is expressed in degrees. A complete turn
gives an angle of 360, a half turn 180, and a quarter turn 90.
In this chapter we will not consider angles greater than 90.
FIG. 55
The line AM which marks the beginning of the revolution
is called the initial line; the line AP which marks the ending
of the revolution is called the terminal line of the angle.
79. Triangle of reference. If from any point B in the
terminal line of the angle a perpendicular BC is dropped to the
initial line, the right triangle formed
is called the triangle of reference for
the angle. The perpendicular BC is
called the opposite side ; A C, the part
of the initial line cut off by the per
pendicular, is called the adjacent
side ; and AB, that part of the ter
minal line which belongs to the triangle of reference, is called
the hypotenuse.
184
FIG. 56
ANGLE FUNCTIONS
135
80. Sine, cosine, and tangent of
an angle. Given the angle A. Con
struct the triangle of reference, and
represent the lengths of the sides by
a, b, and c, set opposite the angles A,
By and C respectively.
BC^a
AB c
__ opposite side
hypotenuse
= sin A (by def
inition).
This ratio, called the
sine of angle A, is a
pure number which
is usually approxi
mate and expressed
as a decimal.
AC ^b
AB~~ c
_ adjacent side
hypotenuse
= cos A (by def
inition).
This ratio, called the
cosine of angle A, is
a pure number which
is usually approxi
mate and expressed
as a decimal.
FIG. 57
BC = q
AC~ b
__ opposite side
adjacent side
= tan A (by def
inition).
This ratio, called the
tangent of angle A, is
a pure number which
is usually approximate
and expressed as a
decimal.
These ratios sin A, cos A, and tan A are called functions of
the angle A because they change in value as the angle changes.
There are other functions of an angle, but as these three seem
to be the more important the discussion will be limited to them.
EXERCISES
1. Make an angle A and construct the triangle of reference.
Letter as before, and measure the sides a, b, and c as accurately
as possible in millimeters. Use the results of the measurement
to find the values of sin A, cos A, and tan A. Carry the divi
sions as far as the errors in the approximation justify, and
no farther.
2. Make another angle A' which differs from A. Calculate
its sine, cosine, and tangent in the same manner. Compare the
values of the two sines, the two cosines, and the two tangents.
If you were to continue the experiment, you would find that
the ratios change in value every time the angle changes in size.
136
APPLIED MATHEMATICS
3. Make an angle and drop perpendiculars from various
points on the terminal line to the initial line. Any one of the
right triangles may be considered a triangle of reference for
the angle. Find sin A from each triangle of reference. Com
pare the values. Should they all be equal? Why? Similarly
for cos A and tan A.
4. In a triangle of reference ABC could BC = 2 in.,
AB = 6 in., and A C = 5 in. ? Why ? Could any two sides
be chosen at random ? Why ? Could
one side be chosen at random?
Why?
81. Functions of 45. Construct
an angle of 45, and in the triangle
of reference make either AC 01 BC
1 unit long. Why is the other side
1 unit long ? Why is the hypotenuse
V2 units long ?
sin 45 =
V2
cos 45 =
V2
= 4(1.414)
= .707.
= i (1.414)
= .707.
FIG. 58
tan 45 = 
= 1.
This ratio is exact.
82. Functions of 30. Construct
an angle of 30, and in the triangle
of reference make the side BC oppo
site 30, 1 unit long. Why is the hy
potenuse AB 2 units long ? Why is
A C V units long ?
VI
2
=  (1732)
= .866.
cos 30 =
= .500.
This ratio is exact.
ANGLE FUNCTIONS
137
83. Functions of 60. Construct an angle of 60, and in
the triangle of reference make the side A C adjacent to 60,
1 unit long. Why is the hypotenuse AB 2 units long? Why
is BC V3 units long?
V3
fos mi" = t ran nu~ = 
1
sin 60 =
= \ (1732)
cos 60 =
= .500.
This ratio is exact.
tan 60 =
= 1.702.
10
Show how the functions of 60 can be
found from the triangle of reference for 30.
84. Table of angle functions. The func
tions of angles have been calculated and
tabulated. In solving problems the func
tions of the angle are taken from the table. The functions
of a few angles, 30, 45, 60, 90, should be memorized.
FIG. 60
PROBLEMS
1. A man standing 110 ft. from a tree on level ground
finds the angle of elevation of the top of the tree to be 37 20'.
How high is the tree, and how far is the man from the top of it ?
SOLUTION. Given
 = tan A.
o
a = I tan A
= 110 (.7627)
= 83.9 ft.
A = 37 20'.
b = 110 ft.
c =
 = cos A.
c
ft = ccos.4.
ft
cos A
110
.7951
= 138 ft.
Check this and all problems by constructing the triangle from the
given parts. Make a goodsized drawing to scale and measure the
computed parts.
138 APPLIED MATHEMATICS
2. A railroad track has a uniform slope of 5 to the
horizontal. How many feet does a train rise in going a mile ?
3. A ladder 24 ft. long rests against a wall. The foot of
the ladder is 4 ft. 4 in. from the wall. Find the height of the
top of the ladder.
4. The shadow of a tree is 38 ft. long when the angle of
elevation of the sun is 42. Find the height of the tree.
5. A ship is sailing northeast 12 mi. per hour. How fast is
she sailing east ?
6. A stick 8 ft. long stands vertically in a horizontal plane,
and the length of the shadow is 6 ft. 'What is the angle of
elevation of the sun ?
7. What is the slope of a mountain path if it rises 118 ft.
in a distance of 835 ft. along the path ?
8. The top of a lighthouse is 152 ft. above sea level. If
the angle of depression of a buoy is 12 15', how far from the
lighthouse is it ?
9. The chord of a circle is 4.4 in. and it subtends at the
center an angle of 38. Find the radius of the circle.
10. At a point 212 ft. from the foot of a column the angle
of elevation of the top of the column is found to be 24 28'.
What is the height of the column ?
11. A man 6 ft. tall stands 4 ft. 6 in. from a lamppost. If
his shadow is 17 ft. long, what is the height of the lamppost ?
12. A cable is attached to a smokestack 10 ft. below the top,
and to a pile 42 ft. from the foot of the stack. If the cable
makes an angle of 62 20' with the horizontal, find the height
of the stack.
13. From the top of a lighthouse 160 ft. above sea level two
vessels appear in line. If their angles of depression are 4 20 f
and 2 45' respectively, how many miles are they apart ?
ANGLE FUNCTIONS 139
14. As the angle of elevation of the sun increases from
35 15' to 64 25', how many feet does the shadow of a church
steeple 120 ft. high decrease ?
15. In the gable shown in the figure
angle BA F= 60, angle GFE = 30,
EG = 6 ft, and GE = 4 ft. Find AF,
16. The base A C of an isosceles
trapezoid is 100 ft., and the equal sides AD and CB make angles
of 60 with the base. The altitude is 40 ft. Compute the length
of the upper base and the area. Draw to scale and check.
17. The pitch of a roof (angle which the rafters make with
the horizontal) is 32. If the house is 22 ft. wide, find the
length of the rafters and the height of the gable.
18. A building 80 ft. long and 40 ft. wide has each side
of its roof inclined 40 to the horizontal. Find the area of
the roof.
19. Two towns A and B are at opposite ends of a lake. It is
known that a station P is 3 mi. from A and 2 mi. from B. If
the angle P^J3==3430' and angle PBA = 62 40', find the
distance between the towns.
20. Make a height or distance problem of your own and
solve it.
85. Logarithmic solutions. In the preceding problems the
numbers involved consist of only two or three figures ; hence
there would be little or no time saved in using logarithms.
However, when there are several figures in the numbers, and
there are three or more multiplications or divisions, logarithms
should be used.
The logarithms of the angle functions are found in exactly
the same way as are the logarithms of numbers. Thus, find
log sin 18 26'.
140 APPLIED MATHEMATICS
Mantissa log sin 18 30' mantissa
log sin 18 20' = 5015  4977
= 38.
38 x.6 = 23.
4977 + 23 = 5000.
..log sin 18 26' = 9.5000  10.
The sine and tangent of an angle increase as the angle
increases, hence the difference for the minutes is added to the
mantissa of the smaller angle taken from the table.
It is to be noted that the cosine of an angle decreases as the
angle increases ; hence the difference for the minutes is to be
subtracted instead of added.
Thus find log cos 24 48'.
Mantissa log cos 24 40' mantissa
log cos 24 50' = 9584  9579
= 5.
6 x .8 = 4.
9584  4 = 9580.
.. log cos 24 48' = 9.9580  10.
Given log tana; = 9.5946  10, find x.
21 20' 5917
21 2 5946
21 30' 5954
37)290(8
/. x = 21 28'.
This work should be done mentally. In class find logarithms
of the functions of many angles, and the angles from the log
arithms of the functions, as quickly as possible until this can
be done readily.
The sine and cosine of an angle are always less than 1.
Why? Hence the characteristic of the logarithm is 9 10,
8 10, and so on. The 10 is not printed in the table, but
should be written in computation.
ANGLE FUNCTIONS
141
PROBLEMS
1. In the right triangle ABC,
given C = 90,
A = 28* 34',
c = 48.32 ft.
Find a and b.
SOLUTION.
a . A
 = sin A.
c
a = c sin A.
logc =
log sin A =
loga =
a =
Before looking up any logarithms always make out an outline
as above.
 = sin A.  = cos A.
c c
a = c sin A. b = c cos A.
log c = 1.6841 log c = 1.6841
log sin A = 9.6796  10 log cos A = 0.9436  10
log a = 1.3637 log b = 1.6277
a = 23.11 ft. ft = 42.43 ft.
Check. It may be as much work to check a problem as to solve
it, but an answer is absolutely worthless unless it is known to be
correct. What is the advantage of knowing how to work problems
if you cannot get correct results ?
5. th.).
= (c  ft) (c + 6).
c= 5.89 log = 0.7701
c + 6 = 90.75 log = 1.9579
log a 2 = 2.7280
log a = 1.3640
a = 23.12.
A difference of 1 in the last figure may be expected since the
logarithms are only approximate*
142 APPLIED MATHEMATICS
2. Two trees M and N are on opposite sides of a river. A
line NP at right angles to MN is 432.7 ft. long and the angle
NPM is 52 27 '. What is the distance from Mto N ?
3. From the top of a building 156.4 ft. high the angle of
depression of a street corner is 18 46'. Find the horizontal
distance from the street corner to the building.
4. To find the height of the Auditorium tower a distance
of 311.2 ft. was measured from the foot of the tower and the
angle of elevation of the tower was found to be 40 57'. Find
the height of the tower.
Solve the following right triangles, two parts being given :
5. a = 146.8, b = 203.3. 9. c = 110.9, a = 64.21.
6. I = 49.74, A = 53 38 f . 10. b = 8.226, c '= 12.15.
7. c = 94.53, B = 62 51'. 11. c = .02936, a = .01153.
8. c = 436.5, A = 74 11'. 12. a = .9681, A = 42 17'.
13. Find the side of an equilateral triangle inscribed in a
circle of radius 52.18 in.
14. The side of an equilateral triangle inscribed in a circle
is 14.26 in. Find the radius of the circle.
15. If a side of a regular pentagon is 30.24 in., find the
radius of the circumscribed circle, and the apothem.
16. A regular pentagon is inscribed in a circle of radius
11.32 in. Find a side and the apothem of the pentagon.
17. The apothem of a regular polygon of 12 sides is 21.26 ft.
What is the perimeter ?
18. The perimeter of a regular octagonal tower is 168.4 ft.
What is the area of the base of the tower ?
19. A regular octagonal column is cut from a circular cylinder
whose diameter is 18.32 in. Find the area of a cross section of
the column.
20. A side of a regular hexagon inscribed in a circle is
28.43 ft. Find a side of a regular decagon inscribed in the
same circle.
ANGLE FUNCTIONS
143
86, Area of triangles. In triangle
ABC, h is the perpendicular from
C to c.
Area triangle
A EC = \ base x altitude
The area of a triangle equals one  = sin.A.
half the product of two sides and the , __ , . .
sine of the included angle.
PROBLEMS
1. Find the area of a triangle ABC, given a = 42.84 ft.,
c = 76.31 ft., and B = 29 18'.
SOLUTION. 2 area = ac sin B.
log a = 1.6318
log c = 1.8826
log sin B = 9.6896  10
log 2 area = 3. 2040
2 area = 1600.
area = 800 sq. ft.
Find the area of the following triangles. Check by finding
the area twice, using different angles :
2. a = 34.36, I = 110.5, c = 98.32, A = 17 43', C = 60 36'.
3. a = 88.48, b = 58.59, c = 54.38, = 40 10', C = 36 47'.
4. a = 1.432, & == 1.583, c = 1.610, 4 = 53 17', = 62 24'.
5. a = 3.207, i = 2.367, c = 1.435, B = 42 55', C = 24 22'.
6. Find the area of a triangle XYZ, given x = 184.2 ft.,
y = 381.3 ft., and Z = 51 24'.
7. The vertical angle of an isosceles triangle is 75 18' and
the equal sides are 16.46 ft. long. Find the area of the triangle.
8. What is the area of a parallelogram if two adjacent sides
are 243.6 yd. and 315.4 yd. and the included angle is 35 40' ?
144
APPLIED MATHEMATICS
9. Two streets make an angle of 53 18' with each other.
The corner lot between them has a frontage of 286 ft. on one
street and 324 ft. on the other. Draw to scale and find the area
of the lot.
10. Two railroads cross at an angle of 21 25'. From a point
on one of them 100 rd. from the crossing how must a fence
be run so that the inclosure shall contain 10 A. ?
11. The survey of a field gave the following data:
EA = 420 ft.
EB = 865 ft.
EC = 875 ft.
E D = 650 ft.
Z BEG = 36.
Z CED = 20.
FIG. 65
Draw the field to scale and
find its area.
12. A surveyor set his transit over the corner A of a field
ABCD and found the angle DAC = 40 12', and angle CAB =
70 54'. A D is 52.8 rd., A C is 86.3 rd., and AB is 38.4 rd. Draw
to scale and compute the area of the field.
87. Law of sines. In the triangle ABC let h be the perpen
dicular from the vertex C to the side c.
 * *
 = sin A. and  = sin B.
b a
By division,
a __ sin A
b sin B
a _ ^
sin A sin B
(i)
(2)
What algebraic operations were used to derive (2) from (1) ?
What theorem in geometry could be used for this purpose ?
ANGLE FUNCTIONS 145
By dropping a perpendicular from A to a we may obtain, in
a similar manner,
sin C sin B
a b c
sin A sin B sin C
LAW OF SINES. In any triangle the sides are proportional to
the sines of the opposite angles.
When a side and two angles of a triangle are given we may
find the other two sides by this law.
PROBLEMS
1. In a triangle ABC given A = 36 56', B = 72 6', and a =
36.74. Find b and c.
SOLUTION. C = 180  (A + B) = 70 58'.
b a c a
sin B sin A sin C sin A
, a sin B a sin C
b = J* c = 
sin A sin ^i
loga= 1.5652 loga = 1.5652
log sin B = 9.9784  10 log sin C = 9.9756  10
11.5486  10 11.5408  10
log sin A = 9.7788  10 log sin ^ = 9.7788  10
log 6= 1.7648 logc= 1.7620
6 = 58.19. c = 57.81.
Solve the following triangles and check by drawing to scale :
2. A = 44 59', B = 62 52 f , a = 7.942.
3. A = 50 24'; C = 68 35', b = 12.63.
4. JB = 7246', C = 4144', c = 203.6.
5. A = 61 18', JB = 58 32', 6 = 84.03.
6. To find the distance from a point A to a point P across a
river, a base line AB 1000 ft. long was measured off from A.
The angles BAP and A BP were found to be 36 18' and 62 35'
respectively. Compute the distance A P.
146
APPLIED MATHEMATICS
7. On board two ships half a mile apart it is found that the
angles subtended by the other ship and a fort are 84 16' and
78 38' respectively. Find the distance of each ship from the
fort.
8. M and N are stations on two hilltops 3684 ft. apart, and
P is a station on a third hill. The angles NMP and MNP are
observed to be 50 42' and 63 24' respectively. Find the dis
tances MP and NP.
C
88. Law of cosines. In triangle ABC,
h is the perpendicular from C to c.
In triangle on left, 6 2 = A 2 + &' 2 , (1)
and
or
In triangle on right,
 = cos^l,
Substituting (2), = ft 2 + &' 2 + c 2  2 be cos ,4.
Substituting (1), a 2 = V + c 2  2 be cos A.
Similarly, by dropping perpendiculars from A and B we get
ft 2 = a 2 + c 2 2ac cos B.
c* = a* + b*2ab cos C.
LAW OF COSINES. In any triangle the square of any side is
equal to the sum of the squares of the other two sides less twice
the product of these two sides and the cosine of the included
angle.
PROBLEMS
1. Find a in the triangle ABC, given b = 6 in., c = 5 in., and
A = 29 15'.
SOLUTION. a 2 = 6* + c 2 2 be cos 4
= 36 + 252 x 6x 5 x .8725
= 8.65.
a = 2.9 in.
ANGLE FUNCTIONS 147
2. In triangle ABC, find A if a = 7, b = 8, = 9.
SOLUTION. a 2 = Z> 2 + c 2 2 fee cos^l.
2 be
^64 + 8149
"" 2x8x9
= .6667.
A = 48 11'.
3. Find B and C in the triangle in Problem 2 and check by
adding the three angles.
Solve the following triangles and check by drawing to scale
or otherwise :
4. a = 10, b = 12, c = 14. 6. b = 21, c = 19, ^1 = 4857 f .
5. a = 4, b = 5, c = 6. 7. a = 14, b = 12, c = 60.
8. Two ships leave a dock at the same time. One sails east
12 mi. per hour and the other northeast 14 mi. per hour. How
far will they be apart at the end of 5 hr. ?
9. From a point 5 mi. from one end of a lake and 4 mi. from
the other end, the lake subtends an angle of 56 8 ; . What is
the length of the lake ?
10. A and B are two stations on opposite sides of a moun
tain, and C is a station on top of the mountain from which A
and B are visible. If CA = 4.2 mi. and CB = 3.1 mi., and angle
A CB = 88 12', find the distance from A to J3, the three stations
being in the same vertical plane.
89. Triangle of forces. The weight W at the end of the
boom is held in position by three forces : (a) the force of gravity
acting downward ; (i) the tension (pull) in the tie ; (c) the
thrust (push) of the boom. The tension in each side of the
triangle is proportional to the lengths of the sides. The ten
sion in the mast is always taken equal to the load W ; and the
tension per foot is the same in each side of the triangle. Thus in
Fig. 68, \tW= 2000 lb., AB = 10 ft., and BC = 16 ft., the tension
in the mast AB = 2000 lb. and the tension per foot = 200 lb
148
APPLIED MATHEMATICS
Therefore the compression in the boom = 16 x 200 = 3200 Ib.
The tie AC = VlO 2 + 16 2 = V366 = 18.9 ft., a,nd the tension
in AC = 18.9 x 200 = 3780 Ib.
~ 7 A 77e
Check.
2000*= 4,000,000
3200 2 = 10,240,000
14,240,000
3780 2 = 14,290,000
FIG. 68. A SIMPLE CRAXE
Exercise. Put two screw eyes in the wall 80 cm. apart and
construct a model of a crane, using a meter stick, string, and a
spring balance, as shown in Fig. 69. Compute the tension for
T
QQcm.
FIG. 69
different weights and check by the readings of the spring
balance. After a weight has been attached the string should
be shortened enough to make the string or the meter stick
perpendicular to the wall in order to form a right triangle.
PROBLEMS
1. The mast of a crane is 12 ft. long and the tie 18 ft. The
boom is horizontal and supports a load of 2400 Ib. Find the
tensions in the boom and tie*
ANGLE FUNCTIONS
149
2. The tie of a crane is horizontal. If it is 24 ft. long and
the boom is 30 ft. long, find the tension in the mast, boom, and
tie for a load of 4 T.
3. The tie of a crane makes an angle of 30 with the mast,
and the boon* is horizontal. If the boom is 20 ft. long and the
load is 3000 lb., find the tension in the mast, tie, and boom.
4. The boom of a crane is 16 ft. long and makes an angle
of 40 with the mast. The tie is horizontal. Find the tension
in the mast, boom, and tie for a load of 2 T.
5. The boom of a crane is 20 ft. long, and when it is hori
zontal the tie is 30 ft. long. If the tie can stand a strain of
4200 lb., find the greatest load that can
be lifted when the boom is horizontal.
6. The bracket BCD carries a load of
400 lb. at D. Find the stresses in BC,
CD, and BD.
7. An arc lamp weighing 20 lb. is
hung on a pole, as shown in Fig. 71.
Find the stresses in MP and NP.
8. A weight of 96 lb. is attached to a
cord which is secured to the wall at a point
A and is pushed out from the wall by a
horizontal stick BC. If ^C = 6ft. and
angle BAG = 38, find the tension in AB
and the pressure on BC.
9. A canal boat is kept 20 ft. from the
towpath and the towline is 72 ft. long. If
there is a pull of 144 lb. on the line, what
is the effective pull ?
FIG. 70
FIG. 71
FIG. 72
SOLUTION. Let C, Fig. 72, be the position of the canal boat.
AB = V72 2  20* = 69.2 ft.
^t = 2 lb., the tension per foot in AC.
.. 69.2 x 2 = 138.4 lb., the effective pull.
150
APPLIED MATHEMATICS
10. The pull on the towline of a canal boat is 400 Ib. and
the line makes an angle of 10 with the direction of the boat.
How much of the pull is effective ? How much is at right
angles to the direction of the boat?
11. A boat is pulled up the middle of a stream 60 ft. wide
by two men on opposite sides, each pulling with a force of
100 Ib. If each rope, attached to the bow of the boat, is 40 ft.
long, find the effective pull on the boat.
12. Each of two horses attached to a load is pulling with a
force of 200 Ib. If they are pulling at an angle of 60 with
each other, what is the effective pull on the load ?
13. Attach two spring balances to the wall, as shown in
Fig. 73, with 10 or 12 ft. of cord be 4 j>
tween them. At the center of the cord
attach an 8lb. weight. Bead each bal
ance for the tension in AC and BC.
Suppose ^C=:6ft. and DC = 4ft.
Compute the stress in AC.
SOLUTION. of 8 = 4 Ib., stress in DC.
^ = 1 Ib. per foot, stress in DC.
1x6 = 6 Ib., stress in A C.
Compare with result of the experiment. Make other experi
ments with different lengths of cord until the reason for the
method of computation is understood. A \ &' C s'
14. A man weighing 180 Ib. sits in
the center of a hammock 12 ft. long.
If the supports are 10 ft. apart, find
the pull on the hammock.
SOLUTION. CD = V6 2  5 2 = 3.32 ft.
of 180 = 90 Ib., pull in CD.
11 * 4.
= pull per foot.
FIG. 73
FIG. 74
3.32
90 X 6
3.32
163 Ib. = pull on hammock.
ANGLE FUNCTIONS 151
Check. cos x = 
= .8333.
x = 33 34'.
Pull in CD .
Pull in CD = ainx x pull in AD
= .553 x 163
= 90.2 Ib.
90.2 x 2 = 180.4 Ib., weight of the man.
15. Two horses attached to a load are pulling with the same
force at an angle of 60 with each other. If the combined
effective pull on the load is 400 Ib., how many pounds is each
horse pulling ?
16. Connect two light strips of wood 60 cm. long, AB and EC
(Fig. 75), by a hinge at B, and put casters at A and C. Put a
cord and spring balance between A B
and C, as shown in the figure. Hold
the frame vertical, measure BD and
AC, and read the balance, when
BD = 48 cm. and AD = 36 cm. At
*~* '
tach an 8lb. weight at B and make FIG. 75
A C = 72 cm. Read the balance, and
subtract the first reading to get the tension in AC due to the
8lb. weight. Compute the tension in A C as follows :
3*g = y 1 ^ Ib., tension per centimeter in BD.
T V X 36 = 3 Ib., tension in AD.
.*. tension in AC = 3 Ib.
Compare with the result of the experiment. Make other ex
periments with different weights and distances AC, until the
reason for the method of computation is understood.
17. A pair of rafters supports a weight equivalent to 800 Ib.
at the ridge. The pitch of the roof is 30 and the width of
the building is 30 ft. Find the tension in the tie through the
foot of each rafter.
152
APPLIED MATHEMATICS
18. The width of a house is 24 ft. and the rafters are 16 ft.
long. If the rafters support a weight equal to 600 Ib. at the
ridge, find the stress in the rafters.
19. A bridge truss ABC supports
a weight of 300 Ib. per foot horizon
tally. The span is 30ft. long. If
CD = 10 ft., find the stresses in AC
and AB. (The load at D equals one
half the total load.)
20. ABC (Fig. 77) is an inverted
kingpost truss. AB = 2$ ft., and the
angles CAB and ABC = 40. If the
load at D is 4 T., find the stresses in
AC and AB.
FIG. 76
CHAPTER XIII
GEOMETRICAL EXERCISES FOR ADVANCED ALGEBRA
90. A figure should be drawn for each exercise, letters or
numbers put on the lines in the figure, and the equations set
up from the figure. Check by drawing to scale and measuring
the required parts. The first exercises involve square roots,
since radicals are reviewed early. Some of the exercises should
be worked out in notebooks, with emphasis placed on accuracy
in drawing and neatness in arrangement.
1. Construct a graph for the squares of numbers from to
13. Units : horizontal, 1 large square = 1 ; vertical, 1 large
square = 10. What is the equation of the curve ? Find V2,
V6, V7J5, V&25, VlO, Vl2, and Vl2J5 to three decimal
places and check by the graph.
2. Find the diagonal of a square whose side is 12 ($).
FIG. 78
SOLUTION. </ a = 12 2 + 12 2 (Pythag.th.) d* = * 2 + x 2 (Pythag. th.)
= 288 = 2 ^
x 2
= 16.97.
158
154 APPLIED MATHEMATICS
3. Find the side of a square whose diagonal is 5 (d).
4. The side of an equilateral triangle is 4(s). Find the
altitude and area.
5. The altitude of an equilateral triangle is 6(&). Find the
side and area.
6. The area of an equilateral triangle is 24 (a). 'Find the
side and altitude.
7. Find the area of a regular hexagon whose side is 3 (s).
8. Find the area of a regular hexagon whose apothem is
2 (A).
9. Find the side and apothem of a regular hexagon whose
area is 36 (a).
10. A starshaped figure is formed by constructing equilateral
triangles outwardly on the four sides of a square. If the area
of the entire figure is 100, find a side of the square.
11. Squares are constructed outwardly on the sides of a
regular hexagon. If the area of the entire figure is 72, find a
side of the hexagon.
12. From a square whose side is 12 (s) a regular octagon is
formed by cutting off the corners. Find a side of the octagon.
13. The edge of a cube is 5(0). Find a diagonal.
14. The diagonal of a cube is S(d). Find an edge.
15. Find the diagonal of a rectangular parallelepiped whose
edges are 4, 5, and 6 (a, 6, and c).
16. Find the side of an equilateral triangle whose area equals
the area of a square whose diagonal is 6 V50.
17. Two sides of a triangle are a and b. Show that the area
is ab when the included angle is 30 or 150.
18. If two sides of a triangle are a and b, show that the area
is J V3a6 when the included angle is 60 or 120.
19. If two sides of a triangle are a and &, show that the area
is \ V2 ab when the included angle is 45 or 135.
GEOMETRICAL EXERCISES
20. The sides of a triangle are 30, 60, and 80 (a, ft, and c).
Find the segments of each side formed by the bisector of the
opposite angle.
21. The shadow cast upon level ground by a certain church
steeple is 27 (37) yd. long, and at the same time the shadow of
a vertical rod .5(7) ft. high is 3(6) ft. long. Find the height
of the steeple.
22. The footpaths on the opposite sides of a street are 30 ft.
apart. On one of them a bicycle rider is moving at the rate
of 15 mi. per hour. If a man on the other side, walking in the
opposite direction, regulates his pace so that a tree 5 ft. from
his path continually hides him from the rider, at what rate
does he walk ?
23. One side of a triangle is divided into two equal parts
and through the point of division a line is drawn parallel to
the base. Into what fractional parts is the triangle divided ?
Similarly, when the side is divided into 3, 4, 5, , n equal parts ?
24. Find the side of an equilateral triangle if the center of
gravity is 2 (x) in. from the vertex.
25. What part of a triangle lies between the base and a line
through the center of gravity parallel to the base ?
26. One side of a triangle is 10 (s) in. Where must a point
be taken in the given side in order that a line drawn through
it, parallel to another side, will divide the triangle into two
areas whose ratio is 3 : 4 (in : ri) ?
27. The bases of a trapezoid are 16 and 10 (6j and ft t2 ) and the
altitude is 6(/*). Find the area of the triangle formed by
producing the noliparallel sides of the trapezoid.
28. Find the side of the square inscribed in the triangle
whose base is 12 (ft) and altitude is 6(Ji).
29. A rectangle whose length is twice its breadth is inscribed
in an equilateral triangle. Find the area of the rectangle if a
side of the triangle is 2.
156 APPLIED MATHEMATICS
30. Find the area of a trapezoid, given the bases 36 and 56
(j and ig) and the altitude 12 (A).
31. The bases of a trapezoid are 73 and 67 (l\ and i a ) and
each of the nonparallel sides is 17 (c). Find the area.
32. One diagonal of a trapezoid ifc 10 (d). The segments of
the other diagonal are 6 and 9(ra and ri). Find the segments
of the first diagonal.
33. A trapezoid contains 480 (65) sq. ft. and its altitude is
20 (10) ft. Find the bases of the trapezoid if one of them is
4 (6) ft. longer than the other.
34. Find the area of a rectangle if its diagonal is 5Q(d) ft.
and the sides are in the ratio 3 : 5 (m : ri).
35. The dimensions of a rectangle are 64 and 58(6 and K)
respectively. If the length is diminished by 10 (m), how much
must the breadth be increased in order to retain the same area ?
36. A rectangle is 8 (h) in breadth and its diagonal is 20 (d).
Upon the diagonal as a base a triangle is constructed whose
area is equal to that of the rectangle. Find the altitude of the
triangle.
37. The ratio of the diagonals of a rhombus is 7 : 5 (m : ri)
and their sum is 16 (&). Find the area of the rhombus.
38. The sides of a right triangle are x, x + 7, and x + 8.
Find them.
39. Two telegraph poles 25 and 30 ft. high are 80 ft. apart
on level ground. Find the length of the wire.
40. The chord of a circle is 8(c) and the height of the seg
ment is 2 (h). Find the radius.
41. In a circle whose radius is 12(r)in. a chord 4(c) in. is
drawn. Find the height of the segment.
42. Two chords 48 and 14 mm. long are on opposite sides of
the center of a circle. If they are 31 mm. apart, what is the
diameter of the circle ?
GEOMETRICAL EXERCISES 157
43. Two parallel chords on the same side of the center of a
circle are 48 and 14 (14 and 4) in. long. If the diameter of
the circle is 50(16) in., find the distance between the chords.
44. Find the common chord of two equal circles of radius
8 (r) in. if each circle has its center on the circumference of
the other.
45. Two chords AB and CD intersect at E within a circle.
If AE = 10(4), BE = 12 (9), and CD = 23(12), find CE and ED.
46. From a point without a circle a secant and a tangent are
drawn. If the external segment of the secant is 6 (m) and the
internal segment is 18 (n), find the tangent.
47. If from a point without a circle two secants are drawn
whose external segments are 3 and 4(3.2 and 5.5), and the in
ternal segment of the latter is 17(4.7), what is the internal
segment of the former ?
48. The distance between the centers of two circles whose
radii are 5 and 8(r t and r 2 ) is 26 (d). How far from the center
of each circle does their common tangent intersect the line of
centers ? (Two solutions.)
49. The radii of two circles are 7 in. and 4 in. The distance
between their centers is 12 in. Find the length of the common
internal and external tangents.
50. Find the radius of a circle if the numerical measure of
the area equals the measure (a) of the radius; (6) of the
circumference.
51. Find the side of the largest square piece of timber that
can be cut from a log 14 ft. in circumference.
52. A rectangle and a circle have equal perimeters. Find the
difference of their areas if the radius of the circle is 9 (12) in.
and the width of the rectangle is f () its length.
53. A circle whose radius is 8(r) has one half of its area
removed by cutting a ring from the outside. What is the width
of the ring ?
158 APPLIED MATHEMATICS
54. Show that the ratio of the square inscribed in a semi
circle to the square inscribed in the entire circle is 2 : 5.
55. Show that the ratio of the square inscribed in a semi
circle to the square inscribed in a quadrant of the same circle
is 8: 5.
56. What is the ratio of the square inscribed in a quadrant
of a circle to the square inscribed in the entire circle ?
57. How much must be added to the circumference of a wheel
whose radius is 2 (r) to make the radius 1 (in) longer ?
58. If an electric cable were laid around the earth at the
equator, how many feet would have to be added if the cable
were raised 10 ft. above the surface of the earth ? ,
59. A quartermile running track is to be laid out with straight
parallel sides and semicircular ends. The track is to be 10 ft.
wide, and the distance between the outer parallel edges is to
be 220 ft. What must be the extreme length of the field so
that a runner may cover the exact quarter of a mile by keeping
in the center of the track ?
60. In any triangle whose sides are a, b, and c derive a
formula for the square of the side opposite an acute angle.
61. Derive a corresponding formula for the square of the
side opposite an obtuse angle.
62. In a triangle whose sides are 7, 8, and 9(4, 5, and 6)
find the projections of the sides 7 and 8(4 and 5) on 9(6).
63. In a triangle whose sides are 10, 12, and 18(40, 80, and
100) find the projections of the sides 18 and 10(100 and 40)
on 12 (80).
64. The sides of a triangle are 4, 5, and 7(70, 90, and 100).
Find the altitude to base 7 (100).
65. Find the area of a triangle whose sides are 8, 12, and 15
(20, 25, and 30).
GEOMETRICAL EXERCISES 159
66. Find the length of the common chord of two circles
whose radii are 5 and 8 (10 and 17) and the distance between
whose centers is 10(21).
67. The base and altitude of a triangle are 8 and 6 (b and A) in.
respectively. If the base be increased 4 (c) in., how much must the
altitude be diminished in order that the area remain the same ?
68. Through the vertex of a triangle whose area is 120
(100) sq. in. a line is drawn dividing it into two parts, one
containing 24 (12) sq. in. more than the other. What are the
segments into which the base is divided if the whole base is
20(14)in.?
69. The base of a triangle is 6 in. and the altitude is 5 in.
Find the change in area if the dimensions are (ct) increased by
3 in. and 2 in. respectively ; (/>) diminished by 3 in. and 2 in.
respectively ; (r) one increased by 3 in. and the other dimin
ished by 2 in. What is the per cent of change in each case ?
70. At a distance of 60 ft. from a building the angles of
elevation of the top and bottom of a tower on the building are
45 and 30 respectively. Find the height of the tower.
71. If the shadow of a tree is lengthened 60 (a) ft. as the
angle of elevation of the sun changes from 45 to 30, how
high is the tree ?
72. A ladder resting against a vertical wall forms an angle
of 60 with the level ground. If the foot of the ladder is drawn
10 ft. farther out from the wall, the angle formed with the
ground is 30. Find the length of the ladder.
73. In a right triangle whose legs are 12 and 16(20 and 40)
find the length of the perpendicular from the vertex of the right
angle to the hypotenuse, and the segments of the hypotenuse.
74. The legs of a right triangle are 9 and 12 (a and i). Find
their projections on the hypotenuse.
75. The projections of the legs of a right triangle on the
hypotenuse are 5f and 9$ (m and ri). Find the legs.
160 APPLIED MATHEMATICS
76. The sum of the three sides of a right triangle is 60 (140) in.
and the hypotenuse is 26 (58) in. Find the legs and the per
pendicular from the vertex of the right angle to the hypotenuse.
77. In a right triangle the perpendicular from the vertex of
the right angle to the hypotenuse is 2 (p) and the ratio of the
segments of the hypotenuse is 4:9(ra:7i). Find the area of
the triangle.
78. The perpendicular from the vertex of the right angle
in a triangle makes the segment of the hypotenuse adjacent to
the longer leg equal to the shorter leg. Find the area of the
triangle when the hypotenuse is 2(c).
79. Two roads cross at right angles at A. 5 mi. from A on
one road a man travels toward A at the rate of 3 ini. per hour.
6 mi. from A on the other road another man travels toward A
at the rate of 6 mi. per hour. When and where will the men
be 2 mi. apart ?
80. Two trains run at right angles to each other, one at 30
and the other at 40 mi. per hour. The first train is 15 mi. from
the crossing and is moving away from it ; the second is 60 mi.
from the crossing and moving toward it. When and where
will the trains be 50 mi. apart ?
81. How much must the length of a rectangle 16 by 12
(b by h) be increased in order to increase the diagonal 4 (c) ?
82. The difference between the diagonal of a square and
one of its sides is 2.071 (a) in. Find one side and the area.
83. Find the sides of a rectangle if the perimeter is 34 (p) in.
and the diagonal is 13 (d) in.
84. The diagonal and longer side of a rectangle are together
5 times the shorter side, and the longer side exceeds the shorter
by 7. What is the area of the rectangle ?
85. The perimeter of a right triangle is 24(216) and the
area is 24(1944). Find the sides. (Solve with one, then with
two, and then with three unknowns.)
GEOMETRICAL EXERCISES 161
86. From a square piece of tin a box is formed by cutting
6in. squares from the corners and folding up the edges. If
the volume of the box is 864 (1944) cu. in., what was the size
of the original piece of tin ?
87. The sum of the volumes of two cubes is 35 (2728) cu. in.
and the sum of an edge of each is 5 (22) in. Find their diagonals.
88. If the edges of a rectangular box were increased by 2, 3,
and 4 in. respectively, the box would become a oube and its
volume would be increased by 1008 cu. in. Find the edges of
the box.
89. The diagonal of a box is 125 in., the area of the lid is
4500 sq. in., and the sum of the three coterminous edges is
215 in. Find the three dimensions.
90. A rectangular piece of cloth shrinks 5 per cent in length
and 2 per cent in width. The shrinkage of the perimeter is
38 in. and of the area 862.5 sq. in. Find the dimensions of
the cloth.
91. If a given square be subdivided into four (n a ) equal
squares and a circle inscribed in each of these squares, the
sum of the areas of these circles will equal the area of the
circle inscribed in the original square.
92. In a square whose side is 16 a square is inscribed by
joining the midpoints of the sides in order. In this square
another square is inscribed in a similar manner. This is re
peated indefinitely. Find the area of the first eight inscribed
squares.
93. In any triangle a triangle is inscribed by joining the
midpoints of the sides. Another triangle is inscribed in this
inscribed triangle in a similar manner, and so on indefinitely.
How does the area of the sixth triangle compare with the area
of the first ?
94. An equilateral triangle is circumscribed about a circle
of radius 4 (r) Find a side of the triangle.
162 APPLIED MATHEMATICS
95. A circle is inscribed in a triangle whose sides are 5, 6,
and 7 (a, b, and c). Find the distances of the points of contact
from the vertices of the triangles.
96. Find the radius of the circle inscribed in an isosceles
trapezoid whose bases are 6 and 18 (b l and i 2 ).
97. A boy places his eyes at the surface of a 'smooth body
of water and finds that the top of a float
1 mi. away is just visible. How far does
the float project above the water ?
SOLUTION, x (8000 + x) = I 2 .
x 2 + 8000 x = 1.
Since x is very small compared with the
diameter of the earth, we may drop x 2 .
5280 x 12 .
98. A man 6 ft. tall standing on the seashore sees an object
on the horizon. How far, in miles, is the object away from
the shore ?
99. From the top of a cliff 60 ft. high is barely visible the
funnel of a steamer, known to be 30 ft. above the surface. How
far is the steamer from the cliff ?
100. The bridge of a steamer is 40 ft. above the water. How
far apart are two such steamers when the bridge of one is just
visible from the bridge of the other ?
101. In a circle whose radius is 5(r) a chord 8(c) is drawn.
Find the length of the chord of one half the arc.
102. Find the side of a regular polygon of twelve sides in
scribed in a circle of radius 6(r).
103. Find the side of a regular octagon inscribed in a circle
of radius 8(r).
104. The area inclosed by two concentric circles is 60 (a) sq. ft.
If the radius of the inner circle is 5 (r) ft., find the radius of
the outer circle.
GEOMETRICAL EXERCISES 163
105. Three men buy a grindstone. If the diameter is 3 (d) ft.,
how much of the radius must each man grind off in order to
obtain his share ?
106. The sum of the circumferences of two circles is 56^ ft.
and the sum of their areas is 141^ sq. ft. Find their radii.
(IT = ^.)
107. The area of a rectangular table whose length is 5 ft.
more than its breadth is equal to the area of a circular table
whose radius is 3 ft. Find the dimensions of the table.
108. On a straight line 8 (w) cm. long as a diameter describe
a semicircle. On each half of the given line as diameters de
scribe semicircles within the other semicircle. Find the radius
of the circle which is tangent to the three semicircles.
109. An increase of 2 ft. in one side of an equilateral triangle
enlarges the area by 4 V3 sq. ft. Find the side of the triangle.
110. The sides of a rectangle are 8 and 12 (b and Ji). Find
the area of an equilateral triangle whose sides pass through
the vertices of the rectangle.
111. The number which expresses the area of a right triangle
is 1 greater than the number which expresses the length of
the hypotenuse. Show that the sum of the legs of the triangle
is 2 greater than the hypotenuse.
112. Find the side of the square inscribed in the common
part of two circles of radius 6(r), if the center of each circle is
on the circumference of the other.
113. Two parallel lines are 8 and 12 in. long respectively,
and are 4 in. apart. Find the area of the two triangles formed
by joining their opposite extremities.
114. How many squares may be inscribed in a triangle
whose sides are 9, 12, and 15?
115. In a triangle whose sides are 3, 3, and 4 (a, a, and c) a
line drawn across the sides 3 and 4 (b and c) bisects both the
perimeter and the area. How far from the vertex does the line
cut the sides ?
CHAPTER XIV
VARIATION
91. Direct variation. If a man earns $25 per week, the
amount he earns in a given time equals $25 multiplied by the
number of weeks.
a = 25 n.
Number of weeks
1
2
3
4
5
Amount earned
25
50
75
100
126
As the number of weeks changes the amount earned changes,
but always the amount earned divided by the number of weeks
equals 25.
^ = 25.
n
We may state this fact in another way and say that the amount
earned varies directly as the number of weeks, or a oc n.
If a steel rail weighs 100 Ib. per yard, the weight of the rail
equals the length in yards multiplied by 100. w = 100 /, or
w
y = 100. Since the weight divided by the length is constant,
100, we may state this fact in the form of variation, and say
that the weight varies directly as the length, or w oc I.
Note that in direct variation an increase in one variable
makes an increase in the other. The greater the length the
greater the weight; the less the length the less the weight.
Double the length and the weight is doubled ; one fourth of
the length gives one fourth the weight,
164
VARIATION 165
92. Definition. One number varies directly as another when
the quotient of the first divided by the second is constant.
Exercise. On a sheet of squared paper take the lines at the
bottom and left for the axes of x and y respectively, and let
one square each way equal one. Draw a straight line from the
lower left corner to the intersection of any two heavy lines.
Make a table for the values of x, y, and ~ for points on this
x .
line, taking x = 1, 2, 3, , 10. Is the quotient of y divided
by x constant ? Does y vary directly as x ? What equation
connects y and x ?
PROBLEMS
1. The weight of a mass of brass varies directly as its
volume. If 150 cu. in. weigh 45 lb., how many cubic inches
weigh 7.5 lb. ?
SOLUTION. Given wccv. (1)
By definition,  = k. (2)
w = kv. (3)
Substitute values, 45 = k 150. (4)
Solving for t, k = .3. (5)
Substitute in (3), 7.5 = .3 r. (6)
<> = 25 cu. in.
Arithmetical solution.
45
The weight of 1 cu. in.=  = .3 lb.
loO
Hence it requires ^ = 25 cu. in. to weigh 7.5 lb.
.3
2. Construct a graph to show the relation between the vol
ume and weight in Problem 1. What is the equation of the
straight line? Read off some sets of values from the graph
and check by the equation.
3. The weight of a mass of gold varies directly as its vol
ume. If 60 cu. in. weighs 42 lb., find the weight of 35 cu. in.
166 APPLIED MATHEMATICS
4. Construct a graph to show the relation between the vol
ume and weight of a mass of gold on the same axes as in
Problem 2. What does the difference in the slope of the two
graphs show ?
5. The distance through which a body falls from rest varies
as the square of the time during which it falls. If a body falls
400 ft. in 5 sec., how far will it fall in 20 sec. ?
Suggestion, d cc< 2 . Check by arithmetical solution. 20 * 5 4.
Since the distance varies as the square of the time, the body will fall
400 x 4 2 = 6400 ft. in 20 sec.
6. Construct a graph to show the relation between distance
and time in the case of a falling body.
93. Inverse variation. A man wishes to lay out a flower
bed containing 120 sq. ft. If he makes it 12 ft. long, it must be
10 ft. wide ; 20 ft. long, 6 ft. wide ; and so on. The greater the
length the less the width. If the length is doubled, the width
120
is halved ; always Ib = 120, or I = We say that the length
. 1
varies inversely as the width, and write it I oc y
94. Definition. One number varies inversely as another
when their product equals a constant.
Exercise 1. Suspend a meter stick at its center so as to bal
ance, and attach a 500g. weight 6 cm. from the fulcrum. Sus
pend on the other side a 100g. weight to balance. How far
from the fulcrum is it ? Suspend other weights to balance, and
make a table for the weights and distances from the fulcrum.
Multiply each weight by its distance from the fulcrum. What
seems to be true ? If w d = 3000 (a constant), we may say
that the distance varies inversely as the weight, dec 
Exercise 2. Locate on squared paper the points from the table
in Exercise 1, and draw a curve through them. Express the
relation between x and y (1) as a variation ; (2) as an equation.
VARIATION 167
PROBLEMS
1. The time it takes to do some work varies inversely as th
number of men at work. If 6 men can do the work in 10 da.,
how long will it take 5 men to do it ?
SOLUTION. Let t number of days.
n = number of men.
Given <oc (1)
By definition, nt = k. (2)
Substitute the given values in (2), 6 x 10 = k. (3)
k = 60. (4)
Substitute in (2), 5 1 = 60. (5)
t = 12. (6)
Check by arithmetic. If 6 men can do the work in 10 da., 1 man
can do it in 60 da. ; and 5 men in ^ of 60 = 12 da.
2. The number of hours in a railway journey varies inversely
as the speed. If it takes 7 hr. to go from Chicago to St. Louis
at 40 mi. per hour, how long would it take at 50 mi. per hour ?
3. The weight of a body varies inversely as the square of
its distance from the center of the earth. If a man weighs
200 Ib. on the surface of the earth (4000 mi. from the center),
how much will he weigh when he is in a balloon 6 mi. from
the surface ?
95. Joint variation. If a carpenter saws a 2in. plank into
strips of various lengths and widths, the volume of each strip
equals twice the length by the width, or /; = 2 U). We may say
that the volume varies jointly a.s the length and width, and write
it in the form v oc Ib.
The number of cubic feet in a rectangular water tank 8 ft.
high varies jointly as the length and width, since the number
of cubic feet = 8 Ib.
96. Definition. One number varies jointly as two others
when the first varies as the product of the other two.
168 APPLIED MATHEMATICS
PROBLEMS
1. The volume of a cylinder varies jointly as the altitude
and the square of the radius of the base. When the altitude
is 20 in. and the radius of the base is 10 in., the volume is
6284 cu. in. Find the volume when the altitude is 8 in. and
the radius of the base is 6 in.
SOLUTION. v Ar 2 .
v = khr*.
6284 = t 20 10 2 .
k = 3.142.
v = 3.142 x 8 x 6 2
= 904.9 cu. in.
2. The pressure of wind on a plane surface varies jointly as
the area of the surface and the square of the velocity of the
wind. If the pressure on 100 sq. ft. is 125 Ib. when the wind
is blowing 16 mi. per hour, what will be the pressure on a plate
glass window 10 by 12 ft. when the velocity of the wind is
70 mi. per hour ?
97. Suggestions for the solution of problems in variation.
1. From the conditions given in the problem write the
variation.
2. Change the variation to an equation.
3. Substitute the given numbers and find the value of the
constant k.
4. In the equation substitute the value of k and the other
numbers given in the problem.
6. Solve this equation for the required number.
6. Check.
While most of the problems in the following list should be
solved by the principles of variation, some of them may be
solved more easily by proportion. All results should be checked,
and as far as possible the meaning of the constant should be
discussed.
VARIATION 169
PROBLEMS
1. The circumference of a circle varies directly as its diam
eter, and when the diameter is 17.5 in. the circumference is
55.0 in. Find the circumference when the diameter is 22.7 in.
2. The velocity acquired by a falling body varies as the
time of falling. If the velocity acquired in 4 sec. is 128.8 ft.
per second, what velocity will be gained in 7 sec. ?
3. The weight of a mass of gold varies directly as its vol
ume. If 5 ccm. weighs 96.3 g., how many cubic centimeters will
weigh 1kg. ?
4. The area of the surface of a cube varies directly as the
square of its edge. What will be the edge of a cube the area
of whose surface is 315$ sq. in., if the area of the surface o f .
a cube whose edge is 3 in. is 73 sq. in. ?
5. The simple interest on a sum of money varies as the
time during which it bears interest. If the interest on a certain
sum is $84.20 for 6 yr., what will be the interest for 8 yr. ?
6. The safe working load on a rope varies as the square of
its girth. If the safe load on a manila rope 6 in. in girth is
1.2 T., find the girth of a rope whose safe load is 3.6 T.
7. If the friction between a wagon and the roadway varies
as the total load on the wheels, and if the friction is 24 Ib.
when the load is 650 Ib., find the friction when the load is 1 J T.
8. The distance a body falls under the action of gravity
varies as the square of the time of falling. If a body falls
403 ft. in 5 sec., in how many seconds will it fall 680 ft. ?
9. The surface of a sphere varies as the square of its radius.
If the surface of a sphere is 616 sq. in., by how much must its
radius of 7 in, be increased in order to double its surface ?
10. Given that the extension of a spring varies as the stretch
ing force, and that a spring is stretched 10 in. by a weight of
5.2 Ib., what weight will stretch the spring 7.5 in. ?
170 APPLIED MATHEMATICS
11. The safe load on a rectangular beam varies jointly as
the breadth and the square of the depth. If a 2 by 4 in. pine
joist of given length supports safely 320 lb., what weight will
a 2^ by 10 in. beam of the same material and length safely
support ?
12. The weight of a disk of copper cut from a sheet of uni
form thickness varies as the square of the radius. Find the
weight of a circular piece of copper 12 in. in diameter, if one
7 in. in diameter weighs 4.42 oz.
13. The volume of a quantity of gas varies as the absolute
temperature when the pressure is constant. If a quantity of
gas occupies 3.25 cu. ft. when the temperature is 14 C., what
will be its volume at 56.5 C. ?
(Absolute temperature = 273 + the reading of the Centigrade
thermometer.)
14. If the volume of a certain gas is 376 ccm. when the
temperature is 12 C., at what temperature will the volume be
533.3 ccm., the pressure remaining the same ?
15. Find the volume of a gas at 23 C., if its volume is
200 ccm. at27C.
16. If the quantity of water that flows through a circular
pipe varies as the square of the diameter of the pipe, and if
1.02 gal. per minute flow through a halfinch pipe, how many
gallons per minute will flow through a 3in. pipe ?
17. The safe load on a wroughtiron chain varies as the
square of the diameter of the section of the metal forming a
link. If the safe load on a chain in which the metal is in.
thick is 900 lb., what diameter of metal will be necessary in a
chain that is to bear a load of 6.4 T. ?
18. What is the safe load for a chain in which the diameter
of a section of the metal forming a link is .9 in. ?
19. The quantity of heat generated by an electric current in
a given conductor for a given time varies as the square of the
VARIATION 171
number of amperes. Find the amount of heat generated by a
current of 25 amperes, if 224 units of heat are generated by
a current of 16 amperes.
20. A current is found to generate 350 units of heat in the
conductor of Problem 19. How many amperes in the current ?
21. The compression of a spring under a given load varies
as the cube of the mean diameter of the coils, other conditions
being the same. When the diameter is 4 in. the compression
is 1.64 in. What is the compression when the diameter is 6 in. ?
22. The deflection by a given load at the middle of a beam
supported at both ends varies as the cube of its length. A
beam 9 ft. long is deflected .135 in. by a certain load. Find the
deflection of a beam 15 ft. long by the same load.
23. The diagonal of a cube varies directly as the edge of
the cube. If the diagonal of a cube is 8.66 in. when its edge
is 5 in., what will be the edge of a cube whose diagonal is
13.4 in. ?
24. A solid sphere of radius 3.5 in. weighs 12 Ib. What is
the diameter of a sphere of the same material that weighs
96 Ib., given that the weight of a sphere varies as the cube of
its radius ?
25. The distance in miles of the offing at sea varies as the
square root of the height in feet of the eye above the sea level.
If the distance is 4 mi. when the height is 10 ft. 8 in., find the
distance when the height is 121.5 ft.
26. According to Boyle's law the volume of a gas varies
inversely as the pressure when the temperature is constant.
If the volume of a gas is 600 ccm. when the pressure is 60 g.
per square centimeter, find the pressure when the volume is
150 ccm.
27. If the volume of a gas is 42.5 cu. in. at a pressure of
12.6 Ib. per square inch, find the pressure when the volume is
35.7 cu. in.
172 APPLIED MATHEMATICS
28. The pressure allowed in a cylindrical boiler varies in
versely as its diameter. When the diameter is 42 in. the
pressure allowed is 104 Ib. per square inch. What pressure is
allowed when the diameter is 96 in. ?
29. Equal quantities of air are on opposite sides of a piston of
a cylinder 16 in. long. If the piston moves 4 in. from the center,
find the ratio of the pressures on the two sides of the piston.
30. The intensity of light varies inversely as the square of
the distance from the source of light. If the illumination of a
gas jet at a distance of 10 ft. is /, what will it be at 20 ft. ?
at 60 ft. ?
31. A student lamp and a gas jet illuminate a screen equally
when it is placed 12 ft. from the former and 20 ft. from the
latter. Compare the relative intensities of the two lights.
32. How far from a lamp is a point that receives three times
as much light as another point 20 ft. away ?
33. How much farther from a gas jet must a book, which is
18 in. away from it, be removed in order that it may receive
two thirds as much light ?
34. An 8 candle power electric lamp at a distance of 6 ft.
from a screen illuminates it with one half the intensity of a
candle at a distance of 1 ft. 6 in. from the screen. What is the
candle power of the candle ?
35. In a given latitude the time of vibration of a pendulum
varies as the square root of its length. If a pendulum 39.1 in.
long vibrates once in a second, what is the length of a pendu
lum that vibrates twice in a second ? three times in a second ?
36. The velocity with which a liquid flows from an orifice
varies as the square root of the head (depth of the liquid above
the orifice). A reservoir 40 ft. high is filled with water, and
when an opening is made in the side at a height of 4 ft., the
water escapes with an initial velocity of 48 ft. per second.
What would be the velocity if the opening were made at a
height of 8 ft. ?
VARIATION 173
37. The weight of a body varies inversely as the square of
its distance from the center of the earth. If a body weighs
100 Ib. at the earth's surface (4000 mi. from the center), what
would be its weight at the summit of the highest mountain,
which is 5 mi. high ?
38. How far above the earth's surface must a body that
weighs 150 Ib. at the surface be removed, in order that its
weight may be reduced to 96 Ib. ?
39. The diameter of the rivets used for a plate varies as the
square root of its thickness. If 1^in. rivets are used for a 1in.
plate, what size of rivets is required for a f in. plate ? What
thickness of plate can be riveted with Jin. rivets ?
40. The volume of a gas varies inversely as the height of
the mercury in a barometer, the temperature being constant.
If a certain mass occupies 32 cu. in. when the barometer reads
28.8 in., what space will it occupy when the reading is 30.4 in. ?
41. Compare the amounts of heat received at two points
whose distances from the source of heat are in the ratio 4 : 3,
assuming that the intensity of heat varies inversely as the
square of the distance from the source of heat.
42. If the attraction of a magnet for a piece of iron varies
inversely as the square of the distance between them, and if
the attraction at the distance of .1 in. is a, what will be the
attraction at .2 in. ? at .3 in. ? at .5 in. ?
43. The attractive force between two oppositely electrified
balls varies inversely as the square of the distance between
them. At a distance of 8 cm. the force is 3.5 g. At what
distance will the force be .64 g. ?
44. The compression of a spring under a given load varies
inversely as the fourth power of the diameter of a cross section
of the steel in the coils, other conditions being the same. If
the compression is 3.5 in. when the diameter is in., what will
bo the compression when the diameter is 1J in. ?
174 APPLIED MATHEMATICS
45. If 7 men in 9 weeks earn $516.60, how many men will
it take to earn $360.80 in 4 weeks, it being given that the
amount earned varies jointly as the number of men and the
number of weeks ?
46. The volume of a circular disk varies jointly as its thick
ness and the square of its radius. Two metallic disks having
thicknesses 5 cm. and 3cm., and radii 12cm.. and 20 cm.
respectively, are melted and recast into a single disk 6 cm.
thick. What is its radius ?
47. The weight of a metal cylinder varies jointly as its length
and the square of its diameter. If a cylinder 12 in. long and
4 in. in diameter weighs 49 lb., what is the diameter of a
cylinder 20 in. long that weighs 135 lb. ?
48. The volume of a cone varies jointly as its altitude and
the square of the radius of its base. If the volume of a cone
is 4.95 ccm. when its altitude is 2.1 cm. and its radius is 1.5 cm.,
find the altitude of a cone whose radius is 3 cm. and volume
33 ccm.
49. How far from a light of 9 candle power will the illumi
nation be 2^ times the illumination at a distance of 24 ft. from
a light of 16 candle power ?
50. The weight of a uniform bar of given material varies
jointly as its length and the area of its feross section. If a steel
bar 1 sq. in. in cross section and 1 ft. long weighs 3.3 lb., what is
the weight of a Trail 2 ft. long and 8 in. in crosssectional area ?
51. An ohm is the resistance offered to the flow of an electric
current through a column of mercury 106 cm. long and 1 sq. mm.
in crosssectional area. What is the resistance of a column of
mercury 3 m. long and 4 sq. mm. in crosssectional area, the
resistance varying directly as the length and inversely as the
crosssectional area ?
52. A wire of diameter .0704 in. has a resistance of 15 ohms.
Find the diameter of a wire of the same length and material
whose resistance is 5.4 ohms.
VARIATION 175
53. If the resistance of 500 yd. of a certain cable is .65 ohm,
what will be the resistance of 1 mi. of a cable of the same
material and of one half the crosssectional area ?
54. Find the resistance of 1000 yd. of copper wire .15 in. in
diameter, if the resistance of 112 yd. of copper wire .06 in. in
diameter is 1 ohm. Solve also by the formula in Exercise 10,
page 73, and compare the results. See Chapter XVII for defi
nitions and explanations.
55. The resistance of a certain wire is 1.82 ohms, and the
resistance of 2 J mi. of the same wire is known to be 3.25 ohms.
Find the length of the first wire.
56. The resistance of 2400 ft. of a certain copper wire of
cross section 11.2 sq. mm. is 1.13 ohms. What is the resistance
of 2 mi. of copper wire of cross section 6.45 sq. mm. ?
57. According to Ohm's law the number of amperes flowing
through an electric circuit varies directly as the number of volts
of electromotive force and inversely as the number of ohms
resistance. If the voltage in a certain circuit is such as to main
tain a current of 10 amperes through a resistance of 40 ohms,
what would be the current if the electromotive force were
doubled and the resistance diminished by one third ?
58. How many amperes are there in the current maintained
by a dynamo whose resistance is 2.4 ohms, that of the rest of
the circuit being 17.6 ohms, and the electromotive force 210
volts ?
59. The resistance offered by the air to the passage of a
bullet through it varies jointly as the square of its diameter
and the square of its velocity. If the resistance to a bullet
whose diameter is .32 in. and whose velocity is 1562.5 ft. per
second is 67.5 oz., what will be the resistance to a bullet whose
diameter is .5 in. and whose velocity is 1300 ft. per second ?
60. From the data of Problem 59 determine the diameter
of a bullet that has a resistance of 50 oz. when its velocity is
900 ft. per second.
176 APPLIED MATHEMATICS
61. If t denotes the time of revolution of a planet in its orbit
about the sun, and d the mean distance of the planet from the
sun, then t 2 varies as d*. Assuming that the earth's period of
revolution is 365 da. and that of Venus 225 da., find the ratio
of the mean distances of these two planets from the sun.
62. The horse power that a solid steel shaft can transmit
safely varies jointly as its speed in revolutions per minute
and the cube of its diameter. A 4in. solid steel shaft making
120 r. p. m. can transmit 240 h.p. How many horse power can
be transmitted if the diameter of the shaft is 3 in. and its speed
100 r. p. m. ?
63. The pressure of the wind on a plane surface varies
jointly as the area of the surface and the square of the wind's
velocity. If the pressure on a square yard is 12^ Ib. when the
velocity of the wind is 17^ mi. per hour, what is the pressure
on a square foot when the velocity of the wind is 45 mi. per
hour?
64. The space s passed over and the time of flight t of a
body projected vertically upward are connected by the rela
tion s = at 16 2 , where a is constant. If s = 676 ft. when
t = 6 J sec., find s when t = 3 sec.
CHAPTER XV
EXERCISES IN SOLID GEOMETRY
98. Use short methods of multiplication and division and
keep the results to a reasonable number of significant figures.
I. NUMERICAL EXERCISES
1. A line 8 ft. long makes with a plane an angle of 45.
Find the length of the projection of the line upon the plane.
2. What will be the length of the projection of the line
in the preceding exercise, if it makes an angle of 30 with
the plane ?
3. Prove that, if a line is inclined to a plane at an angle of
60, its projection upon the plane is equal to half the line.
4. In a swimming tank the water is 5^ ft. deep and the
ceiling is 11 ft. above the water ; a pole 22 ft. long rests
obliquely on the bottom of the tank and touches the ceiling.
How much of the pole is above the water ?
5. From a point P 6 in. from a plane a perpendicular PQ
is drawn to the plane ; with Q as a center and a radius of
4^ in. a circle is described in the plane ; at any point It of
this circle a tangent RT 10 in. long is drawn. Find the dis
tance from P to 7*.
6. With a 12ft. pole marked in feet how can you deter
mine the foot of the perpendicular let fall to the floor from
the ceiling of a room 9 ft. high ?
7. If a point is 20 cm. from each of the vertices of a right
triangle whose legs are 12 cm. and 16 cm. respectively, find
the distance from the point to the plane of the triangle.
177
178 APPLIED MATHEMATICS
8. Determine the relation between (a) the edge and the
diagonal of a face of a cube ; (b) the edge and a diagonal of
a cube.
9. The sum of the squares of the three edges of a rectangular
parallelepiped is 2166 and the three edges are to each other as
1:2:3. Find the edges.
10. The dimensions of a rectangular bin are 4 ft., 4J ft., and
10 ft., and it is desired to treble its capacity. How can this be
done if only one dimension is changed ? two dimensions ? all
three dimensions ?
11. Make a geometrical application of the equation (x + y) 8 =
12. How much will it cost, at 40 cents per cubic yard, to dig
an open ditch 80 rd. long, 6 ft. wide at the top, 2 J ft. wide at
the bottom, and 3 ft. deep ?
13. How many square feet of lead will be required to line
a rectangular cistern 10 ft. long, 7 ft. wide, and 4J ft. deep ?
What will be the weight of the lead if it is ^ in. thick and a
cubic inch weighs .411 Ib. ?
14. What is the weight of the water received upon an acre
of ground during a storm in which rain falls to the depth of
an inch ?
15. Allowing 30 cu. ft. of air per minute for each person in
this classroom, how much air must be driven into the room
and how many times must the air be changed during the
recitation period to insure good ventilation ?
16. The cross section of a trough 12 ft. long is an equilateral
triangle. When 20 gal. of water are poured into the trough,
whose edges are in the same horizontal plane, how deep will
the water be ?
17. A room is 10 ft. high and its length is one half greater
than its width. If the area of the ceiling and walls is 816 sq. ft.,
find the other two dimensions.
EXERCISES IN SOLID GEOMETRY 179
18. A block of ice 1 J ft. by 2 ft. by 3 ft. is placed in a box
4 ft. long and 2 ft. wide. What will be the depth of water in
the box after the ice melts, the specific gravity of ice being
.917?
19. How large a cubical reservoir will be required to hold
the water that falls on the roof of a house covering 548 sq. ft.
of ground, during a shower in which  of an inch of rain falls ?
20. How many square yards of canvas will be required to
make a tent 10 ft. by 16 ft., if the sides are 6 ft. high and the
roof has pitch ?
21. An oblique prism whose altitude is h has for its base a
rhombus whose diagonals are k and I. Find its volume.
22. Two rectangular parallelepipeds are to each other as
5 : 18. The dimensions of the first are 5, 13, and 18. Find
the dimensions of the other, if they are to each other as
1:2:3.
23. The base of a prism whose altitude is 15 cm. is a quadri
lateral whose sides are 10 cm., 18 cm., 12 cm., and 16 cm., the
last two forming a right angle. Find its volume.
24. A prism has for its base a triangle whose sides are to
each other as 5 : 12 : 13. If its altitude is 4 m. and its volume
is 4.8 cu. m., find the sides of the base.
25. The Great Pyramid is 762 ft. square at the base and 484 ft.
high. Compute its volume and its lateral area.
26. The lateral area of a regular square pyramid of wood is
144 sq. in., and one side of the base is 8 in. Find its weight,
if its specific gravity is .53.
27. Determine the volume of a pyramid, one of whose lateral
faces is an equilateral triangle on a side of 18 in., and whose
third lateral edge is perpendicular to the other two and is
24 in. long.
28. A section of a pyramid parallel to the base contains
96 sq. ft., and its distance from the base whose area is 120 sq. ft
is 4 ft. Find the altitude of the pyramid.
180 APPLIED MATHEMATICS
29. The lateral area of a regular triangular pyramid is
64 sq. ft. and one side of the base is 8 ft. Find the altitude.
30. If a section of a pyramid parallel to the base is so taken
that its area is that of the base, what part of the pyramid is
that portion above the section ?
31. If the sides of the base of a pyramid are 4, 6, 7, and 9,
and the solid is cut by a plane parallel to the base so that the
section is ^ of the base, what will be the lengths of the sides
of the section ?
32. A granite obelisk in the form of a frustum of a regular
quadrangular pyramid, surmounted by a pyramid of slant
height 15 in., has each side of one base 1 ft. 4 in. and each side
of the other base 2 ft. 3 in., and the slant height is 12 ft. If
the specific gravity of the granite is 2.6, find the weight of the
obelisk.
33. What will be the expense of polishing the faces of the
obelisk in the preceding exercise at 50 cents per square foot ?
34. What is the capacity in gallons of a reservoir 12 ft. in
depth and 300 ft. long by 160 ft. wide at the top, the slope of
the walls being 3:2?
35. A granite monument in the form of a prismoid is 16 ft.
high and the dimensions of its ends are 42 in. by 28 in. and
18 in. by 12 in. respectively. What is its weight if the specific
gravity of the granite is 2.7 ? See Kent's " Mechanical Engi
neers' PocketBook" for the definition of a prismoid and for
the prismoid formula.
36. A milldam of earth with plane sloping sides and rec
tangular bases is 80 m. by 6 m. at the top and 66 m. by 18 in.
at the bottom. If its height is 5.4 m., find its cubic contents.
37. How many cubic yards of earth will it be necessary to
remove in making a cut for a railroad, which must be 14 ft. deep,
24 ft. wide, 240 ft. long at the bottom, and 170 ft. long at the
top, the slope of the sides being 7 : 10 ?
EXERCISES IN SOLID GEOMETRY 181
38. Apply the prismoid formula to the regular octahedron
whose edge is e.
39. The volume of a wedge whose base is 7.5 cm. by 12 cm.,
and whose height is 3.5 cm., is 142 ccm. Find the length of its
edge, regarding the solid as a prismoid.
40. Find the weight of a steel wedge whose base measures
3 in. by 7 in., the edge 5 in., and the height 6 in., if a cubic
inch of steel weighs .283 Ib.
41. If a cubic foot of steel weighs 490 Ib., what is the weight
of a hollow steel beam 10 in. square at one end, 7 in. at the
other end, and 18 ft. long, the metal being  in. thick ?
42. Find the cost of painting the lateral surface of an octag
onal tower whose slant height is 40 ft., if the short diameter
of the lower base is 12 ft. and of the upper base 3 ft., at 24
cents per square yard.
43. At what distance from the vertex of any pyramid must
a lateral edge 12 ft. long be cut by planes parallel to the base,
in order that the areas of the sections formed may be to each
other as 2 : 3 : 5 ?
44. What must be the height of a prism of iron equal in
weight to the sum of three other prisms of iron of the same
shape, the height of the latter being 2 in., 3 in., and 4 in. re
spectively ?
45. A block of granite weighs 2 T. and its width is 3 ft.
What is the width of a block of granite of the same shape
whose weight is 6 T. ?
46. A block of wood of specific gravity .675 weighs 72.4 Ib.,
and a block of steel of specific gravity 7.84 and of the same
shape weighs 13,14 Ib. Find the ratio of their corresponding
dimensions.
47. Of two bodies of the same form, one weighs 2 Ib. and
its specific gravity is .24, while the other weighs 56 Ib. and its
specific gravity is 2.32. If one dimension of the first body is
60 cm., what is the corresponding dimension of the second ?
182 APPLIED MATHEMATICS
48. An irregular mass of iron, specific gravity 7.2, weighs
42J Ib. What is the weight of a mass of gold of the same form,
specific gravity 19.3, if two corresponding lines of the two
masses have the ratio 2:3?
49. How much tin will be required to make an open cylin
drical vessel of altitude 65 cm. which shall contain 160.2 1.,
taking no account of seams ?
50. What is the amount in cubic feet of evaporation daily
from a circular fishpond 6 rd. in diameter, if the loss in depth
is .04 in. ?
51. How many board feet of lumber 16 in. wide can be made
from a round log 20 in. in diameter and 16 ft. long ?
52. The areas of two sections of a cylinder of revolution
4 ft. high, which are parallel to the axis and to each other, are
6 sq. ft. and 4 sq. ft. respectively. If the sections are 2 in.
apart, what is the volume of the cylinder ?
53. If a cubic foot of copper is drawn into a wire ^ in. in
diameter, what will be its length ?
54. An irregular mass weighing 21.07 kg. is dropped into a
cylindrical vessel 42 cm. in diameter, partially filled with water.
If the water rises 80 cm., find the volume and specific gravity
of the body.
55. How many cubic yards of stone will be required for a
semicircular culvert under a railroad bank 112 ft. wide, the
throat of the culvert being 6 ft. high and the walls 2 ft. thick ?
56. A hollow cylindrical iron column is 14 ft. 4 in. long, 6 in.
in diameter, and 1 in. thick. What is its weight if the specific
gravity of iron is 7.2 ?
57. A steel shaft is reduced in diameter in a lathe from
5 in. to 4.5 in. Find to the nearest hundredth what part of its
weight is lost.
58. In what time will a 1in. circular pipe in which a flow
of water of 1 ft. per second is maintained, fill a rectangular
cistern of dimensions 3 ft. by 4 ft. by 7 J ft. ?
EXERCISES IN SOLID GEOMETRY 183
59. The plunger of a certain singleacting pump is 10 in. in
diameter, has a 10in. stroke, and makes 15 strokes per minute.
How many gallons of water pass through it in 12 hr. ?
60. A Holley pump has an hourly capacity of 145,800 gal.
of water. If the plunger has a 40in. stroke and makes 18 strokes
per minute, what is its diameter ?
61. When a pump is required to furnish 2,800,000 gal. of
water in 24 hr., how many strokes per minute must the plunger
make if its diameter is 30 in. and its stroke is 40 in. ?
The following rule is sometimes used to calculate the horse power
of a steam boiler. To the heating surface afforded by the flues is to
be added two thirds of the lateral siirface of the boiler, and two
thirds of one flue sheet diminished by the ends of the flues. In gen
eral practice 12 sq. ft. of heating surface are considered to afford
Ih. p.
62. Compute the horse power of a steam boiler whose length
is 16 ft. and diameter 6 ft., if there are 136 flues, each 16 ft.
long and 3 in. in interior diameter.
63. What must be the length of the flues of a steam boiler
of diameter 2 ft., containing 34 2in. flues, in order that it may
afford 12 h. p. ?
64. A conical heap of grain 4 ft. high covers a space 12 ft. in
diameter on the floor. How large must be a cubical bin to
hold it ?
65. How many square yards of canvas are required to make
a conical tent 10 ft. high, such that a man 6 ft. tall may stand
without stooping anywhere within 4 ft. of the center ?
66. A conical vessel whose angle is 60 is filled to the depth
of 8 in. with water,, and when a solid cube of wood is submerged
in it, the water rises 1 in. Find the edge of the cube.
67. How much ground is covered by a conical tent 9 ft. in
height, which contains 162 sq. ft. of canvas ?
68. A square whose side is 4 cm. revolves around one of its
diagonals. Find the volume generated.
184 APPLIED MATHEMATICS
69. A rectangle 6 in. by 8 in. revolves around one of its
diagonals. Determine the volume and the area of the surface
generated.
70. If a sector of 120 is cut out of a circular piece of canvas
28 ft. in diameter, what are the dimensions of the conical tent
that can be made out of the remainder ?
71. A hollow iron cone is 4 in. long and 4 in. in diameter,
and the metal is J in. thick. Find its weight if a cubic inch of
iron weighs .261 Ib.
72. The altitudes of a cylinder and an equivalent cone are to
each ather as 16 : 27. Find the ratio of their other dimensions.
73. At 15 cents per square foot, what will be the cost of
cementing the walls and bottom of a cistern in the form of an
inverted frustum of a cone of revolution whose clepth is 7 ft.
and diameters 6 ft. and 3 ft., the lid l ft. square not being
cemented ?
74. A cone whose slant height is 16 cm. is to be divided into
three parts in the ratio of 1 : 2 : 3. At what distance, measuring
from the vertex, must the slant height be cut by planes parallel
to the base ?
75. In a sphere of radius 6 ft., what is the area of the circle
whose plane is 4 ft. from the center ?
76. In a sphere of radius 6 ft. how far from the center is
the plane of a circle whose area is 50f sq. ft. ?
77. What is the length of an arc of 120 in the circumference
of a circle whose plane is 4^ ft. from the center of a sphere of
radius 5 ft. ?
78. On a sphere of radius 6 in. what is the polar distance of
a small circle whose latitude is 60 ? What is the radius of the
circle ?
79. How many degrees in each angle of an equilateral
spherical triangle whose area is T 5 F of that of the sphere ?
80. If a birectangular triangle is ^ of the surface of its
sphere, what is the third angle of the triangle ?
EXERCISES IN SOLID GEOMETRY 185
81. If the diameter of the moon is 2162 mi., find its surface
'
in square miles and its volume in cubic miles.
82. The diameter of the earth is 7918 mi. and that of the
planet Mercury 3030 mi. If the density of the latter is 2.23
times that of the former, show that the mass of Mercury is
nearly  that of the earth.
83. If the mean diameters of the earth and the moon are
7918 and 2162 mi. respectively, show that the ratio of their
surfaces is 27 : 2 nearly.
84. What is the diameter of a sphere of which a wedge of
11 15' contains 359.3 cu. dm. ?
85. How many bullets 1 in. in diameter can be made of 3 ft.
of lead pipe 1 J in. in exterior diameter and in. thick ?
86. A steel ball 6 in. in diameter is dropped into a cylindrical
vessel 8 in. in diameter, which is filled within 2 in. of the top
with water. How much water will overflow ?
87. If 400 lead balls each in. in diameter are melted
and run into a disk ^ in. thick, what will be the radius of
the disk?
88. How many bullets of caliber .32 (.32 in. in diameter) can
be made from a bar of lead 2 in. by 4 in. by 6 in. ?
89. A marble f in. in diameter is dropped into a conical glass
whose diameter is 2 in. and depth 3 in., and is just covered by
the water that it contains. What was the depth of the water
at first ?
90. What is the altitude of that zone of a sphere which
equals a trirectangular triangle in area ?
91. Find the surface of the zone of a sphere of radius 8 in.
cut off by a plane 3 in. from the center of the sphere.
92. What is the altitude of a zone of 120 sq. in. surface, if
the radius of the sphere is 10 in. ?
93. How far from the surface of a sphere must a lamp be
placed in order that one sixth of the surface may be illuminated?
186 APPLIED MATHEMATICS
94. Show that the portion of the earth's surface that is
O 2 Z.
visible to an aeronaut at a height h above the surface is >
r + h
r being the radius of the earth. When h is small it may
be dropped in the denominator, giving the approximate area
277TA.
95. On a globe of radius 7 cm. it is desired to mark off a
zone whose area shall be 6.16 sq. cm. What opening of the
compasses shall be used ?
96. On a globe of radius 9 in. a small circle is described
with an opening of the compasses of 6 in. Find the length of
the circumference.
97. The altitude and radius of the base of a right cone
are 12 and 9 in. respectively. Find the radius of the circle of
tangency of the inscribed sphere.
98. How does the specific gravity of a spherical body com
pare with that of a liquid in which it floats, with one half its
surface above the surface of the liquid ? one third ? when it is
just submerged ?
99. If a sphere of oak 6 in. in diameter floats in water with
.3 of its surface above the surface of the water, what is the
specific gravity of the oak ?
100. What portion of the surface of a ball of iron of diameter
1 in. and specific gravity 7.2 will remain visible when it is
dropped into a dish of mercury whose specific gravity is 13.6 ?
II. GRAPHICAL EXERCISES
99. A few of these exercises should be worked out carefully
in the notebook.
1. Construct a graph to show the change in the volume of a
cube as its edge increases from to 12 in. What is the equation
of the graph ?
EXERCISES IN SOLID GEOMETRY
187
2. On the same axes as in Exercise 1 show graphically the
change in the surface of the cube. How do the graphs show
(a) when the surface equals the volume numerically ? (i) when
a cube has a greater surface than volume ? Write on each graph
its equation.
FIG. 80
3. The altitude of a regular square pyramid is 12 ft. and
each side of the base is 18 ft. Show graphically (a) the volume,
(b) the lateral surface of the pyramids cut off from the vertex
by planes parallel to the base. Find the ratio of the surface
of any of the pyramids to its volume, and use the result to check
the table of values.
188 APPLIED MATHEMATICS
4. The altitude of a right cone is 12 ft. and the radius of
the base is 9 ft. Show graphically (a) the volume, (&) the lat
eral surface of the cones cut off from the vertex by planes
parallel to the base.
5. On the same axes construct graphs to show the change
(a) in volume, (6) in lateral surface of a right cylinder the radius
of whose base is 6 in., as its altitude* increases from to 15 in.
6. Represent graphically the change in the area of a section
parallel to the base of a regular triangular pyramid, the side
of whose base is 8 cm. and whose altitude is 12 cm.
7. On the same axes represent graphically the change (a) in
volume, (i) in surface of a sphere as the radius increases from
to 10 in.
8. The volume of a pyramid is 60 cu. in. Construct a graph
to show the relation between the base and altitude as the altitude
increases from to 180 in.
9. The volume of a cylinder is 440 cu. in. Construct a curve
to show the relation between the radius of the base and the
altitude, as the radius increases from to 10 in.
10. From each corner of a square piece of tin 12 in. on a
side a smaller square is cut, the remainder of the sheet being
bent so as to form a rectangular open box. Determine the side
of each small square in order that the capacity of the box may
be as great as possible.
11. If the sheet of tin in the preceding exercise had been
rectangular, 20 in. by 12 in., what then would have been the
size of each small square ?
12. A bin with a square base and open at the top is to be
constructed to contain 400 cu. ft. of grain. What must be its
dimensions to require the least amount of material ?
13. A closed cylindrical oil tank is required to hold 100 bbl.,
each of 42 gal. What dimensions will necessitate the least
steel plate in the making ?
EXERCISES IN SOLID GEOMETRY 189
14. An open rectangular tank whose length is to be twice
its width is to hold 200 gal. of water. What dimensions will
require the least amount of lining for the tank ?
15. The strength of a rectangular beam is proportional to
the product of its breadth and the square of its depth. What
are the dimensions of the strongest beam that can be cut from
a round log 2 ft. in diameter ?
16. If the slant height of a right cone is 12 ft., what must
be the radius of its base in order that its volume may be as
great as possible ?
17. Determine the right cylinder of greatest lateral surface
that can be inscribed in a cone of revolution whose altitude is
14 in. and radius of base 8 in.
18. Find the dimensions of the smallest cone of revolution
that can be circumscribed about a cylinder whose altitude and
radius are respectively 9 dm. and 3 dm.
19. The stiffness of a rectangular beam varies as the product
of its breadth and the cube of its depth. Find the dimensions of
the stiff est beam that can be sawed from a log 20 in. in diameter.
20. Determine the dimensions of the largest right cone that
can be inscribed in a sphere of radius 5 in.
21. Find what radius of the base of a conical tent of 375 cu. ft.
capacity will require the least amount of canvas in the making.
Also find the relation between the altitude and the radius.
22. Find the radius of the right cylinder of greatest lateral
surface that can be inscribed in a sphere whose diameter is 12 in.
23. Find the relation between the radius of the base and
the altitude of a right cone whose convex surface contains
264 sq. ft., in order that the volume may be as great as possible.
24. Determine the altitude of the least cone of revolution
that can be circumscribed about a sphere of radius 2 dm.
25. What must be the altitude of the cone of revolution of
least lateral surface that can be circumscribed about a sphere
whose radius is 4 in. ?
190 APPLIED MATHEMATICS
III. ALGEBRAIC PROBLEMS
100. Make a sketch for each problem. Put the given dimen
sions on the figure and set up the equations from the sketch.
1. What are the other two dimensions of a rectangular
parallelepiped whose length is 8 in., if its volume is 160 cu. in.
and its total surface is 184 sq. in. ?
2. If the three face diagonals of a rectangular solid are
respectively 6, 7, and 9 cm., what must be the dimensions of
the solid ?
3. One dimension of a rectangular parallelepiped is 6 in.,
one diagonal is 12 in., and the area of one of the wholly unknown
faces is 44 sq. in. What are the other two dimensions ?
4. The sum of the three dimensions of a rectangular solid
is 12 and the diagonal of the solid is 5 v2. Find its total surface.
5. The sum of a diagonal and an edge of a cube is 6. Find
an edge of the cube.
6. The area of one face of a rectangular solid is 10 sq. cm.,
that of another is 15 sq. cm., and the total area is 100 sq. cm.
Find the dimensions.
7. What are the dimensions of a rectangular solid whose
entire surface is 392 sq. in., if its top contains 96 sq. in. and
one end 40 sq. in. ?
8. Given the diagonal of a cube equal to k . Find the volume
of the cube and its surface.
9. Given the volume v and the altitude h of a regular hexag
onal prism. Find s, the length of one side of the base.
10. The sides of the base of a triangular prism are as 3 : 4 : 5,
and its volume is 432 cu. ft. If the altitude is 4 ft., find the
sides of the base.
11. What must be the altitude of a pyramid in order that
its total area may be equal to the sum of the areas of two similar
pyramids whose altitudes are respectively 6 and 4 in. ?
EXERCISES IN SOLID GEOMETRY 191
12. What is the altitude of a pyramid whose base contains
98 sq. in., if a section parallel to the base and 4 in. from the
vertex contains 32 sq. in. ?
13. The volume of a pyramid with a rectangular base is 76.8
cu. in., one side of the base is 9.6 in., and the altitude exceeds
the other side of the base by 2 in. Find the altitude and the
other side of the base.
14. If a square pyramid has each basal edge equal to e and
each lateral edge equal to e v show that the volume will be
15. Given v, the volume, and s, one side of the square base
of a regular quadrangular pyramid, find the lateral surface.
16. Derive an expression for the volume of a regular tetra
hedron in terms of its edge e.
17. An iron plate 8 in. long and 2 in. thick has squared ends
but uniformly and equally beveled sides, and contains 122 cu. in.
If the difference of the widths of the two flat faces is 2.8 in.,
find those widths.
18. The lateral area of a frustum of a regular quadrangular
pyramid is 281.2 sq. in., the slant height is 15.2 in., and a side
of the lower base exceeds a side of the upper base by 3.75 in.
Find a side of each base.
19. What must be the diameter of a cylindrical gas holder
which is to hold 6,000,000 feet of gas, if its height is to be
of its diameter ?
20. The sum of the numerical measures of the volume and
lateral area of a cylinder of revolution is 231. If the altitude
is 14, what is the diameter ?
21. Write the formula that gives f, the total surface of a
cylinder of revolution, in terms of A, the altitude, and r, the
radius of the base, and solve it for h and r.
In case of a cylinder of revolution :
22. Given t and r, find h and v.
192 APPLIED MATHEMATICS
23. Given v and r, find h and t.
24. Given v and A, find r and (the lateral area).
25. Given I and i;, find h, r, and .
26. Given I and A, find r, v, and .
27. Given t and #, find r, A, and
Suggestion. Find r by trial from 2 ?rr 8 /r f 2 r = for any given
numerical values of t and v (see sect. 58) ; then find A from v = irr 2 ^,
and then ? from / = 2 ?rrA.
28. How far from the axis of a cylinder of revolution whose
height is h ft. and diameter d ft. must a plane parallel to the
axis be passed, in order to make a section of area k sq. ft. ?
29. If the total surface of a cone of revolution is 21 TT and
the slant height is 4, find the radius and the volume of the cone.
30. The sum of the altitude and the radius of the base of a
cone of revolution is 11 and their product is 10. What is the
volume of the cone ?
31. The lateral area of a right cone is 9 VlOTr, and its alti
tude is equal to 3 times the radius of its base. Find its volume.
32. Find the slant height and the radius of the base of a
cone of revolution whose total surface is 462 sq. in., and the
sum of the slant height and the radius is 21 in.
33. The lateral surface of a right cone whose slant height is
5 exceeds the base by 12f . Find the radius of the base.
34. What is the radius of the upper base of a frustum of a
right cone, if its volume is .516 TT cu. dm., its altitude 1.2 dm.,
and the radius of its lower base .8 dm. ?
35. The lateral area of a frustum of a cone of revolution is
77 TT, the slant height is 7, and the altitude is 2 V6. Find the
radii of the bases.
36. What is the volume of a frustum of a right cone the sum
of the radii of whose bases is 11 and their product 28, the
altitude being 7 ?
EXERCISES IN SOLID GEOMETRY 193
37. Find the radii of the bases of a frustum of a right cone,
given the lateral area as 1068^ sq. ft., the slant height as 17 ft.,
and the altitude as 15 ft.
38. The volumes of two spheres are to each other as 8 : 125,
and the sum of their radii is 12 in. Find the radii.
39. The product of the radii of two spheres is 22.5 and the
ratio of their surfaces is 25 : 64. What are the radii ?
40. If the surface of a sphere is equal to the sum of the
surfaces of two spheres whose radii are 2 in. and 4 in. respec
tively, how does its volume compare with the sum of their
volumes ?
41. What is the radius of a sphere of which a zone of
24 sq. in. is illuminated by a lamp placed 18 in. from its
surface ?
42. What relation must the radius of a given sphere bear to
the radii of two other spheres if its surface is a mean propor
tional between their surfaces ?
43. Compare the expression for the volume of a sphere with
that for its surface, and determine how long the radius must
be in order that the volume may be numerically greater than
the surface.
44. In a sphere of radius 8 the radius of one small circle is
a mean proportional between the radius of the sphere and the
radius of another Small circle, and the sum of the radii of the
two small circles is 10. Find the radii of the small circles.
45. Derive an expression in one variable for the volume of
a right cone inscribed in a sphere of radius r.
46. Find an expression in terms of the altitude for the total
surface of a cylinder of revolution inscribed in a sphere of
radius r.
47. What is the expression for the volume of a right cylinder
inscribed in a right cone, altitude A, radius of base r, in terms
of the radius of the cylinder ?
194 APPLIED MATHEMATICS
48. Find an expression in one variable for the total surface
of a right cone circumscribed about a given right cylinder.
49. What expression in one variable denotes the volume of
a right cone circumscribed about a given sphere ?
50. Derive the expression in one variable for the lateral
surface of the cone in Exercise 49.
51. Find an expression in one variable for the volume of a
right cone circumscribed about a given right cylinder.
Problems 4551 furnish good exercises in maxima and minima
by giving numerical values to the dimensions of the constant solids.
Since some of the expressions are rather complicated the work of
computing the table of values may be divided among the members
of the class, each one computing the value of the function for a
single value of the variable.
CHAPTER XVI
HEAT
101. Thermometers. Though the Fahrenheit scale is in
general use in everyday life and in ordinary engineering work,
the Centigrade scale is used in laboratories and all scientific
work to such an extent that one should become acquainted with
it. Fahrenheit (Danzig, Germany) devised his scale about
1726. He thought that the lowest possible degree of cold was
obtained by mixing salt and ice ; hence he took as zero the
position of the mercury when placed in such a mixture. It is
not known why he marked the boiling point of water 212.
The Centigrade scale was proposed by Anders Celsius (Upsala,
Sweden) about 1741.
In the Fahrenheit thermometer the boiling point of water at
sea level is taken at 212 and the freezing point of water at 32.
In the Centigrade thermometer the boiling point is taken at
100 and the freezing point at 0. Hence 180 on the Fahrenheit
scale equals 100 on the Centigrade scale.
180 F. = 100 C.
l F.gC. (1)
1 C. =  F. (2)
It should be remembered that a division on the Centigrade
scale is longer than a division on the Fahrenheit scale. Hence
in changing from degrees Centigrade to degrees Fahrenheit we
get a greater number of degrees, and from Fahrenheit to Centi
grade we get a smaller number of degrees.
Equations (1) and (2) enable us to change readily from one
scale to the other.
195
196
APPLIED MATHEMATICS
05
05
185
PROBLEMS
1. Construct a graph to change a number of degrees of one
scale to degrees of the other scale. Why .is it necessary to
locate only two points and draw a straight line through them ?
2. Change (a) 90 F. to C. ; (6) 200 F. to C. ; (c) 40 C. to
F. ; (d) 80 C. to F. ; (e) 150 F. to C. ; (/) 112 F. to C. Check
by the graph.
3. The sum of a number of degrees F. and a number of
degrees C. is 121. When the degrees F. are changed to degrees
C. and added to the number of degrees C. the result is 85. Find
the number of degrees F. and C.
4. The suin of a number of degrees F. and a
number of degrees C. is 53. If each number of
degrees is changed into the other scale, the
sum is 73. Find the number of degrees F. and C.
102. To change thermometer readings from
one scale to the other. In the above problems
we were dealing with degrees not with ther
mometer readings. When we change thermom
eter readings from one scale to the other we
must take account of the difference in position
of the zeros on the two scales.
Thus find the C. reading when the F. reading
is 80. Looking at Fig. 81, we see that by tak
ing 32 from 80 we get 48, the number of
degrees the F. reading is above 0C. Then
48F. = 48x C. = 26.7C.
Similarly, to find the F. reading when the C.
reading is 70,
45
\
50
0
= 126F.
FIG. 81
But 126 F. takes us only to 32 F. opposite 0C. Hence we
add 32 to get the F. reading, 158, corresponding to 70 C.
HEAT 197
To change from F. to C. readings, subtract 32 and multiply
the difference by j. To change from C. to F. readings, multiply
by  and add 32 to the product.
C. = f(F.32).
F. = C. + 32.
103. To determine the relation of the corresponding read
ings of the two thermometers by experiment. Take several
readings of the two thermometers on different days, or obtain
the readings by putting the thermometers into water at dif
ferent temperatures.
Readings obtained :
F. 32 47 70 96 118 151
C. 8 21 36 48 66
Locate these points on squared paper. Units : C., horizontal,
1 large square = 10 ; F., vertical, 1 large square = 10. On
stretching a thread along these points it will be found that
they lie nearly in a straight line. Draw a straight line among
the points so that they are distributed evenly above and below
it. This line is the graph of the equation which connects the
corresponding readings. To find the equation we will suppose
that it is of the form
ij . = a C. + 0> (1)
where a and b are unknown numbers which must be determined.
Taking the second and fifth points and substituting the read
ings in (1), we have
Solving these equations, we get a = 1.77, b = 32.8.
Substituting these values in (1), we get F. = 1.77 C. + 32.8.
The readings in the experiment were not taken with sufficient
exactness to give a close result (see sect. 69).
Exercise. Take several corresponding readings on the two
thermometers and find the relation as above.
198
APPLIED MATHEMATICS
PROBLEMS
1. Construct a graph to change the readings of one ther
mometer to those of the other. Units : horizontal, 1 large
square = 20 F. ; vertical, 1 large square = 10 C. Take the
lower lefthand corner as the origin and mark it 40. Show
that 40 is the same reading on both scales. Locate one
other point. Why is the graph a straight line ?
2. Change the reading of one thermometer to that of the
other, and check by the graph :
(a) 78 F. to C. 0) 195 F. to C.
(6) 18 F. to C. (/)  20 F. to C.
(c) 88 C. to F. (g)  30 C. to 'F.
(d) 60 C. toF. (A) 0F. to C.
3. The melting point of the following metals is given in
degrees F. Change to the Centigrade scale:
Tin . . . 442 to 446 Copper .
Lead . . . 608 to 618 Cast iron
Silver . . . 1733 to 1873 Steel . .
Gold . . . 1913 to 2282 Platinum
1020 to 1996
1922 to 2075
2372 to 2532
3227
4. The following record of temperature was taken from
The Chicago Daily News.
3 P.M
... 69
3 A.M. . . .
07
4 p M
... 68
4 A.M. .
GO
5 P.M
... 68
5 A.M. .
(]5
6 P.M
... 67
6A.M. . . .
04
7 P.M
... 66
7A.M. . . .
05
8 P.M
... 67
8 A.M. . . .
...'.. 66
9 P.M
... 67
9 A.M. .
07
10 P.M
... 68
10A.M. . . .
67
11 P.M
... 68
11 A.M. . . .
68
12 midnight
. . . 68
12 noon .
70
1 A.M
... 69
1 P.M. . . .
73
2 A.M
... 68
HEAT 199
Change the readings to Centigrade by using the graph, and
on the same sheet of squared paper plot a curve for each of
the two sets of readings. Are the curves parallel ? Why ?
5. What temperature is expressed by the same number on
the F. and C. scales ?
6. A Fahrenheit and a Centigrade thermometer are placed
in a liquid and the F. reading is found to be double the C.
reading. What is the temperature of the liquid in degrees C.?
EXPANSION OF SOLIDS BY HEAT
104. Linear expansion. When a solid is subjected to
changes of temperature its length changes ; in general, the
length increases as the temperature rises, and decreases as it
falls. For ordinary temperatures the amount of change is
nearly the same for each degree of increase or decrease. The
following table gives results that have been secured by experi
ment ; they are only approximate.
LINEAR EXPANSION OF SOLIDS FOR 1 DEGREE, BETWEEN
AND 212 F.
Aluminum . . . .00001234 Lead 00001571
Brass, plate . . . .00001052 Platinum 00000479
Copper 00000887 Steel, cast 00000636
Glass, white . . . .00000478 Steel, tempered . . .00000689
Iron, wrought . . .00000648 Tin 00001163
Iron, cast 00000556 Zinc 00001407
The amount of expansion is seen to be very small. Thus
when we say that the linear expansion of wrought iron is
.00000648 we mean that the length of a wroughtiron rod
100 ft. long increases 648 millionths of a foot when the tem
perature of the rod rises 1 degree. However, provision must be
made for this expansion in all construction work ; for example,
a little space is left between the ends of the rails in laying
railway track, hotwater pipes have telescopic joints, and so on.
200 APPLIED MATHEMATICS
PROBLEMS
1. Find the linear expansion of copper, wrought iron, and
tinforlC.
2. A brass wire is 200 ft. long at ,0. Find its length at 85.
SOLUTION. 200 x .00001052 x 85 = .179 ft.
200 + .179 = 200.179 ft. = 200 ft. 2.2 in.
3. A steel chain is 66ft. long at 77. What will be its
length at 32?
4. The iron girders of a railway bridge are 100 ft. long at
a temperature of 10. What will be the length of the girders
at 90 ?
5. A lead pipe is 80 ft. long at 10. How Idng will it be
at 100 ?
6. A brass rod is 5 m. long at C. What is its length at
38 C. ?
7. What is the length of a wroughtiron rod at C. if it
is 1.56 m. long at 100 C. ?
8. What is the length of a copper wire which increases in
length 1.2 in. when its temperature is raised 200 ?
9. What is the area of a brass plate at 100 C. which
measures 8.35cm. by 4.16cm. at 0C. ?
10. One brass yardstick is correct at 32 and another at 68.
What is the difference in their lengths at the same temperature ?
11. A bar of metal 10ft. long at 200 increases in length .31 in.
when heated to 362. Find the expansion of 1 ft. for 1.
12. A plateglass window is 10 ft. by 12 ft. How much will
it change in area when its temperature changes from 20 to
90, if its linear expansion for 1 is .000005 ?
13. An iron steam pipe 200 ft. long at 190 ranges in tem
perature from 190 to 4. What must be the range of motion
of an expansion joint to provide for the change in length ?
HEAT
201
14. A platinum wire and a brass wire are each 100 ft. long
at 30. How much must they be heated to make the brass wire
1 in. longer than the platinum wire ?
Suggestion. Let x the number of degrees.
.00001052 x 100 x 12 x  .00000470 x 100 x 12 x x  1.
15. A copper bar is 10 ft. long. What must be the length of
a castiron bar in order that the two may expand the same
amount for 1 ?
16. A steel tape 100 ft. long is correct at 32. On a day when
its temperature was 88 a line was measured and found to be
1 mi. long. What was the error and what was the true length
of the line ?
17. An iron casting shrinks about \ in. per linear foot when
cooling down to 70. What is the shrinkage per cubic foot ?
18. The Chicago and Oak Park Elevated Railway is about
9 mi. long from Wabash Avenue to Oak Park Station. What
is the difference in the length of the rails for a change in
temperature from  20 to 80 ?
19. Construct a graph to show the expansion of a steel wire
100 ft. long as its temperature rises from to 2000.
20. The following table shows the change in the volume of
water as its temperature rises from to 17 C. Construct the
graph. How does the graph show an exception to the law that
the volume increases with a rise in temperature ?
Temp.
Volume
Temp.
Volume
Temp.
Volume
1.000000
6
.999914
12
1.000334
1
.990948
7
.1)99952
13
1.000462
2
.999911
8
1.000003
14
1.000593
3
.999889
9
1.000008
15
1.000735
4
.999883
10
1.000147
16
1.000890
5
.999891
11
1.000239
17
1.001'057
202 APPLIED MATHEMATICS
MEASUREMENT OF HEAT
105. Units. When a definite quantity of heat is applied to
various substances different effects are produced, depending on
the nature and condition of the substance. An amount of heat
may be expressed by any of its effect^ which can be measured ;
but it has been found convenient to measure heat by considering
the change in temperature it produces.
Two heat units in general use are the British thermal unit
(B. t. u.) and the calorie. For ordinary engineering work the
unit is the British thermal unity the amount of heat required
to raise 1 Ib. of water 1 F. A smaller unit used in laboratory
investigations is the calorie, the amount of heat required to
raise 1 g. of water 1 C. The amount of heat required to raise
a quantity of water 1 degree varies with the temperature ; but
the variation is so small that in practical work we may neg
lect it and say that the same amount of heat will raise 1 Ib. of
water from 10 to 11 or from 211 to 212.
106. The relation between heat and work. In sawing
wood, boring iron, and so on, a part of the energy of work
becomes heat. It has been found possible to determine the
number of foot pounds of work required to raise the temperar
ture of a quantity of water a certain number of degrees. The
famous experiments of Joule, in England, in the years 1843""
1850, showed that 772 ft. Ib. of work raised the temperature of
water at 60 F., 1 degree,
His apparatus consisted of a brass cylinder in which water
was churned by a brass paddle wheel, which was made to re
volve by a falling weight. Later experiments by other methods
have given results more nearly exact, and by general consent
it is now considered that 778 ft. Ib. of work are required to
raise the temperature of 1 Ib. of water 1 F.
1 B. t. u. = 778 ft. Ib.
1 ft. Ib. = .00129 B. t. u.
HEAT 203
PROBLEMS
1. How many British thermal units are required to raise
the temperature of 120 Ib. of water from the freezing point to
the boiling point?
2. On a cold day in winter a tank 1 ft. by 2 ft. by 8 ft. was
filled with water at a temperature of 100. When the water had
reached the freezing point, how much heat had been given out ?
3. If 1 Ib. of coal contains 13,500 B. t u. of heat, how many
pounds of coal would be required to raise the temperature of
12 cu. ft. of water 50 if there was an efficiency of 10 per cent ?
4. Find the number of British thermal units required to
raise the temperature of 20 Ib. of lead from 70 to the melting
point, 608. (It takes only .03 as much heat to raise 1 Ib. of
lead 1 as it does to raise 1 Ib. of water 1.)
5. A steel ingot weighing 1 T. is redhot (1200). How much
heat is given off as it cools to 70 ? (It takes only .12 as much
heat to raise 1 Ib. of steel 1 as it does to raise 1 Ib. of water 1.)
6. How many foot pounds of work are required to raise the
temperature of 20 Ib. of water 12 ?
SOLUTION. 778 x 20 x 12 = 186,720 ft. Ib.
7. The temperature of 1 cu. ft. of water was raised from 32
to 70. How many foot pounds of work did it require ?
8. Through how many feet would a weight of 1200 Ib. have
to fall to generate enough energy to raise the temperature of
81b. of water 15?
9. Find the distance through which a weight of 2 T. could
be raised by the expenditure of an amount of heat that would
raise the temperature of 2 Ib. of water 30.
10. How many horse power would it take to raise the tem
perature of 10 cu. ft. of water from 70 to 120 in 1 hr. ?
62.4x10x778x50 10Q ,
SOLUTION '
60 x 33,000
204 APPLIED MATHEMATICS
11. Find the number of British thermal units per minute
required for an engine of the following dimensions : diameter
of cylinder, 50 in. ; stroke, 36 in. ; revolutions per minute, 54 ;
mean effective pressure, 28 Ib. per square inch. Find also the
number of pounds of coal required per houj, if 1 Ib. of coal
contains 13,500 B. t. u. and only 10 per cent of the heat of the
coal reaches the piston.
12. How many calories are required to raise the temperature
of 40 g. of water 20 C. ?
13. If 126 calories of heat raised the temperature of a
quantity of water 49 C., how many grams of water were there ?
14. The temperature of 1 1. of water was raised from 40 C.
to the boiling point. How many calories were required ?
15. How many calories are there in a British thermal unit ?
16. Construct a graph to change calories to British thermal
units.
SPECIFIC HEAT
107. Exercise. Put equal weights of water and mercury in
similar dishes. Note the temperature of each. Place the dishes
on an electric stove or in a dish of boiling waiter. After a time
it will be found that there is a considerable difference in the
temperatures of the mercury and water.
Since the mercury and the water have received the same
amount of heat, it is evident that it takes less heat to raise the
temperature of 1 Ib. of mercury 1 than is required for 1 Ib. of
water. It is found by experiment that equal weights of differ
ent substances require different amounts of heat to raise their
temperatures the same number of degrees. Thus 1 Ib. of water
requires 1 B. t. u. to raise its temperature 1 F. ; 1 Ib. of glass
requires .2 B. t. u. ; 1 Ib. of cast iron requires .12 B. t. u. ; and
1 Ib. of ice requires .5 B. t. u.
108. Definition. The specific heat of a substance is the quo
tient obtained by dividing the amount of heat required to raise
the temperature of a given weight of it 1 by the amount of
HEAT 205
heat required to raise the temperature of an equal amount
of water 1. (Note the similarity to specific density.)
The specific heat of substances varies a little with the tem
perature, but in practice it is considered to be constant.
TABLE OF SPECIFIC HEAT
Aluminum 0.21 Silver 0.06
Brass 0.09 Steel 0.12
Copper 0.09 Tin 0.06
Glass 0.20 Zinc 0.09
Iron, cast 0.12 Water 1.00
Iron, wrought .... 0.11 Ice 0.50
Lead 0.03 Steam 0.48
Mercury 0.03 Air 0.24
PROBLEMS
1. How many British thermal units are required to raise
the temperature of 10 Ib. of copper from 70 to 200 ?
SOLUTION. 200  70 = 130.
It would require 1300 B. t. u. to raise the temperature of 10 Ib. of
water 130. Specific heat of copper = .09.
.. 1300 x .09 = 117B.t.u.
2. How many calories are required to raise the temperature
of 500 g. of lead 40 C.?
SOLUTION. 500 x 40 x .03 = 600 calories.
3. Find the amount of heat required to raise the tempera
ture of (a) 20 Ib. of silver from 70 to the melting point, 733 ;
(ft) 30 Ib. of aluminum from 70 to the melting point, 1157 ;
(c) 25 Ib. of ice from to 32 ; (d) 1 kg. of mercury 80 C.
4. How many British thermal units are given off by an iron
casting which weighs 50 Ib., as it cools from 2000 to 70 ?
5. If 1 Ib. of water at 70 and 1 Ib. of mercury at 70 are
given the same amount of heat, how hot will the mercury
become when the water is at 73 ?
206 APPLIED MATHEMATICS
6. Equal weights of tin and cast iron are put into a tank
of boiling water. When the tin has been heated 10, how much
has the iron been heated ?
7. If 15 Ib. of water at 200 and 8 Ib. of water at 70 are
mixed together, what is the resulting temperature ?
SOLUTION. Let / = the resulting temperature.
15 (200 t) = number of British thermal units lost.
8 (t 70) = number of British thermal units gained.
8 (t  70) = 15 (200  t).
Solving, t = 154.8.
Check. 15 (200  154.8) = 8 (154.8  70).
15 x 45.2 = 8 x 84.8.
678 = 678.4.
8. 20 Ib. of water at the freezing point are poured into 25 Ib.
of water at the boiling point. What is the temperature of the
mixture ?
9. A tank 2 ft. by 3 ft. by 6 ft. is two thirds full of water
at 60. If the tank is filled with water at 120, what is the
temperature of the mixture ?
10. How many pounds of water at 40 must be mixed with
60 Ib. of water at 100 to obtain a temperature of 80 ?
SOLUTION. Let p = number of pounds at 40.
p (80 40) = number of British thermal units gained.
60 (100 80) = number of British thermal units lost.
p (80 40) = 60 (100 80).
p = 30 Ib.
Check. 30 (80  40) = 1200 B. t. u. gained.
60 (100  80) = 1200 B. t. u. lost.
11. How many pounds of water at 180 must be mixed with
1 cu. ft. of water at 56 to obtain a temperature of 112 ?
12. How many grams of water at C. must be mixed with
1 kg. of water at 100 C. to obtain a temperature of 80 C. ?
13. How much water at 212 and how much water at 32
must be taken to make up 36 Ib. at 97 ?
HEAT 207
SOLUTION. Let x = number of pounds at 212.
y = number of pounds at 32.
x + y = 36. (1)
x (212  97) = y (97  32). (2)
115 x = 65 y. (3)
* = * (4)
Substitute (4) in (1), }f y + y = 36. (5)
y = 23. (6)
Substitute (6) in (1), x = 13. (7)
CAcct. 13 (212  97) = 23 (97  32).
13 x 115 = 23 x 65.
115 = 115.
14. How much water at 180 and at 81 must be taken to
fill a tank which contains 90 lb., if it is desired to have the
temperature of the mixture 125 ?
15. Into a dish containing some water at 4C. was poured
some water at 75 C. How many grams of each were taken if
there were in all 250 g. at a temperature of 60 C. ?
16. An iron casting when redhot (1300) was put into a tank
containing 2 cu. ft. of water at 170. If the temperature of the
water rose to 200, what was the weight of the casting ?
SOLUTION. Let x = number of pounds of cast iron.
Specific heat of cast iron = .12.
2 cu. ft. of water = 62.4 x 2 = 124.8 lb. of water.
(1300 200) X .12 x = number of British thermal units lost
by the iron.
(200 170) x 124.8 = number of British thermal units
gained by the water.
(1300  200) x .12 x = (200  170) x 124.8.
132 x = 3744.
x = 28.4.
Check. 28.4 x 1100 x .12 = 3749 B.t.u. lost.
124.8 x 30 = 3744 B. t. u. gained.
17. If a mass of lead at 500 was put in a gallon of water
(8^ lb.) and the temperature of the water rose from 77 to 80,
what was the weight of the lead ?
208
APPLIED MATHEMATICS
18. An 80lb. mass of steel at 1000 is placed in a tank con
taining water at 60. If the final temperature is 70, how
many pounds of water are in the tank ?
19. If 20 Ib. of brass at 300 were placed in a tank containing
1 cu. ft. of water at 72, what would be the final temperature ?
20. If 500 g. of brass at 100 C.' were placed in 188 g. of
water at 17.5 C. and the final temperature was 33.5 C., find
the specific heat of the brass.
SOLUTION. Let .9 = the specific heat of the brass.
500 (100 33.5). <? = number of calories lost by the brass.
188(33.5 17.5) = number of calories gained by the
water.
500(100  33.5)* = 188(33.5  17.5).
188 x 16
S ~ 500 x 66.5
= .0905.
Check. 500 x 66.5 x .0905 = 3010 calories lost.
188 x 16 = 3008 calories gained.
21. The following data were obtained by experiment. Find
the specific heat of each metal.
No.
Water
(grams)
Tempera
ture of
Water
Material
Weight
used
Initial
Temperature
Final Tem
perature
1
188
18.5
Zinc
250 g.
100 C.
28.5 C.
2
188
11.0
Cast iron
750 g.
100 C.
39.0C.
3
188
16.5
Lead
700 g.
100 C.
26.0 C.
22. 45 g. of zinc at 100 C. were immersed in 52 g. of water
at 10 C. If the temperature of the water rose to 17 C., find
the specific heat of the zinc, assuming that no heat was absorbed
by the dish containing the water.
23. A room 20 ft. by 30 ft. by 10 ft. is to be heated from a
temperature of 32 to 72. Assuming that 1 cu. ft. of air at 32
weighs .08 Ib., that the specific heat of air is .24, and that 8 per
HEAT 209
cent of the fuel is available for raising the temperature, how
many pounds of hard coal (1 Ib. coal = 13,500 B. t. u.) would
be required ?
LATENT HEAT
109. Exercise. Place a dish of melting ice on a stove. Though
the melting ice and water receive heat continuously, a ther
mometer placed in the dish will stand at 32 F. till all the ice
is melted. Then the mercury will rise till the boiling point is
reached. The temperature will remain at 212 till all the water
is evaporated.
110. Latent heat. This heat which goes into a substance
and produces a change in form rather than an increase in
temperature is called latent (hidden) heat.
The following table gives the approximate number of British
thermal units absorbed by 1 Ib. of the substance in changing
from solid to liquid or liquid to solid.
LATENT HEAT OF FUSION LATENT HEAT OF VAPORIZATION
Bismuth 22.75 Alcohol . . 363 at 172 F.
Cast iron 42.5 Ether . . . 162 at 93 F.
Ice 142.6 Mercury . . 117 at 580 F.
Lead 9.66 Water. . . 965.7 at 212 F.
Silver 43. Water . . . 1044.4 at 100 F.
Tin 27. Water. . . 1091.7 at 32 F.
Zinc 54.
PROBLEMS
1. Find the number of British thermal units required to
melt the following masses of metal after they have been
brought to the melting point : (a) 120 Ib. of iron ; (b) 24 Ib. of
lead ; (c) 55 Ib. of silver ; (d) 40 Ib. of tin.
SOLUTION, (a) 42.5 x 120 = 5100 B. t. u.
2. How much heat is given out by 50 Ib. of molten zinc as
it becomes solid ?
210 APPLIED MATHEMATICS
3. How much heat is required to melt 16 Ib. of tin at 70
if its melting point is 442 ?
SOLUTION. Specific heat of tin = .06.
442  70 = 372.
16 x 372 x .06 = 357 B.t.u. to raise to 442.
16 x 27 = 432 B. t. u. to melt.
789 B. t. ii., total.
4. How much heat is required to melt 150 Ib. of lead at 70
if its melting point is 622 ?
5. 1 T. of molten iron at 2200 cooled to 70. How much
heat was given off if the melting point was 2000 ?
6. A cake of ice weighing 50 Ib. is at 0. How many British
thermal units are required to melt it and bring the water to
the boiling point ?
7. If 1 Ib. of ice at 32 is put into 2 Ib. of water at 80, how
much of the ice will melt ?
SOLUTION. (80  32) 2 = 96 B. t. u. available to melt the ice.
142.6 = number of British thermal units re
quired to melt 1 Ib. of ice at 32.
= .67 Ib. ice melted.
142.6
Check. 142.6 x .67 = 96.
Y + 32 = 80.
8. How much boiling water will be required to melt 12 Ib.
of ice at 32 ?
9. What would be the final temperature of the water if 16 Ib.
of ice at 32 were put into 40 Ib. of boiling water ?
SOLUTION. Let t = the final temperature.
142.6 x 16 = number of British thermal units to melt
the ice.
40 (212 t) = number of British thermal units lost.
16 (t  32) + 142.6 x 16 = number of British thermal units gained.
40 (212  = 16 (t  32) + 142.6 x 16.
Solve and check.
HEAT 211
10. 5 Ib. of molten lead at the melting point 610 were poured
into 50 Ib. of water at 70. What is the resulting temperature ?
11. 1 Ib. of lead at 212 is placed on a cake of ice at 30.
How much ice will it melt ?
12. How many pounds of steam at 212 will melt 20 Ib. of
ice at 32?
13. How many pounds of zinc at 500 must be added to
100 Ib. of water at 75 to heat it to 100 ?
14. 20 Ib. of ice at are immersed in 200 Ib. of water at
200. What is the temperature of the water when the ice has
just melted ?
15. How many pounds of water at 70 would be evaporated
at 212 by 1 T. of coal, assuming an efficiency of 12 per cent,
and 13,500 B. t. u. per pound of coal ?
16. The temperature of 1 Ib. of water in a teakettle rises
from 32 to 212 in ten minutes. () How long before the
kettle will boil dry ? (ft) If the kettle contained 5 Ib. of water,
how many British thermal units would be needed to boil it dry ?
SOLUTION, (a) 212  32 = 180.
180 1QTJ , . .
= 18 B. t. u. per minute.
965.7 , Q 7 . ,
= 53.7 minutes.
18
17. If 1 Ib. of ice at is put on an electric stove which gives
out 8 B. t. u. per minute, find the number of British thermal
units and the number of minutes required (r/) to raise the ice
to 32; (ft) to melt the ice; (c) to raise the water to 212;
(rf) to evaporate the water ; (e) to raise the steam to 312. Con
struct a graph and write the results on it. Units : horizontal,
1 large square == 10 minutes ; vertical, 1 large square = 20.
CHAPTER XVII
ELECTRICITY
111. Exercise. Into a tumbler two thirds full of water pour
2 ccm. of sulphuric acid. Stand in this solution a strip of zinc
and a strip of copper each well sandpapered. Take 6 ft. of No. 20
insulated copper wire and wind about 25 turns around a large
lead pencil, leaving about a foot uncoiled
at each end. Cut the insulation from the
ends of the wire and wrap the ends
around the strips, as shown in Fig. 82.
To get good connections it may be neces
sary to cut into the edge of the strips
and wedge the wire under the pieces
lifted.
Take a piece of soft wrought iron and
sprinkle some iron filings on each end.
Kesult ? Place the iron within the coil,
as shown in Fig. 82, and drop some iron
filings on the ends. Result? Is the iron
magnetized ? If so, we have generated a current of electricity and
magnetized the iron. (See Shepardson's "Electrical Catechism.")
112. Nature of electricity. The exact nature of electricity
is not known. Some scientists think it is a condition of the
ether. Others think that it is a form of energy or force. How
ever, much is known about the laws of electricity and about
methods of applying it to useful work.
113. Electromotive force. When the strips of copper and
zinc were placed in the solution of sulphuric acid, the acid dis
solved the zinc strip faster than it did the copper strip. We
212
FIG. 82
ELECTRICITY 213
say that this caused an electrical flow from the zinc to the
copper ; that is, an electromotive force exists between the two
strips. Whatever produces or tends to produce an electrical
flow is called an electromotive force (e. m. f .). When the two
strips are connected by the wire this action takes place con
tinuously and there is said to be a flow of electricity from the
zinc to the copper and through the wire to the zinc again.
Though we cannot perceive this flow by any of our senses, we
can see the effects it produces.
114. The electrical units. It is not possible nor is it neces
sary to give exact definitions here. However, definitions can
be given which are readily understood and are sufficiently exact
for practical purposes.
The volt. We may think of the electromotive force existing
between the strips of zinc and copper in the cell described
above, as pressure. It takes pressure to force a current of elec
tricity through a wire, just as it takes pressure to drive a stream
of water through a pipe. To measure this pressure we have
the unit of electromotive force called a volt (from Volta, an
Italian physicist who lived from 1745 to 1827). The pressure
of a gravity or crowfoot cell is about 1.1 volts. When a wire
is moved across the magnetic lines of force which exist between
the poles of a magnet, an electrical flow is produced in the
wire. A volt is the electromotive force set up in a wire that
crosses magnetic lines of force at the rate of one hundred mil
lion per second. In a dynamo the armature may be thought of
as a bundle of wires which cut across the lines of force of the
field magnet as the armature revolves.
The ohm. The pressure (electromotive force) produces a
flow of electricity which meets with resistance in the conductor.
Just as the frictional resistance in a water pipe opposes the
flow of water, so the electrical resistance of a conductor opposes
the flow of electricity. The unit of resistance is the ohm (from
Ohm, a German mathematician who lived from 1789 to 1854).
214 APPLIED MATHEMATICS
The ohm is nearly equal to the resistance of 1000 ft. of copper
wire .1 in. in diameter. Different substances have different de
grees of resistance. The resistance of metals increases slightly
as the temperature rises, but the resistance of carbon (incandes
cent lamp filament) decreases with a rise in temperature. Thus
the resistance of a 16 candle power 1 incandescentlamp carbon
filament is about 220 ohms when hot, but it may, be as high as
400 ohms when cold.
Resistance varies directly as the length and inversely as the
cross section of a conductor. Thus if the resistance of 100 ft.
of wire is 2 ohms, the resistance of 300 ft. of the same wire
is 6 ohms ; if the resistance of a wire .3 in. in diameter (cross
section, 7.07 sq. in.) is 8 ohms, the resistance of a wire of the
same material and length .6 in. in diameter (cross section,
28.27 sq. in.) is 2 ohms.
The ampere. The unit for measuring the rate of the electri
cal flow is the ampere. An ampere (from Ampere, a French
physicist who lived from 1775 to 1836) may be defined as the
current which an electromotive force of 1 volt will send through
a conductor whose resistance is 1 ohm.
The number of amperes of current corresponds quite closely
to the rate of flow of a stream of water. We may say that at a
certain point in an electrical circuit the rate of flow is 5 amperes,
just as we would say that at a certain point in a water pipe the
rate of flow of the water is 10 gal. per minute.
Given an electromotive force of 1 volt, a circuit of 1 ohm
resistance, and we have a current of 1 ampere.
115. Ohm's law. A very simple relation exists between the
electromotive force, resistance, and current in a closed circuit.
Let V = the number of volts of electromotive force,
R = the number of ohms of resistance,
A = the number of amperes of current,
y
and we have Ohm's law, = A.
R
ELECTRICITY 215
In words this law may be stated as follows : The number of
volts of electromotive force divided by the number of ohms
of resistance gives the number of amperes of current flow
ing through a circuit. This law was first formulated by Ohm
in 1827.
PROBLEMS
1. How many amperes are there in a circuit of 20 ohms
resistance if the dynamo generates 110 volts ?
F 110
SOLUTION. =  = 5.5 amperes.
j\, \)
2. A battery sends a current of 5 amperes through a circuit.
If the electromotive force is 10 volts, find the total resistance
of the circuit.
3. If a cable has a resistance of .004 ohm and a current of
20 amperes passes through it, what is the electromotive force ?
4. Find the resistance of an incandescent lamp which takes
a current of .5 ampere when connected to a 110volt main.
5. If a telegraph wire has a resistance of 200 ohms, how
many amperes will be sent through it by a pressure of 10 volts ?
6. The wires in an electric heater will stand 8 amperes
without becoming unduly heated. What must be the resistance
for 110 volts ?
7. A dynamo generates 110 volts. What is the total resist
ance of the circuit if there is a current of 40 amperes ?
8. A 32 candle power lamp for a 220volt circuit has a
resistance of 330 ohms, and a 16 candle power lamp for a
110volt circuit has a resistance of 180 ohms. Which lamp has
the greater current ?
9. Construct a curve to show the relation between the
electromotive force and the resistance of a circuit in which
the current is 20 amperes, as the resistance varies from 1 to
10 ohms.
216 APPLIED MATHEMATICS
10. If the electromotive force of a dynamo remains constantly
at 120 volts, construct a curve to show the changes in the
current as the resistance increases from 10 to 120 ohms.
116. Resistances in combination. In the preceding problems
the resistance of the circuit was considered as a single resist
ance, but in practical work the circuit is made up of several
parts. Thus in an electric lighting system the total resistance
is made up of the resistances of the dynamo, lamps, and con
necting wires. The parts of a circuit may be combined in two
distinct ways.
117. Series circuits. When the different parts of a circuit
are joined end to end and the whole current flows through each
part, the circuit is called a series cir
cuit. Let D in Fig. 83 be a dynamo
maintaining an electromotive force of
110 volts measured across the termi
nals AB. This means that 110 volts A ^ j< IG 33
are continuously generated and used
up in forcing the current through the circuit BCEFA . Hence we
may say that from B to A there is a drop in voltage of 110 volts.
Let C, J5J, and F be arc lights having resistances of 4.2, 4.6,
and 4.8 ohms respectively, and let the resistance of the line
be 4 ohms.
Total resistance = 4.2 + 4.6 + 4.8 + 4
= 17.6 ohms.
By Ohm's law, = r=^ = 6.25 amperes.
H 17. b
At any point in the circuit the current is 6.25 amperes, since
in a series circuit the current is constant. But there is a con
tinual drop in the voltage along the circuit as the voltage is
used up in forcing the current along its path. This drop in
voltage, or drop of potential, as it is sometimes called, follows
Ohm's law.
ELECTRICITY 217
The drop in lamp C = A  R = 6.25 x 4.2 = 26.2 volts.
The drop in lamp E = A  It = 6.25 x 4.6 = 28.8 volts.
The drop in lamp F = A  R = 6.25 x 4.8 = 30.0 volts.
The drop in the line = A R = 6.25 x 4 = 25.0 volts.
Total drop = 110.0 volts. Check.
The arc lights in general use to light city streets are connected
in series, and the entire current goes through each lamp.
118. Ammeter. The number of amperes of current is meas
ured by an ammeter. It consists of a coil of wire suspended
FIG, 84
between the poles of a magnet so that it rotates through a
small angle when the current passes through. The coil carries
a light needle. The instrument is graduated by passing through
it currents of known strength, and marking on the scale the
position of the needle. The type of ammeter commonly used
is cut into the circuit when a measurement is made.
119. Voltmeter. Voltage (electromotive force, drop of poten
tial) is measured by the voltmeter. Most voltmeters are simply
special forms of ammeters. The voltmeter also is graduated
by experiment. It is put on. circuits of known voltage and the
218
APPLIED MATHEMATICS
position of the needle is marked on the scale. In using the
voltmeter its terminals are connected to the ends of the parts
of the circuit in which the voltage is to be measured; the
FIG. 85
reading of the voltmeter is the number of volts of electro
motive force, or drop of potential. If a voltmeter is connected
across the terminals of an arc light and the reading is 47 volts,
it means that 47 volts are used up in running that arc light.
In Fig. 86 the ammeter A is arranged to measure the current
produced by the dynamo I) ; and the voltmeter V is connected
FIG. 86
FIG. 87
to show the electromotive force between the terminals of the
dynamo. Fig. 87 shows an ammeter and a voltmeter arranged
to measure the current and drop in voltage in an arc lamp L.
ELECTRICITY 219
PROBLEMS
1. Three wires having resistances of 2, 5, and 8 ohms
respectively are joined end td end and a voltage of 90 volts is
applied. How many amperes of current are there ?
2. Two wires of resistances 6 and 8 ohms respectively are
joined in series. If the current is 1.8 amperes, find the voltage.
3. Two incandescent lamps are in series and one has twice
as great resistance as the other. If the voltage is 110 and the
current is ^ of an ampere, find the resistance of each lamp.
SOLUTION. Let R = the resistance of one lamp.
2 R = the resistance of the other lamp.
V = 110 = 1
R 3/J~3 '
R = 110 ohms.
2JFJ = 220 ohms.
Total = 330 ohms.
F 110 1
4. Find the internal resistance of a battery which gives a
current of 1.5 amperes with an electromotive force of 5 volts,
if the external resistance is 1.33 ohms.
5. An iron wire and a copper wire are in series. If the
voltage is 12 volts, the current 2.8 amperes, and the copper wire
has a resistance of 3 ohms, find the resistance of the iron wire.
6. A circuit consists of two wires in series. An electro
motive force of 30 volts gives a current of 3.2 amperes. If the
length of one wire is doubled and the other is made 5 times
as long, the current is .84 ampere. Find the resistance of the
two wires.
7. What voltage is necessary to furnish a current of 9.6
amperes, if the circuit is made up of 2 mi. of No. 6 Brown &
Sharpe gauge copper wire (resistance of 1000 ft., .395 ohm) and
10 arc lights in series, each of 4.8 ohms resistance ? Find also
the drop in voltage in the wire and in the lamps.
220 APPLIED MATHEMATICS
8. A dry cell is used to ring a door bell. The resistance of
the wire in the bell is 1.5 ohms, of the line .5 ohm, and of the
cell 1 ohm. If the electromotive force of the cell is 1.4 volts,
what current will flow when the circuit is closed ?
9. What is the resistance per mile of No. 20 Brown & Sharpe
gauge copper wire, if the voltmeter connected to the ends of
100 ft. of the wire reads 5.13 volts and the ammeter reads
6 amperes ?
10. An arclight dynamo of 30 ohms resistance supplies a
current of 6.8 amperes through 12 mi. of No. 6 Brown & Sharpe
gauge copper wire to a series of 60 arc lights, each adjusted to
47 volts. Find the electromotive force of the dynamo.
Suggestion. The drop in voltage in the lamps = 47 X 50. Find
the drop in voltage in the dynamo and in the line by V=R A. The
total voltage is the sum of the drop in voltage in the three parts of
the circuit. Check by finding the total resistance of the circuit and
dividing it into the total electromotive force; this should give 6.8
amperes.
11. In an electric lighting system there are 6 mi. of No. 6
Brown & Sharpe gauge copper wire, and 80 arc lights, each
having a resistance of 4.5 ohms. The resistance of the dynamo
is 3 ohms and the electromotive force is 3725 volts. Find
(a) the total resistance ; () the current ; (c) the fall of voltage
in the dynamo, line, and lamps.
12. The voltage across the mains of an electriclight circuit
is 110 volts. If a voltmeter is connected across the mains in
series with a resistance of 6000 ohms, it reads 70 volts. What
is the resistance of the voltmeter ?
SOLUTION. Since there is a drop of 70 volts in the voltmeter,
there is a drop of 110 70 = 40 volts in
the resistance.
V 40 1
ELECTRICITY . 221
Since the current is the same in all parts of the circuit, =  =
10,500 ohms, resistance of the voltmeter.
150
Check. 10,500 + 6000 = 16,500 ohms.
V 110 1
R = 16^00 = 150
13. A coil of wire is placed in series with a voltmeter having
a resistance of 18,000 ohms across 110volt mains. If the volt
meter reading is 60 volts, find the resistance of the coil of wire.
14. A voltmeter has a resistance of 10,000 ohms. What
will be the reading of the voltmeter when connected across
110volt mains with a man having a handtohand resistance
of 5000 ohms ?
120. Multiple circuits. When the branches of a circuit are
connected so that only a part of the current flows through each
of the several branches, the circuit is ^ _
called a multiple, parallel, or divided s*\ /S xk
circuit. Fig. 89 shows three incandes III
cent lamps connected in multiple. The F IO . 89
ordinary incandescent lamps used in
houses are connected in multiple between mains from the
terminals of the dynamo. The full electromotive force of the
dynamo, except the drop in voltage in the wires, acts upon each
lamp ; but only a part of the current goes through each lamp.
121. To find the total resistance of a multiple circuit. In
Fig. 90 let the drop in voltage from B to C be 12 volts, and
a and b have resistances of 2 and 4 ohms respectively. The
pressure (electromotive force) in each branch is 12 volts ; just
as in a water pipe of similar construction the pressure would
be the same in each branch.
V 12 A
= 5 = 6 amperes in a.
K i
12
7 = 3 amperes in b.
* FIG. 00
222
APPLIED MATHEMATICS
Hence the total current is 6 + 3 = 9 amperes.
The total resistance of a and b is given by
V 12 4
We will now obtain a general formula for the total resistance
of a multiple circuit.
Let V = the drop in voltage from B to C.
r x = resistance of a.
r^ = resistance of b.
R = total resistance.
= current in a.
r i
v
= current in b.
V V V fa + r a ) _
But
= total current.
R
or
R
Jn a multiple circuit of two branches the total resistance is the
product of the resistances divided by their sum.
In a similar manner let the student work out the formulas
for three and four branches, obtaining :
i8
4 r, V, + r 2 r $ r 4 +
/o\
^ >
ELECTRICITY 223
When two, three or four equal resistances are combined in
multiple, we have from (1), total resistance = = 5 >
from (2), total resistance = 2 = ^ >
J T O
t* t*
from (3), total resistance = jj = 
Thus when ten 16 candle power lamps of 220 ohms resistance
are connected in multiple the total resistance is
220
10
= 22 ohms.
122. Graphical method of finding the total resistance. The
total resistance of a multiple circuit can be readily determined
by a graph.
EXERCISES
1. Construct a graph to find the result of combining resist
ances of 20 and 30 ohms in multiple.
Take OX any convenient length, and with convenient units lay off
OM=30ohms, and XN = 20
ohms. Draw ON and XM, inter
secting at A. AB = 12 ohms, the
total resistance.
That is, ^45 =
OMXN
OM+XN
(1)
Prove geometrically. The two
pairs of similar triangles OB A,
OXN ; and XBA , X OM give two
equations. Eliminating XB from
these equations gives (1). (See
Problem 14, p. 77.)
Y
30
35
26
15
10
5
M
\
\
N
\
\
x
/
p
\
AS
/
/
' X ^
^
%\
/
X
/
\
O
DO *
A similar construction gives IT IG> 92
the total resistance of any
number of resistances in multiple. Thus, given the resistances
20, 30, and 18 ohms, combine 20 and 30 ohms, as above. Then
224
APPLIED MATHEMATICS
lay off X P = 18 ohms. Draw
PJ3, intersecting XA at C, and
CD is the total resistance.
2. What resistance must be
combined with 24 ohms to
obtain a total resistance of
3 ohms?
Y
3O
35
20
15
10
5
P
<s
X
x
"V
X
^
M
,^^
^
***^~
X
^
***"
*^"
1
>>
X
B X
Take OX any convenient
length and with a convenient unit
lay off OM = 24 ohms. Draw
3/X. On MX take a point ^4,
such that AB = 8 ohms. Draw
O.4, and extend to meet XP at
A r . A" 2V 12 ohms, the required resistance.
The graphical method should be used to solve and check some of
the following problems.
PROBLEMS
FIG. 93
1. Three resistances of 20, 30, and
40 ohms are joined in multiple. Find
the total resistance.
20.3040
2O ohms
SOLUTION. R =
20.30 + 20.40
= 9.2 ohms.
3040
FIG. 94
2. If 110 volts be applied to the circuit in Problem 1,
what is the total current and the current in each branch ?
F 110
SOLUTION.
= 12 amperes.
R 9.2
jO. 5.5 amperes.
jO. = 3.7 amperes.
(T = 2.8 amperes.
12 amperes.
Check.
3. Three lamps having resistances of 60, 120, and 240 ohms
are connected in multiple. If they are supplied with 110 volts,
find the total resistance, the total current, and the current in
each lamp.
ELECTRICITY
225
4. Two lamps of 100 and 150 ohms are put in parallel with
each other, and the pair is joined in series with a lamp of
100 ohms. If the electromotive force is 200 volts, what will
be the current?
5. A resistance of 10 ohms is put in parallel with an un
known resistance. If an electromotive force of 120 volts gives
a current of 20 amperes, find the unknown resistance.
SOLUTION. Let
Then
r = the unknown resistance.
= the total resistance.
10 r
10 +
120
= 6 = the total resistance.
10 r
= 6.
Check.
10 + r
10 r = 60 + 6 r.
r 15 ohms.
10 x 15 150
10 + 15 25
= 6 ohms.
6. A lamp of unknown resistance is put in parallel with a
lamp of 220 ohms resistance. If a voltage of 110 volts gives
a current of 1.6 amperes, what is the unknown resistance ?
7. The total resistance of three wires in multiple is 1.52
ohms. If the resistance of two of the wires is 3 and 5 ohms
respectively, what is the resistance of the third ?
8. The total resistance of two conductors in multiple is
4.8 ohms, and the sum of the two resistances is 20 ; find them.
JCohms
5 ohms
FIG. 96
9. The total resistance between A and B in Fig. 95 is
5.25 ohms. Find the resistance x.
226 APPLIED MATHEMATICS
10. Three resistances in parallel are in the ratio 1:2:3.
If an electromotive force of 120 volts gives a current of
11 amperes, find each resistance.
11. Twenty 16 candle power 110volt lamps are in multiple.
If the resistance of each lamp is 220 ohms, what is the total
resistance, and what is the current ?
12. A 110volt incandescent lighting circuit divides into
three multiple circuits of 5, 8, and 10 lamps respectively. If
the resistance of each lamp is 220 ohms, find (a) the resistance
of each branch ; (&) the total resistance ; (c) the current ; (d) the
current in each branch.
13. Construct a curve to show the change in the resistance
of a multiple circuit consisting of a number of incandescent
lamps of 220 ohms, as the number of lamps increases from
1 to 20.
14. Two conductors of 12 and 18 ohms respectively are in
multiple. What resistance must be placed in series with the
multiple circuit to give a current of 5 amperes with an electro
motive force of 110 volts ?
WORK AND POWER
123. The watt. When an electromotive force overcomes the
resistance of a conductor and causes a current to flow, work is
done. This is analogous to the case where work is done by the
pressure of steam on the piston of an engine. The number of
pounds pressure multiplied by the number of feet through
which the piston is moved gives the number of foot pounds of
work. Power is the rate of doing work. The unit of mechanical
power is the horse power, the rate of doing work equal to
33,000 ft. Ib. per minute. The unit of electrical power is the
watt (James Watt, Scotland, 17361819, practically the in
ventor of the modern steam engine), the rate of doing work
equal to 44J ft. Ib. per minute.
ELECTRICITY 227
Power in watts equals the number of volts multiplied by the
number of amperes.
W=V.A. (1)
Thus if a dynamo supplies a current of 50 amperes at a voltage
of 110 volts, the power delivered is 110 x 50 = 5500 watts.
V
From Ohm's law  = A, (1) may be written
.iC
W = A 2 R. (2)
72
Wg. (3)
In words : watts equal volts multiplied by amperes; (1)
watts equal current squared multiplied by resistance; (2)
watts equal volts squared divided by resistance. (3)
To express watts in horse power :
Since 1 h. p. = 33,000 ft. Ib. per minute,
and 1 watt = 44 ft. Ib. per minute,
_ 33,000 .,
1 h. p. = ' watts.
44J
1 h. p. = 746 watts.
124. The kilowatt. For many purposes a larger unit than
a watt is convenient. Hence 1000 watts, called a kilowatt (kw.),
is sometimes taken as the unit of power.
125. The kilowatt hour. A kilowatt hour is a practical unit
used in measuring electrical energy. It is the energy expended
by 1 kw. in 1 hr. Thus 20 kw. hr. might mean 2 kw. for 10 hr.,
6 kw. for 4 hr., 1 kw. for 20 hr., and so on.
1 kw. hr. = 44J x 1000 x 60 ft. Ib.
1 h. p. hr. = 33,000 x 60 ft. Ib.
i i v 44 i X 1000 x 60 , .
Hence 1 kw. hr. = x 60 h ' P ' hr '
1 kw. hr, = 1.34 h. p. hr.
228 APPLIED MATHEMATICS
PROBLEMS
1. An arc light requires 10 amperes at 45 volts. How much
power does it absorb ?
SOLUTION. W = VA = 45 x 10 = 450 Watts.
2. A 16 candle power incandescent lamp is on a 110volt
circuit and takes J ampere. How many watts per candle
power are required ?
SOLUTION. W = V A = 110  = 55 watts.
55
= 3.5 watts per candle power.
16
3. A dynamo has a voltage of 550 volts and is producing
40 kw. How many amperes in the current ?
4. How many watts will be lost in forcing the current
through the armature of a dynamo, if the resistance is .035 ohm
and the current is 30 amperes ?
5. A 150kw. dynamo was supplying 273 amperes. What
was the voltage of the dynamo ?
6. How many horse power are required to send a current of
65 amperes through 10 mi. of No. 6 B. & S. gauge copper wire ?
7. A current of 15 amperes flows through 100 ft. of iron
wire whose resistance is ohm per foot. How many watts are
lost in the wire ?
8. With a current of 50 amperes 450 watts are absorbed in
the conductor. Find the drop in voltage in the conductor.
9. A voltmeter has a resistance of 17,000 ohms. If placed
in a circuit of 110 volts, how much power is required to
operate it?
10. A 200 volt lamp takes J ampere. How many watts are re
quired for 30 such lamps ? How many horse power are required
to drive the dynamo if it has an efficiency of 90 per cent ?
SOLUTION. 200 x x 30 = 2000 watts.
2000
ELECTRICITY 229
= 2000 watts.
= 2.68 h. p. 100 per cent efficiency.
746
2.68
.90
= 2.98 h. p. 90 per cent efficiency.
11. In a room there are thirty 16 candle power incandescent
lamps, each taking .52 ampere at 110 volts ; and 3 arc lights,
each taking 6.8 amperes at 50 volts. How many watts and how
many horse power are required to operate these lights ?
12. How many incandescent lamps, each having a resistance
of 220 ohms and requiring a current of .5 ampere can be run
by a 10kw. generator ?
13. A 25 h. p. dynamo is running at 550 volts. How many
amperes in the current ? How many 16 candle power incandes
cent lamps can be placed on the circuit if each lamp takes
55 watts and there is a loss of 10 per cent on the line ?
14. In an electriclighting circuit there are 60 arc lights, each
taking 50 volts, and 15 mi. of wire having a resistance of
2.1 ohms per mile. If the current is 9.6 amperes, how many
watts are required to run the lights ?
15. Find the energy in foot pounds expended per candle
power in a 16 candle power incandescent lamp in 1 hr., if it
takes ampere at 110 volts.
16. If a 500 candle power arc light requires 50 volts with
9.6 amperes, how many foot pounds per candle power are ex
pended in 1 hr. ? How does this compare with the result in
Problem 15 ?
17. A 10kw. dynamo has an efficiency of 88 per cent. How
many horse power are required to drive it ?
18. The efficiency of a dynamo is 85 per cent. How many
horse power are required to drive it when there are 200
16 candle power lamps on the circuit, each lamp taking J
ampere at 110 volts?
230 APPLIED MATHEMATICS
19. How many amperes at 120 volts must be furnished a
hoisting motor which is to lift 900 Ib. 70 ft. per minute, if it
has an efficiency of 70 per cent ?
20. A motor operates a pump which in 1 hr. lifts 20,000 gal.
of water (1 gal. = 8^ Ib.) 400 ft. If the combined efficiency of
the pumping system is 72 per cent, what current will the motor
require at 550 volts ?
21. An electric street car with its load weighs 8 T. ; on a
level track the pull required is 20 Ib. per ton. How much power
is necessary at the axle to move the car 10 mi. per hour ? If
the motor and gearing have an efficiency of 75 per cent, how
many amperes are required on a 550volt circuit ?
22. To perform a certain amount of work 30 kw. hr. are
required. If the dynamo gives a current of 125 amperes at
220 volts, how long must it be used to perform this work ?
125 x 220   .
SOLUTION. = 27.5 kw.
30 = 1.09 hr.
27.5
23. A 5kw. motor is used to operate a printing press 8 hr.
What will be the cost of the power at 12 cents per kilowatt
hour ?
24. What is the cost of running a motor which requires
15 amperes at 110 volts, at 12 cents per kilowatt hour ?
25. An incandescent lamp takes .6 ampere at 110 volts. If
power costs 15 cents per kilowatt hour, what is the cost of
operating the lamp 12 hr. ?
26. An inclosed arc lamp takes 80 volts on a current of 6.6
amperes. How much does it cost to operate the lamp 12 hr.
at 15 cents per kilowatt hour ?
27. How many watts per candle power are required in each
of the following lamps ? If power costs 10 cents per kilowatt
hour, how much would it cost per hour to keep each lamp at
ELECTRICITY
231
full candle power? Construct a curve to show the relation
between the candle power of each lamp and the cost per
candle power.
Volts
Candle power
Amperes
Ohms
110
10
.32
344
110
16
.51
216
110
20
.64
172
110
24
.76
145
110
32
1.02
108
28. From the equation V x A = W construct a series of
curves on the same axes for W equal to 1, 2, 3, 4, 5 kw. Know
ing the voltage and current in a circuit, by means of these
curves the approximate power can be determined readily.
HEAT GENERATED BY A CURRENT
126. Heat loss in a conductor. We have seen that it takes
pressure (voltage) to drive a current through a conductor, and
we have computed this fall of potential. Thus if a current of
10 amperes flows through a resistance of 2 ohms, the amount
of voltage required to send the current is F = ^4/2 = 10 x 2 =
20 volts. We have also computed the loss of power. Thus the
number of watts lost is V X A = 200 watts. This power or
energy which is lost in the conductor is changed into heat.
We may say in the above problem that the heat loss is 200
watts per minute.
Hence to find the heat loss in a conductor we simply find the
watts lost, and, if desired, change the watts into calories or
British thermal units.
Hence
1 watt minute = 44.25 ft. lb. per minute.
1 B. t. u. = 778 ft. lb.
1 watt minute = .057 B. t. u. per minute
232 APPLIED MATHEMATICS
PROBLEMS
1. Find the heat loss due to a current of 60 amperes through
a resistance of 10 ohms.
SOLUTION. W = A*  R = 60 2 x 10 = 36,000 watts in 1 min.
2. A conductor having a resistance of 5 ohms carries a cur
rent of 18 amperes. How much heat is developed in 1 hr. ?
3. How much heat is developed in a wire of 15.2 ohms re
sistance by a current of 8 amperes in 15 min., (a) in watts ?
() in calories ? (c) in British thermal units ?
4. A current of 36 amperes is sent over a line of 2 ohms
resistance. What is the drop in voltage ? What is the heat
loss per hour (a) in watts ? (II) in British thermal units ?
5. A current of 12 amperes flows through a, resistance of
3.2 ohms for 15 min., and another current of 8 amperes flows
through a conductor of 2.5 ohms resistance. How long must
the second current flow in order that the amount of heat gen
erated may be the same as in the first case ?
6. Construct a curve to show the heat loss in a conductor
as the resistance changes from 1 to 10 ohms while the current
remains constantly 5 amperes.
7. Construct a curve to show the heat loss in a conductor
of 1 ohm resistance as the current varies from 10 to 20 amperes.
8. In a conductor of 10 ohms resistance the voltage increases
from 10 to 1000 volts. Construct a curve to show the heat loss.
9. A Leclanch^ cell used to ring a doorbell has an electro
motive force of 1.6 volts and the current is .75 ampere. If the
wire has a resistance of .4 ohm, what per cent of the power
is the heat loss in the line ?
SOLUTION. 1.6 x .75 = 1.2 watts, total power.
A*>R = .75 a x .4 = .23 watts, heat loss in line.
^ = 19 per cent.
ELECTRICITY 233
10. The dynamo of an arclight system furnishes a current
jf 9.6 amperes at 3000 volts. The circuit is made up of 16 mi.
of No. 6 B. & S. gauge copper wire. What per cent of the power
is the heat loss in the line ?
11. A 6in.plate stove requires 5.5 amperes at 110 volts.
What is the cost of running it 30 min. if the current costs
6 cents per kilowatt hour ?
12. Find the cost of heating a 6lb. flatiron for 3 hr., if it
takes 4 amperes at 110 volts, at 6 cents per kilowatt hour.
13. An electric radiator takes 13.6 amperes at 110 volts.
Find the cost for 8 hr. at 6 cents per kilowatt hour.
14. In an electric heater there is a coil of iron wire 224 ft.
in length having a resistance of  ohm per foot. If it is con
nected to a 110volt circuit, how much heat is generated ?
15. It is desired to make an electric soldering iron to be
heated by a coil of No. 27 German silver wire of resistance
1.25 ohms per foot. How many feet will be required to give
200 watts on a 500volt circuit ?
WIRING FOR LIGHT AND POWER
127. The mil. In electrical calculations involving the diam
eter of wire, the mil is usually taken as the unit of length.
1 mil = .001 in.
A circular mil is a circle whose diameter is 1 mil.
A circular mil = TT r 2 = TT x .5 2 = .7854 sq. mils.
1 circular mil = .7854 sq. mils.
1 sq. mil = 1.273 circular mils.
Circles are to each other as the squares of their diameters.
Hence to find the area of a circle in circular mils, square its
diameter expressed in mils.
A mil foot of wire is 1 ft. long and 1 mil in diameter. In
practice the resistance of 1 inilft. of copper wire is usually
taken as 10.7 ohms.
234 APPLIED MATHEMATICS
PROBLEMS
1. The diameter of a wire is in. Find (a) its diameter in
mils ; (ft) its cross section in circular mils.
2. How many circular mils in the cross section of a wire of
diameter (a) in. ? (ft) .125 in. ? (o) .06 in. ?
3. Find the diameter and area in square mils of a wire
whose cross section is (a) 10,381 circular mils ; (ft) 26,250
circular mils ; (c) 105,590 circular mils.
4. A copper bar is 1 in. by in. Find the area of a cross
section in square mils and in circular mils.
5. Find the resistance of 1000 ft. of copper wire 40 mils in
diameter.
SOLUTION. The cross section = 40 2 = 1600 circular mils.
Resistance of 1 mil foot = 10.7 ohms.
Resistance of 1 ft. of wire of 40 mils diameter
10 7
= ^  .00669 ohm.
1600
Resistance of 1000 ft. of wire of 40 mils diameter
= .00669 x 1000 = 6.69 ohms.
6. Find the resistance of 1000 ft. of copper wire that has a
diameter of (a) 460 mils ; (ft) 289.3 mils ; (c) .1 in. ; (d) 40.3
mils ; (e) 20 mils.
7. A current of 75 amperes is sent through 1 mi. of copper
wire 229.4 mils in diameter. Find the drop in voltage.
SOLUTION. Cross section = 229.4 2 = 52,620 circular mils.
Resistance of 1 ft. = = .000203 ohm.
52,620
Resistance of 1 mile = .000203 x 5280
= 1.07 ohms.
V = A . R = 75 x 1.07 = 80.3 volts.
8. What is the drop in voltage in a circuit of 5 mi. of copper
wire 162 mils in diameter if the current is 40 amperes ?
ELECTRICITY 235
9. With a current of 210 amperes what will be the drop in
voltage in 2500 ft. of copper wire 460 mils in diameter ?
10. How many circular mils are required in a power line
500 ft. long with a current of 150 amperes, if a drop of 12 volts
is allowed ?
SOLUTION. Let n = number of circular mils required.
10.7 x 500 = 5350 ohms per circular mil for 500 ft.
= number of ohms resistance of n circular mils
n for 500 ft.
5350 x 150
n
5350 X 150
= number of volts in drop.
= 12.
n
n = 66,880 circular mils.
, io/7
Check.  " = .00016 ohm, resistance of 1 ft. of the wire.
66,880
.00016 x 500 = .08 ohm, resistance of 500 ft.
x .08 = 12.
From the above solution we may obtain the following formula,
which is in general use for finding the size of conductor required
to carry a given load.
vr . , ., 21.4 x distance one way in feet x amperes
No. circular mils =  ,, , ,   
volts lost
11. A motor is 300 ft. from the dynamo. How many circular
mils are required for a current of 90 amperes, if a drop of
6 volts is allowed?
12. Find the number of circular mils required to deliver
10 kw. to a motor at a distance of 200 ft., with 100 volts pressure
at the motor, if a drop of 5 volts is allowed in the line.
13. Find the number of circular mils required to transmit
25 kw., with a 20 per cent drop in the voltage, a distance of
10 mi., if the voltage at the load is to be (a) 100 volts ; (b) 500
volts ; (c) 1000 volts.
100 *
Suggestion.  = 125 volts at the dynamo,
.80
236 APPLIED MATHEMATICS
14. It is required to deliver 120 h. p. to a motor 2 mi. away,
from a dynamo which has a voltage of 550 volts. If the line
loss is to be not more than 16 per cent, find the cross section
of the wire in circular mils and the number of pounds of
copper required.
DYNAMOS AN* MOTORS
128. Construction. When a closed wire is rotated between
the poles of a magnet so as to cut the lines of force, a current
flows in the wire. The dynamo is
constructed on this principle. The
armature is the part of the ma
chine in which the current is gen
erated, and in most machines the
armature revolves. The field is
the space between the poles of the
magnets in which the armature
revolves. The magnets are pieces of soft iron, which are mag
netized by a current from the machine itself or from a sepa
rate dynamo. This current flows in coils which are placed
around the magnets. In Fig. 96 the armature is represented
by a single wire revolving in the field.
129. The field coils connected in three ways. Direct current
dynamos generally excite their own fields ; and there are three
ways of connecting the fieldmagnet coils.
1. Serieswound dynamos. The fieldmagnet
coils are connected to the armature so that
the whole current generated passes through
them. FIG. 97
2. Shuntwound dynamos. The fieldmag
net coils are connected in multiple with the
terminals of the armature ; hence only a
part of the current goes through them.
These coils consist of many turns of comparatively fine wire.
ELECTRICITY 237
3. Compoundwound dynamos. The field magnets are wound
with two sets of coils, one in series and
one in multiple with the armature.
Motors are also wound in these three
ways.
FIG. 99
130. Electrical efficiency of dynamos
and motors. Since it requires pressure (voltage) to drive a
current through the armature and field coils, there is a loss
of power in a dynamo and in a motor. This loss is sometimes
called the copper loss. Electrical efficiency takes into account
only the copper loss.
ni i i as j Power given out
Electrical efficiency of a dynamo =    ^
Power generated
^ . , ~, . , Power left for useful work
Electrical efficiency of a motor =    
Power supplied to motor
PROBLEMS
1. The output of a serieswound dynamo is 5 kw. at a voltage
of 110 volts. The resistance of the armature is .06 ohm and oi
the field coil .072 ohm. Find (a) the copper loss ; (&) the elec
trical efficiency ; (c) the total electromotive force generated.
= 45.5 amperes.
.06 + .072 = .132 ohm, total resistance.
(a) A*R =
45.5 a x .132 = 273 watts, copper loss. Fio. 100'
5000 + 273 = 5<J73, total power generated.
(J) g^ = 95 per cent, electrical efficiency.
(c) V=AR =
45.5 x .132 = 6 volts, loss in armature and field coils.
110 + 6 = 116, total electromotive force generated.
2. A serieswound motor has a resistance of .68 ohm. Wher
supplied with 15 amperes at a voltage of 105 volts, fine
238 APPLIED MATHEMATICS
(a) the copper loss ; (&) the electrical efficiency ; (c) the volts
lost in the motor.
Suggestion. Find the copper loss as in Problem 1 and subtract it
from the number of watts supplied to the motor. ,
Electrical efficiency = ^fff Drop, in voltage = 15 x .68.
3. A shuntwound dynamo furnishes 5 kw. #,t a voltage of
110 volts. The shunt resistance is 45 ohms and the armature
resistance is .06 ohm. Find (a) the copper loss ; () the electrical
efficiency.
d A W A K e v> *5OOOW?r7&
SOLUTION. A = = 45.5 amperes. $~ /^_
In the shunt, % W^T^ UOVOL73
A V 110 nAA
A = = = 2.44 amperes.
*5 FIG. 101
45.5 f 2.44 = 47.9 amperes.
W = VA = 110 x 2.44 = 268 watts, loss in shunt.
W = A*R = 47.9 2 x .06 = 137 watts, loss in armature,
(a) 405 watts, total loss.
5000 + 405 = 5405 total watts.
(6) ITST 93 per cent, electrical efficiency.
4. The armature of a shunt motor has a resistance of .02 ohm,
and the shunt a resistance of 62 ohms. If the input is 5 h. p.
at 124 volts, find (a) the copper loss ; (b) the electrical efficiency.
SOLUTION. 5 x 746 = 3730 watts.
A W 3730 QA1
A = = = 30.1 amperes.
T , . . V 124
In shunt, A =   = 2 amperes. ^ m
30.1  2 = 28.1 amperes.
W = VA = 124 x 2 = 248 watts, loss in shunt.
VA^R = 28.1 2 x .02 = ^16 watts, loss in armature,
(a) 264 watts, total loss.
3730  264 = 3466 watts for useful work.
(&) f ^ff == 93 per cent, electrical efficiency.
Note that the current in the armature of a shunt motor equals
the total current less the current in the field coils.
ELECTRICITY 239
5. A 50kw., 125volt, compoundwound dynamo has a shunt
resistance of 62.5 ohms, a seriescoil resistance of .001 ohm, and
an armature resistance of .002 ohm. . soooowrrs
Compute the copper losses and the
electrical efficiency.
SOLUTION. *$t$ SL = 400 amperes.
1 ft <*
In shunt, Z = IE!L = 2 amperes. FlG  103
R 62.5
400 + 2 = 402 amperes, total current generated.
402 2 x .002 = 323 watts, loss in armature.
402* x .001 = 162 watts, loss in series coil.
125 x 2 = 250 watts, loss in shunt.
735 watts, total loss.
fcirHJ^ = 98.6 per cent, electrical efficiency.
Note that the total current generated by a shunt dynamo equals
the sum of the currents in the armature and in the field coils.
6. A compound motor is supplied with 50 amperes of current
from 110volt mains. If the armature resistance is .09 ohm, the
seriescoil resistance .078 ohm, and the shuntcoil resistance
55 ohms, find (a) the copper loss ; (i) the electrical efficiency.
SOLUTION. 50 x 110 = 5500 watts.
= = 2 amperes in shunt.
R 55 F
50 2 =48 amperes in armature.
Find loss in shunt, armature, and series coil to be 220, 207, and
180 watts respectively, and the electrical efficiency 89 per cent.
7. The output of a series dynamo is 20 amperes at 1000 volts.
The resistance of the armature is 1.4 ohms and of the field coil
1.7 ohms. Find the copper loss, the electrical efficiency, and
the volts lost in the dynamo.
8. The armature of a shunt motor has a resistance of .3 ohm,
and the shunt a resistance of 120 ohms. When running at full
load on a 110volt circuit the motor takes a current of 8 amperes.
Find the copper loss and the electrical efficiency.
240
APPLIED MATHEMATICS
Find the copper losses and electrical efficiency of the follow
ing dynamoelectric machines :
DYNAMOS
RESISTANCE, OHMS
No
TYPE
OUTPUT
' VOLTS
AMPERES
Armature
Series coil
Shunt coil
Series
2
2.5
10 kw.
1000
10
Compound
.003
.002
55
60 h. p.
110
11
Shunt
.29
57.5
6.5 kw.
115
12
Series
.15
.12
110
50
13
Compound
.04
.03
20
10 kw.
110
14
Shunt
.006
12
50 kw.
500
15
Compound
.023
.012
19.4
' 111
220
16
Shunt
.0117
52,7
410
590
MOTORS
RESISTANCE, OHMS
No.
TYPE
INPUT
VOLTS
AMPERES
Armature
Series coil
Shunt coil
17
Shunt
.15
48
110
10
18
Series
.39
.35
Ikw.
80
19
Shunt
.14
44
110
50
20
Shunt
.018
200
80 kw.
400
21
Series
.112
.113
220
100
22
Compound
.14
.02
55
5.5 kw.
110
131. Commercial or net efficiency. The commercial effi
ciency of a dynamo or motor takes account of all the losses in
the machine ; it is equal to the output divided by the input.
Commercial efficiency = r
Input
ELECTRICITY 241
PROBLEMS
1. A motor is supplied with a current of 20 amperes at
110 volts. If 2.8 h. p. are developed at the pulley, find the
commercial efficiency of the motor.
SOLUTION. Input = 110 x 20 watts.
Output = 746 x 2.8 watts.
n . , ffi . 74G x 2.8
Commercial efficiency =
J 110 x 20
= 95 per cent.
Check. 110 x 20 x .95 = 2090 watts = 2.8 h. p.
2. A motor generator takes a current of 14 amperes at 220
volts and supplies a current of 112 amperes at 25 volts. Find
its efficiency.
3. A 220volt electric hoist is raising coal at the rate of 1 T.
270 ft. per minute. If the current is 90 amperes, what is the
efficiency of the hoist ?
4. A 3kw. motor is used to operate a lathe. Find its effi
ciency if it takes 30 amperes at 110 volts.
5. The output of a generator is 50 kw. If it requires 76 h. p.
to drive it, what is its efficiency ?
6. A 550volt generator supplies a current of 300 amperes.
If the generator has an efficiency of 85 per cent, how many
horse power are required to drive it ?
7. It takes 25 h. p. to operate a dynamo which supplies power
for 40 arc lights in series at 7 amperes. The resistance of each
lamp is 8 ohms and the line resistance is 25 ohms. Find the
efficiency of the dynamo.
8. A lighting circuit consists of 1200 ft. of No. 6 B. & S.
gauge copper wire and eighty 16 candle power incandescent
lamps in multiple, each having a resistance of 220 ohms. If
the voltage is 110 at the lamps and 7.5 h. p. is supplied to
the generator, find its efficiency.
242
APPLIED MATHEMATICS
9. In testing a motor the following results were obtained.
Find the efficiency given by each test.
No.
Volts
Amperes
Brake horse power
1
224
96.5
24.6
2
221
101
25.7
3
222
103
27.2
4
230
109
29.1
5
227
123
32.6
10. The following data were obtained in a test of a motor
generator.
Construct a curve showing the relation between output and
efficiency.
Volts
225
225
229
228
228
228
Input
Amperes
5.9
7.7
9.6
11.7
13.7
15.9
Volts
21
20.8
21
20.6
20.2
20
Output
Amperes
20
40
60
80
100
CHAPTER XVIII
LOGARITHMIC PAPER
132. Description of logarithmic paper. In many engineering
problems where it is necessary to compute a set of values from
a formula, it is found that the required values can be secured
quickly and easily by using paper ruled on the logarithmic
scale. This paper is used both as a " ready reckoner," to read
off tables of values and to find the law connecting the two
variables in the problem. The advantage of logarithmic paper
lies in the fact that many formulas which are represented by
curves on squared paper are represented by straight lines on
logarithmic paper. Hence while many pairs of values must be
worked out to construct a curve on the former, only two or
three pairs are required for the latter.
Fig. 104 shows the way in which logarithmic paper is ruled.
The xaxis and the yaxis are laid off in divisions exactly like
those of the slide rule. That is, OX and OF are each divided
into 1000 equal parts ; 2 is placed at the 301st division
(log 2 = 0.301) ; 3 is placed at the 477th division (log 3 =
0.477) ; 4 is placed at the 602d division (log 4 = 0.602),
and so on.
Exercise. Construct a graph to read off the area of a circle of
any given radius.
In order to learn the properties of logarithmic paper we will
construct the graph by locating points. Later it will be shown
that the whole graph can be constructed easily by locating only
one point.
243
244 APPLIED MATHEMATICS
The formula for the area, a = m 2 , gives the following table :
Radius .
Area . .
1
3.14
1.2
4.62
1.5
7.07
2
12.6
3
28.3
4
60.3
6
78.6
6
113
7
164
8
201
10
314
z
Z
Z
Y
A
c/
FIG. 104
Locating the points as shown in Fig. 104, we see that the
points lie on the straight lines AB, CD, and EF. Hence
ABCD EF is the graph required. From it we see that
when the radius is 2.5 the area is 19.6 ; when the area is 38.5,
the radius is 3.5, and so on.
LOGARITHMIC PAPER
245
133. Properties of logarithmic paper. Some properties of
the paper may now be noted. The equation a = Trr 2 js in the
form y = mx n . AB, CD, and EF are parallel to one another.
FX
BD=CE = $ YZ. FX = 2EX; hence = 2, the exponent of r.
b*X
The graph can be drawn mechanically as follows : Find P,
the midpoint of YZ. Tack the sheet of paper on a drawing
board so that the Tsquare, in position, lies on O and P. Set
the Tsquare on A (making OA = 3.14) and draw AB. Set the
Tsquare at C on OX directly below B and draw CD. Similarly,
draw EF. Check ; F should be directly opposite A, that is,
FX = 3.14.
It will be found that these are general properties of logarith
mic paper, which may be used to construct graphs for formulas
of the form y = mx n ; that is, a formula in which y equals an
expression consisting of only one term in which the variable
is raised to any power (n, being positive, negative, or fractional)
and multiplied by any number. This form alone will be con
sidered in the following discussion, and some of the properties
of the paper which lead to simple and accurate constructions
will be considered.
I. EQUATIONS OF THE FORM y = mx
EXERCISES
1. Construct the graph of y = x.
X
y
i
i
2
2
3
3
4
4
6
5
Locating the points from the table, we see that they lie on the
straight line OZ (Fig. 105). Hence OZ is the graph of y = x.
2. Construct on the same sheet of paper the graph of (1) y =
'
246
APPLIED MATHEMATICS
It is seen that all these lines are parallel. When we plot
y = x (1) and y = 2x (2), we are really plotting the logarith
mic equations log y = log x (!') and log y = log 2x, or log y =
\og# + log 2 (2'). Comparing (!') and (2'), we see that they
9 O
differ only by the constant term log 2 on the right side ; that
18, every point of the graph of (2) is 2 above the correspond
ing point of the graph of (1). Note that the graph of each of
these equations, except y = #, is made up of two lines ; and all
the lines are parallel to OZ. Hence to graph any equation of
the form y = mx> for example, y = 5x, proceed as follows. From
LOGARITHMIC PAPER
247
5 on OF draw MN parallel to OZ. Take OP = YN and draw
PQ from P to 5 on XZ. MNPQ is the required graph.
The slope of a graph. We shall find that each graph we are
to consider (except y = x and y = or 1 ) consists of two or more
parallel lines, and that one line in each graph cuts OX and XZ
or OX and OY. Thus in the graph of y = 5x, PQ cuts OX and
XQ
XZ. We will call r the sZope o/ A0 graph ; that is, the tan
gent of the angle which the line makes with OX, always taking
the angle on the righthand side of the line.
II. EQUATIONS OF THE FORM y = mx*
A. When n is a positive whole number.
EXERCISES
1. Construct the graph of ?/ = x 2 .
X
y
i
i
2
4
3
9
4
16
6
25
6
36
7
49
8
64
9
81
10
100
Locating these points, we get the graph OABZ (Fig. 106).
Note that A and B are the midpoints of YZ and OX respectively.
2. Construct the graph of y = x 8 .
Locating points, we get OD FG HZ (Fig. 106). Note that
/>and <7, and Fand H divide YZ and OX respectively into three
equal parts.
3. Construct the graphs of y = # 4 and y = x 6 without locating
points.
JSoofc of numbers. From the graphs of y = # 2 , y = a 8 , y = x 4 ,
and so on we can read off roots of numbers. Thus in the graph
of y = # 2 , OA gives the square roots of numbers from 1 to
9, 100 to 999, 10,000 to 99,999, . . . ; that is, of numbers con
taining 1, 3, 5  figures. BZ gives the square root of numbers
containing 2, 4, 6 figures. To find the square root of 2, read
248
APPLIED MATHEMATICS
from 2 on OY to OA, 1.41 ; for the square root of 20 read front
2 on OF to BZ) 4.47. Similarly, y = x* gives cube roots ; OD
gives the cube root of numbers containing 1, 4, 7 figures, FG
9 to
Em. 106
of numbers containing 2, 5, 8 figures, and HZ of numbers
containing 3, 6, 9 figures.
4. Construct the graph of y = 2 x 2 .
X
y
i
2
2
8
3
18
4
32
6
50
6
72
7
98
8
128
9
162
10
200
LOGARITHMIC PAPER
249
Note that each y is twice as great as the corresponding y in
y = x*. On locating the points and drawing the lines of the
graph it will be seen that the lines are parallel to the lines of
y = x 2 and 2 units above them. Hence the graph of y = 2 x 2
may be constructed mechanically as follows : Tack the sheet of
paper on a drawing board so that the edge of the Tsquare, in
position, lies on and the midpoint of YZ. Move the Tsquare
up to 2 on OF and draw a line from 2 to YZ. Move the
Tsquare to a point on OX directly below the point already
determined on YZ and draw a line to YZ. Continue in the
same manner and the graph will end at 2 on XZ if accurately
drawn. I 1 his method holds for all cases where x* has a coeffi
cient. Note that the exponent of x is the slope of the graph.
5. Construct the graph of (a) y = 2 x* ; () y = .5 x 4 ;
(e) y = 1.68 z a ; (d) y = .0625 x\
B. When n is a positive fraction.
EXERCISES
1. Construct the graph of y = xL
X
y
i
i
4
8
9
27
16
64
25
125
36
216
49
343
64
512
81
729
100
1000
Locating points from the table, we get the graph OA BC
DE FZ (Fig. 107). A study of the graph shows that it could
be drawn in the following manner : Divide OX and YZ each
into three equal parts by the points JP, JB, E, and A ; and OY
and XZ each into two equal parts by the points Z> and C. Join
to A, the second point of division on YZ. This gives the
correct slope, $. Directly below A is J5, draw BC ; opposite C is
D, draw DE ; below E is F, draw FZ.
A similar construction holds for any positive fractional value
of n. Thus for y = x% , divide OX and YZ each into 3 (the
numerator of the exponent) equal parts, and OY and XZ each
250
APPLIED MATHEMATICS
into 5 (the denominator of the exponent) equal parts, and join
the points so as to make the slope f .
If x has a coefficient, for example, y = 6 i, start the graph at
6 on OF and draw it parallel to OA, thus making the slope $.
: yx*
y
v
7
iH x
FIG. 107
2. Construct the graphs of (a) y = x\ ; (J) y = 2 ai ; (c) y =
(d)y = 5 x ; (e) y = 2.5 x 8  2 ; (/) y = .06 x 1  1 .
3. Construct a graph to show the distance passed over by a
falling body in 1 to 10 sec.
4. Construct graphs to find (a) the surface, (i) the volume
of spheres of radii from 1 to 10 in.
LOGARITHMIC PAPER
C. When n is negative.
EXERCISES
1. Construct the graph of y = x~ l or y = 
x
251
V
1C
FIG. 108
X
y
i
i
2
.5
4
.25
8
.125
10
.1
Locating the points from the table, we get the graph YX
(Fig. 108). The graph of y = mx~ l is parallel to YX, and we f
252 APPLIED MATHEMATICS
begin to draw it from m on OF. Thus, to graph y = 4 x" 1 , from
4 on OF draw a line parallel to YX cutting OX at a point A ;
from B on FZ* directly above A draw a line parallel to YX
cutting XZ at C.
2. Construct the graph of y = art .
Divide OX and YZ into 2 (numerator of the exponent) equal
parts, and OF and XZ into 3 (denominator of the exponent)
equal parts. Draw lines as shown in Fig. 108, and we get the
graph YA  BC  DE  FX.
3. Construct the graphs of :
. (e) y==
(c) y = 4o5i (g) y = .006 ar".
y = 8ai (A) y = 2800 or 1  12 .
PROBLEMS
1. If in a gas engine the gas expands without gain or loss
of heat, the law of expansion is found to be pv lt2B = 3060. Con
struct the curve to show the pressure as the volume increases
from 10 cu. in. to 26 cu. in.
Locate only one point (Fig. 109) ; when v = 10, p = 180.
Mark this point by A on OF. The exponent of v is #$, when
the equation is in the form p = 3060 1;" 1  28 .
Measure OM = 123 mm. on OF, and ON = 100 mm. on OX.
Tack the paper on a drawing board so that the Tsquare, in
position, lies on M and N. Move the Tsquare to A and draw
AB. Move the Tsquare to C on YZ directly above B and draw
CD. AB CD is the graph ; from this graph pressures can be
read off for volumes from 10 cu. in. to 100 cu. in,
Given that steam expands without gain or loss of heat;
construct graphs on logarithmic paper for volumes from 10 to
100cu.in.:
2. pv 1  11 = 3000. 4. pvll = 3200. 6. pv* = 250.
3. X' 25 = 2840. 5. X' 81 = 3420.
LOGARITHMIC PAPER
253
7. The diameter d of wroughtiron shafting to transmit A
horse power at 100 r. p. m. is given by d = .85 Ai Construct
the graph and make a table for horse power from 10 to 80.
v
to
\
\
\
\
Volume
FIG. 109
8. The number of gallons of water per minute flowing over
a rectangular weir 6 in. wide is given by g = 17.8 A3, where
g = the number of gallons per minute, and h = the depth in
inches from the level of free water to the sill of the weir.
Construct the graph and make a table showing the number of
gallons per minute for depths 1, 1.5, 2, 2.5, , 6 in.
254 APPLIED MATHEMATICS
9. The number of cubic feet of water per minute discharged
over a Vnotch, or triangular weir, is given by Q = 18.5 bh% ,
where Q = the number of cubic feet per minute, b = breadth of
notch in feet at the free surface, and h = de^th in inches from
the free level to the bottom of the notch. Construct a graph
and make a table for the quantity of water discharged for
depths from 6 to 15 in. when b = 1 f t.
10. The diameter of a copper wire which will be fused by an
electric current is given by d = .00212 A$, where d = the diam
eter in inches, and A = the number of amperes. Construct a
graph and make a table of diameters of wire which will be
fused by currents of 10, 20, 30, , 100 amperes.
11. The weight in pounds that a rectangular steel beam,
supported at both ends, can sustain at its center, is given by
bd?
w = 890 9 where w = the weight in pounds, b = the breadth
(/
of beam in inches, d = the depth of beam in inches, and I = the
length of beam in feet.
Find the number of pounds that can be supported at the
middle of a steel beam 4 in. in breadth and 15 ft. long for
depths from 4 to 10 in.
12. In accordance with the building laws of Chicago the safe
load in tons, uniformly distributed, for yellowpine beams is
.08 6d 2 . . , . , . , ,,, f
given by w =  > where w = load in tons, b = breadth or
V
beam in inches, d = depth of beam in inches, and I = length of
beam in feet between the supports.
Find the safe load for yellowpine beams 25 ft. long, 4 in.
in breadth, and depths from 8 to 18 in.
13. The number of cubic feet of air transmitted per minute
in pipes of various diameters is given by q = .327 vd*, where
q = number of cubic feet of air per minute, v = velocity of flow
in feet per second, and d = diameter of pipe in inches.
Make a table showing the volume of air transmitted in pipes
of diameters from 2 to 10 in. with a flow of 12 ft. per second.
LOGARITHMIC PAPER
255
14. The following formula is used for computing the surface
curvature in paving streets : y = 5 # 2 , where x = horizontal dis
tance in feet from center of street, y = vertical distance in inches
below grade, a = one half the width of the street in feet, ft =
depth of gutter in inches below center of street.
FIG. 110
Construct a graph to read off the vertical distances below
grade at points 2, 4, 6 ft. from the center of a street 60 ft.
wide, if the gutter is 15 in. below the center of the street.
Find the equation connecting x and y when the following
corresponding values are given :
15.
Suggestion. Locate the points and draw a line through them, cut
ting OX at A and YZ at B. From C on YZ directly above A draw
a line parallel to BA, cutting OY at D. OD = 3.5 = in. The slope of
A B is 2 ; hence the required equation is y = 3.5 x 2 .
X
y
2
14
2.5
21.9
3
31.5
3.5
42.9
4
56
16.
x
y
2
32
3
108
4
256
5
500
6
864
17.
X
y
4
4
5
4.47
6
4.90
7
5.29
8
6.66 .
256
APPLIED MATHEMATICS
X
y
1.61
220
2.01
230
3.05
250
4.48
270
7.59
300
X
y
20
1099
30
2248
40
3826
50
5717
60
7943
18.
Suggestion. The line through the points cuts OF at 2. The values
of y, however, suggest that it should be read 200$ and this will be
found to be correct on checking.
19.
Suggestion. Let the line through the points cut OX at A and YZ
at B. From C on OX directly below B draw CD to XZ parallel to
AB; and from E on YZ directly above A draw EF to OY parallel
to AB. FE A B CD is the part of the graph for values of x from
10 to 100. To find m construct the part of the graph for values of
x from 10 to 1.
20.
Find the law connecting the two variables in the following :
21. In a test of castiron columns 6 ft. long, both ends
rounded, the following results were obtained, where d = diame
ter of column in inches, and t = load in tons under which the
column broke by bending.
x
y
15
486
20
589
25
684
30
772
64
1280
d
t
2
10.7
2.5
24.9
3
49.4
3.5
88.2
4
146
22. The bearing end of a vertical shaft is called a pivot. For
slowmoving steel pivots the following table of values is given,
where d = diameter of pivot in inches, and p = total vertical
pressure on the pivot in pounds.
d
P
I
816
1.5
1886
2
3265
2.5
5102
3
7347
3.5
10,000
4
13,061
4.5
16,530
LOGARITHMIC PAPER
257
23. The following table gives the absolute temperature
(F.) of air at different pressures when it is compressed without
gain or loss of heat, t = absolute temperature (F.), and p =
pounds per square inch.
P
t
15
530
30
649
45
730
60
792
90
892
24. The following results were obtained in a test in towing
a canal boat, p = pull in pounds, and v = speed of boat in miles
per hour.
P
V
76
1.68
160
2.43
240
3.18
320
3.60
370
4.03
In the following examples find the law connecting p and v.
The expansion is without gain or loss of heat, and p and v are
corresponding values of the pressure and volume.
25. Steam.
v 1 2 3 5 7 9
p 100 37.7 21.3 10.4 6.48 4,54
26. Steam.
v
P
3
118
4
90.8
6
63.3
8
48.9
10
40
27. Superheated steam.
t>
P
2
105
3
61.8
5
52
7
20.7
9
15
28. Mixture in cylinder of a gas engine.
v
P
2
57
4
21.2
6
11.8
8
8.1
10
5.9
258
APPLIED MATHEMATICS
WIRE TABLE COPPER WIRE
o88 BROWN AND SHARPE GAUGE
o o o o
Area in
circular mils
Diameter in mils
Resistance,
ohms per 1000 ft.
Weight, pounds
per 1000 ft.
2,000,000
1,750,000
1,600,000
1,250,000
1,000,000
1414
1323
1225
1118
1000
.06519
.00593
.00692
.00830
.01038 '
6044
6289
4633
3778
3022
950,000
900,000
850,000
800,000
750,000
974.7
948.7
922.0
894.4
866.0
.01093
.01153
.01221
.01298
.01384
2871
2720
2669
2418
2266
700,000
650,000
600,000
550,000
500,000
836.7
806.2
774.6
741.6
707.1
.01483
.01597
.01730
.01887
.02076
2115
1964
1813
1662
1511
450,000
400,000
350,000
300,000
250,000
670.8
632.5
591.6
547.7
500.0
.02307 ,
.02595
.02966
.03460
.04152
1360
1209
1058
906.5
755.5
225,000
211,600
167,805
133,079
105,592
474.3
460.00
409.64
364.80
324.95
* .04614
.04906
.06186
.07801
,09831
680.0
639.33
607.01
402.09
319.04
1
2
3
4
83,694
66,373
52,634
41,742
289.30
257.63
229.42
204.31
.12404
.15640
.19723
.24869
252.88
200.54
159.03
126.12
5
6
7
8
33,102
26,251
20,816
16,509
181.94
162.02
144.28
128.49
.31361
.39546
.49871
.62881
100.01
79.32
62.90
49.88
9
10
12
14
13,094
10,381
6,529.9
4,106.8
114.43
101.89
80.808
64.084
.79281
1.0000
1.5898
2.5908
39.66
31.37
19.73
12.41
16
18
19
20
2,582.9
1,624.3
1,288.1
1,021.6
50.820
40.303
35.890
31.961
4.0191
6.3911
8.2889
10,163
7.81
4.91
3.89
3.09
22
24
28
32
36
40
642.70
404.01
159.79
63.20
25.00
9.89
25.347
20.100
12.641
7.950
5.000
3.144
16.152
26.695
64.966
164.26
415.24
1049.7
1.94
1.22
.48
.19
.08
.03
TABLES
UNIT EQUIVALENTS
PRESSURE
1 pound per square inch . . . 2.042 inches of mercury at 62 F.
1 pound per square inch . . . 2.309 feet of water at 62 F.
1 atmosphere 14.7 pounds per square inch.
1 atmosphere 30 inches of mercury at 62 F.
1 atmosphere 33.95 feet of water at 62 F.
1 foot of water at 62 F 433 pound per square inch.
1 inch of mercury at 62 F. . . .491 pound per square inch.
LENGTH
1 mil 001 inch.
1 inch 2. 54 centimeters.
1 mile 1.609 kilometers.
1 centimeter 3937 inch.
1 kilometer 3280.8 feet.
AREA
1 circular mil 7854 square mil.
1 square mil 1.273 circular mils.
1 square inch 645.16 square millimeters,
1 square centimeter 155 square inch.
VOLUME
1 cubic inch 16.387 cubic centimeters.
1 cubic foot 7.48 gallons (liquid, U. S.).
1 pint (liquid, U. S.) 473.18 cubic centimeters.
1 pint (liquid, U. S.) 28.875 cubic inches.
1 gallon (liquid, U. S.) . . . . 231 cubic inches.
1 bushel 2150.4 cubic inches.
1 cubic centimeter 061 cubic inch.
1 liter 61.02 cubic inches.
1 liter 2.113 pints (liquid, U. S.).
259
260 APPLIED MATHEMATICS
WEIGHT
1 ounce (avoirdupois) .... 437, 6 grains.
1 ounce (avoirdupois) 28.35 grams.
1 pound (avoirdupois) .... 453.6 grams.
1 ton (2000 pounds) 907. 185 kilograms.
1 cubic centimeter of water . . 1 gram.
1 gram 0&53 ounce (avoirdupois).
1 cubic foot of water 62.4 pounds.
1 cubic inch of water 0361 pounds.
1 gallon of water (liquid, U. S.) . 8.345 pounds.
ENERGY, WORK, HEAT
1 British thermal unit (B. t. u.) . 1 pound water 1F.
1 British thermal unit .... 778 foot pounds.
1 British thermal unit 293 watt hour.
1 horse power hour 746 watt hours.
1 horse power hour 2544.7 British thermal units.
1 kilowatt hour 8412.66 British thermal units.
1 kilowatt hour 1.341 horse power hours.
POWER
1 watt 44.25 foot pounds per minute.
1 watt 0669 B. t. u. per minute.
1 horse power 83,000 foot pounds per minute,
1 horse power 746 watts per minute.
1 horse power 42.41 B. t. u. per minute.
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YOUNG. The Teaching of Mathematics, pp. 851. 1907. Longmans, Green
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KENT. Mechanical Engineers 1 PocketBook, pp. 1129. 1906. John Wiley
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SEAVER. Mathematical Handbook, pp. 365. McGrawHill Book Com
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SUPLEE. The Mechanical Engineer's Reference Book, pp. 859. 1905.
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TRAUTWINE. Engineers' PocketBook, pp. 1267. 1909. John Wiley &
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262 APPLIED MATHEMATICS
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CASTLE. Manual of Practical Mathematics, pp. 548. 1904. The Macmillan
Company. $1.50.
CONSTERDINE AND BARNES. Practical Mathematics, pp. 332. 1907.
Murray. 2s. 6d.
CRACKNELL. Practical Mathematics, J>p. 378. 1906. Longmans, Green
&Co. $1.10.
GRAHAM. Practical Mathematics, pp. 276. 1899. Arnold. 3s. 6d.
HOLTON. Shop Mathematics, pp. 212. 1910. The TaylorHolden Com
pany. $1.25.
JESSOP. Elements of Applied Mathematics, pp. 344. 1907. Geo. Bell &
Sons. 4s. 6d.
KNOTT AND MACKAY. Practical Mathematics, pp. 627. 1903. Chambers.
4s. 6d.
OLIVER. Elementary Practical Mathematics, pp. 240. (Reissue) 1910.
Oliver. Is. 6d.
ORMSBY. Elementary Practical Mathematics, pp. 442. 1900. Spon &
Chamberlain. 7s. 6d.
PERRY. Practical Mathematics, pp. 183. 1907. Wyman. 9d.
SAXELBY. Practical Mathematics, pp. 438. 1905. Longmans, Green &
Co. $2.25.
SAXELBY. Introductory Practical Mathematics, pp. 220. 1908. Long
mans, Green & Co. 80 cents.
STAINER. Junior Practical Mathematics, pp. 360. 1906. Geo. Bell &
Sons. 3s. 6d.
STARLING AND CLARKE. Preliminary Practical Mathematics, pp. 168.
1904. Arnold. Is. 6d.
STERN AND TOPHAM. Practical Mathematics, pp. 376. 1905. Geo. Bell
& Sons. 2s. 6d.
GIBSON. Treatise on Graphs, pp. 181. 1905. The Macmillan Company.
$1.00.
HAMILTON AND KETTLE. Graphs and Imaginaries, pp. 41. 1904. Long
mans, Green & Co. 50 cents.
LIGHTFOOT. Studies in Graphical Arithmetic, pp. 63. Normal Corre
spondence College Press. London. Is. 6d.
MORGAN. Elementary Graphs, pp. 76. 1903. Blackie. Is. 6d.
NIPHER. Introduction to Graphic Algebra, pp. 60. 1898. Henry Holt
and Company. 60 cents.
BIBLIOGRAPHY 263
PHILLIPS AND BEEBE. Graphic Algebra, pp. 156. 1904. Henry Holt
and Company. $1.60.
SCHULTZE. Graphic Algebra, pp. 93. 1908. The Macmillan Company.
80 cents.
TURNER. Graphics applied to Arithmetic, Mensuration, and Statics,
pp. 398. 1908. The Macmillan Company. $1.60.
EDSER. Measurement and Weighing, pp. 120. 1899. Chapman & Hall.
2s. 6d.
GRAVES. Forest Mensuration, pp. 458. 1906. John Wiley & Sons. $4.00.
LAMBERT. Computation and Mensuration, pp. 92. 1907. The Macmillan
Company. 80 cents.
LANGLEY. Treatise on Computation, pp. 184. 1895. Longmans, Green
& Co. $1.00.
LARARD AND GOLDING. Practical Calculations for Engineers, pp. 455.
1907. Griffin. 6s.
CHIVERS. Elementary Mensuration, pp. 344. 1904. Longmans, Green &
Co. $1.25.
EDWARDS. Mensuration, pp. 304. 1902. Arnold. 3s. 6d.
EGGAR. A Manual of Geometry, pp. 325. 1906. The Macmillan Company.
8s. 6d.
HARRIS. Plane Geometrical Drawing, pp. 270. 1907. Geo. Bell & Sons.
2s. 6d.
MYERS. Geometrical Exercises for Algebraic Solution, pp. 71. The
University of Chicago Press. 75 cents.
STONEMI LLIS. Elementary Plane Geometry, pp. 252. 1910. Benj. H.
Sanborn & Co. 80 cents.
WRIGHT. Exercises in Concrete Geometry, pp. 84. 1906. D. C. Heath
& Co. 30 cents.
BOHANNAN. Plane Trigonometry, pp. 374. 1904. Allyn & Bacou. $2.50.
PLAYNE AND FAWDRY. Plane Trigonometry, pp. 176. 1907. Arnold.
2s. 6d.
MILLER. Progressive Problems in Physics, pp. 218. 1909. D. C. Heath
& Co. 60 cents.
SANBORN. Mechanics Problems, pp. 194. John Wiley & Sons. $1.50.
SNYDER AND PALMER. One Thousand Problems in Physics, pp. 142.
1902. Ginn and Company. 55 cents.
264 APPLIED MATHEMATICS
ATKINSON. Electrical and Magnetic Calculations, pp. 810. 1908. D. Van
Nostrand Company. $1.60.
HOOPER AND WELLS. Electrical Problems, pp. 170. Ginn and Company.
$1.25.
JAMES AND SANDS. Elementary Electrical Calculations, pp. 224. 1905.
Longmans, Green & Co. $1.25.
SHEPARDSON. Electrical Catechism, ppi 417. 1908. McGrawHill Book
Company. $2.00.
WHITTAKER. Arithmetic of Electrical Engineering, pp. 159. Whittaker.
25 cents.
BHREND8ENG5TTiNG. Lehrbuch der Mathematik, 254 S. 1909. Teubner.
M. 2.80.
EHRIG. Geometric ftir Baugewerkenschulen, Teil I., 138 S. 1909. Leine
weber. M. 2.80.
FENKNER. Arithmetische Aufgaben, Ausgabe A.,TeilL,274S. M. 2.20.
Teil II a., 114 S. M. 1.50. Teil II &., 218 S. M. 2.60. Salle.
GEIGENMI)LLER. Htfhere Mathematik, 1. 290 S. 1907. Polytechnische
Buchhandlung. M. 6.
MULLER UND KuTNEWSKY. Sammlung von Aufgaben aus der Arithmetik,
Trigonometric und Stereometric, Ausgabe B., 2ter Teil, 312 S. 1910.
Teubner. M. 3.
SCHULKE. AufgabenSammlung, Teil I., 194 S. 1906. Teubner. M. 2.20.
WEILL. Sammlung Graphischer Aufgaben fttr den Gebrauch an hflhere
Schulen, 64 S. 1909. J. Boltzesche Buchhandlung. M. 1.80.
I. FOURPLACE LOGARITHMS OF THREEFIGURE NUMBERS
II. THE NATURAL SINES, COSINES, TANGENTS, AND COTAN
GENTS OF ANGLES DIFFERING BY TEN MINUTES, AND THEIR
FOURPLACE LOGARITHMS
265
266
APPLIED MATHEMATICS
1
1
2
8
4
5
6
7
8
9
'oooo
0000
3010
4771
6021
6990
7782
8451
9031
9542
1
0000
0414
0792
1139
1461
1761
2041
2304
2553
2788
2
3010
3222
3424
3617
3802
3979
4150
4314
4472
4624
3
4771
4914
6051
5185
5315
5441
5568
5682
5798
5911
4
6021
6128
6232
6335
6435
6532
6628
6721
6812
6902
5
6990
7076
7160
7243
7324
' 7404
7482
7559
7634
7709
6
7782
7853
7924
7993
8062
8129
8195
8261
8325
8388
7
8451
8513
8573
8633
8692
8751
8808
8865
8921
8976
8
9031
9085
9138
9191
9243
9294
9345
9395
9445
9494
9
9542
9590
9638
9685
9731
9777
9823
9868
9912
9956
10
0000
0043
0086
0128
0170
0212
0253
0294
0334
0374
11
0414
0453
0492
0531
0569
0607
0645
0682
0719
0755
12
0792
0828
086i
0899
0934
0969
1004
1038
1072
1106
13
1139
1173
1206
1239
1271
1303
1335
1367
1399
1430 .
14
1461
1492
1523
1553
1584
1614
1644
1673
1703
1732
16
1761
1790
1818
1847
1875
1903
1931
1959
1987
2014
16
2041
2068
2095
2122
2148
2175
2201
2227
2253
2279
17
2304
2330
2355
2380
2405
2430
2455
2480
2504
2529
18
2553
2577
2601
2625
2648
2672
2695
2718
2742
2765
19
2788
2810
2833
2856
2878
2900
2923
2945
2967
2989
20
3010
3032
3054
3075
3096
3118
3139
3160
3181
3201
21
3222
3243
3263
3284
3304
3324
3345
3365
3385
3404
22
3424
3444
3464
3483
3502
3522
3541
3560
3579
3598
23
3617
3636
3655
3674
3692
3711
3729
3747
3766
3784
24
3802
3820
3838
3856
3874
3892
3909
3927
3946
3962
25
3979
3997
4014
4031
4048
4065
4082
4099
4116
4133
26
4150
4166
4183
4200
4216
4232
4249
4265
4281
4298
27
4314
4330
4346
4362
4378
4393
4409
4425
4440
4456
28
4472
4487
4502
4518
4533
4548
4564
4579
4594
4609
29
4624
4639
4654
4669
4683
4698
4713
4728
4742
4757
30
4771
4786
4800
*H
4829
4843
4857
4871
4886
4900
31
4914
4928
4942
4955
4969
4983
4.997
5011
5024
5038
32
5051
5065
6079
5092
5105
5119
6132
5145
5159
5172
33
5185
5198
5211
5224
6237
5250
5263
6276
"5289
5302
34
5315
5328
5340
5353
5366
5378
6391
5403
5416
5428
35
5441
5453
5465
5478
5490
5502
5514
5527
5539
6551
36
5563
5575
5587
5699
5611
5623
6635
6647
6658
5670
37
5682
5694
6706
5717
5729
6740
6752
5763
5775
5786
.38
5798
5809
6821
5832
5843
6855
6866
5877
5888
5899
39
5911
5922
5933
5944
5956
5966
5977
5988
RQQQ
UJ/I/JJ
6010
40
6021
6031
6042
6053
6064
6075
6085
6096
6107
6117
41
6128
6138
6149
6160
6170
6180
6191
6201
6212
6222
42
6232
6243
6253
6263
6274
6284
6294
6304
6314
6325
43
6335
6345
6355
6365
6375
6385
6395
6405
6415
6425
44
6435
6444
6454
6464
6474
6484
6493
6603
6513
6522
45
6532
6542
6551
6561
6571
6580
6590
6599
6609
6618
46
6628
6637
6646
K*ff{t
OOOO
Of^Ojf
OOOO
6675
6684
6693
6702
6712
47
6721
6730
6739
6749
6758
6767
6776
6785
6794
6803
48
6812
6821
6830
6839
6848
6867
6866
6875
6884
6893
49
6902
6911
6920
6928
6937
6946
6955
6964
6972
6981
50
1
2
3
4
5
6
7
8
9
FOURPLACE LOGARITHMS
267
50
1
2
3
4
5
6
7
8
9
60
6990
6998
7007
7016
7024
7033
7042
7050
7059
7067
51
7076
7084
7093
7101
7110
7118
7126
7135
7143
7152
52
7160
7168
7177
7185
7193
7202
7210
7218
7226
7235
53
7243
7251
7259
7267
7275
7284
7292
7300
7308
7316
54
7324
7332
7340
7348
7356
7364
7372
7380
7388
7396
55
7404
7412
7419
7427
7435
7443
7461
7469
7466
7474
56
7482
7490
7497
7505
7513
7620
7528
7536
7543
7551
57
7559
7566
7574
7582
7589
7597
7604
7612
7619
7627
58
7634
7642
7649
7657
7664
7672
7679
7686
7694
7701
59
7709
7716
7723
7731
7738
7746
7752
7760
7767
7774
60
7782
7789
7796
7803
7810
7818
7825
7832
7839
7846
61
7853
7860
7868
7875
7882
7889
7896
7903
7910
7917
62
7924
7931
7938
7945
7952
7959
7966
7973
7980
7987
63
7993
8000
8007
8014
8021
8028
8035
8041
8048
8055
64
8062
8069
8075
8082
8089
8096
8102
8109
8116
8122
65
8129
8136
8142
8149
8156
8162
8169
8176
8182
8189
66
8195
8202
8209
8215
8222
8228
8236
8241
8248
8264
67
8261
8267
8274
8280
8287
8293
8299
8306
8312
8319
68
8325
8331
8338
8344
8351
8357
8363
8370
8376
8382
69
8388
8395
8401
8407
8414
8420
8426
8432
8439
8446
70
8451
8457
8463
8470
8476
8482
8488
8494
8500
8606
71
8513
8519
8525
8531
8537
8643
8549
8566
8561
8567
72
8573
8579
8585
8591
8697
8603
8609
8615
8621
8627
73
8633
8639
8645
8G51
8657
8663
8669
8675
8681
8686
74
8692
8698
8704
8710
8716
8722
8727
8733
8739
8746
75
8761
8756
8762
8768
8774
8779
8785
8791
8797
8802
76
8808
8814
8820
8825
8831
8837
8842
8848
8864
8869
77
8865
8871
8876
8882
8887
8893
8899
8904
8910
8916
78
8921
8927
8932
8938
8943
8949
8964
8960
8966
8971
79
8976
8982
8987
8993
8998
9004
9009
9016
9020
9025
80
9031
9036
9042
9047
9053
9058
9063
9069*
9074
9079
81
9085
9090
9096
9101
9106
9112
9117
9122
9128
9133
82
9138
9143
9149
9154
9159
9165
9170
9175
9180
9186
83
9191
9196
9201
9206
9212
9217
9222
9227
9232
9238
84
9243
9248
9253
9258
9263
9269
9274.
9279
9284
9289
85
9294
9299
9304
9309
9315
9320
9326
9330
9336
9340
86
9345
9350
9355
9360
9365
9370
9375
9380
9385
9390
87
9395
9400
9405
9410
9415
9420
9425
9430
9435
9440
88
9445
9450
9455
9460
9465
9469
9474
9479
9484
9489
89
9494
9499
9504
9509
9513
9518
9623
9628
9633
9538
90
9542
9547
9552
9557
9562
9566
9571
9576
9581
9586
91
9590
9595
9600
9605
9G09
9614
9619
9624
9628
9633
92
9638
9643
9647
9652
9G57
9661
9666
9671
9675
9680
93
9685
9689
9694
9G99
9703
9708
9713
9717
9722
9727
94
9731
9736
9741
9745
9750
9754
9759
9763
9768
9773
95
9777
9782
9786
9791
9795
9800
9805
9809
9814
9818
96
9823
9827
9832
9836
9841
9845
9850
9854
9869
9863
97
9868
9872
9877
9881
9886
9890
9894
9899
9903
9908
98
9912
9917
9921
9926
9930
9934
9939
9943
9948
9952
99
9956
9961
9965
9969
9974
9978
9983
9987
9991
9996
100
1
2
3
4
6
7
8
9
268
APPLIED MATHEMATICS
ANGLE
SINES
COSINES
TANGENTS
COTANGENTS
ANGLE
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
QO (XX
.0000
00
1.0000
0.0000
.0000
00
00
00
90 (XX
10
.0029
7.4637
1.0000
0000
.0029
7.4637
2.5363
343.77
50
20
.0058
7648
1.0000
0000
.0058
7648
2352
171.89
40
30
.0087
9408
1.0000
0000
.0087
9409
0591
114.59
30
40
.0116
8.0658
.9999
0000
.0116
8.0658
1.9342
85.940
20
50
.0145
1627
QQQQ
tt/Wtf
0000,,
.0145
1627
8373
68.750
10
10 (XX
.0175
8.2419
,9998
9.9999
.0175
8.2419
1.7581
67.290
89000'
10
.0204
3088
.9998
QQQQ
Wuy
.02OI
3089
, 6911
49.104
50
20
.0233
3668
.9997
yyyy
.0233
3669
6331
42.964
40
30
.0262
4179
.9997
9999
.0262
4181
5819
38.188
30
40
.0291
4637
.9996
9998
.0291
4638
5362
34.368
20
50
.0320
5050
.9995
9998
.0320
5053
4947
31.242
10
2<> (XX
.0349
8.5428
.9994
9.9997
.0349
8.5431
1.4569
28.636
880 0<X
10
.0378
5776
.9993
9997
.0378
5779
4221
26.432
50
20
.0407
6097
.9992
9996
.0407
6101
3899
24.542
40
30
.0436
6397
.9990
9996
.0437
6401
3599
22.904
30
40
.0465
6677
.9989
9995
.0466
6682
3318
21.470
20
50
.0494
6940
.9988
9995
.0495
6945
3055
20.206
10
3 00'
.0523
8.7188
.9986
9.9994
.0524
8.7194
1.2806
19.081
87000'
10
.0552
7423
.9985
9993
.0553
7429
2571
18.075
50
20
.0581
7645
.9983
9993
.0582
7652
, 2348
17.169
40
30
.0610
7857
.9981
9992
.0612
7865
2135
16.350
30
40
.0640
8059
.9980
9991
.0041
8067
1933
15.605
20
50
.0669
8251
.9978
9990
.0670
8261
1739
14.924
10
4000'
.0698
8.8436
.9976
9.9989
.0699
8.8446
1.1554
14.301
80o (XX
10
.0727
8613
.9974
9989
.0729
8624
1376
13.727
50
20
.0756
8783
.9971
9988
.0758
8795
1205
13.197
40
30
.0785
8946
.9969
9987
.0787
8960
1040
12.706
30
40
.0814
9104
.9967
9986
.0816
9118
0882
12.251
20
50
.0843
9256
.9964
9985
.0846
9272
0728
11.826
10
5 (XX
.0872
8.9403
.9962
9.9983
.0875
8.9420
1.0580
11.430
85 (XX
10
.0901
9545
.9959
9982
.0904
9563
0437
11.059
50
20
.0929
9682
.9957
9981
.0934
9701
0299
10.712
40
30
.0958
9816
.9954
9980
.0963
9836
0164
10.385
30
40
.0987
9945
.9951
9979
.0992
9966
0034
10.078
20
50
.1016
9.0070
.9948
9977
.1022
9.0093
0.9907
9.7882
10
60 (XX
.1045
9.0192
.9945
9.9976
.1051
9.0216
0.9784
9.5144
840 (XX
10
.1074
0311
.9942
9975
.1080
0336
9664
9.2553
50
20
.1103
(H26
.9939
9973
.1110
0153
9547
9.0098
40
30
.1132
0539
.9936
9972
.1139
0567
9433
8.7769
30
40
.1161
0648
.9932
9971
.1169
0678
9322
8.5555
20
50
.1190
0755
.9929
9969
.1198
0786
9214
8.3450
10
TOGO'
.1219
9.0859
.9925
9.9968
.1228
9.0891
0.9109
8.1443
83 (XX
10
.1248
0961
.9922
9966
.1257
0995
9005
7.9530
60
20
.1276
1060
.9918
9964
.1287
1096
8904
7.1704
40
30
.1305
1157
.9914
9963
.1317
1194
8806
7.5968
30
40
.1334
1252
.9911
9961
.1346
1291
8700
7.4287
20
50
.1363
1345
.9907
9959
.1376
1385
8615
7.2687
10
8 (XX
.1392
9.1436
.9903
9.9958
.1405
9.1478
0.8522
7.1164
82<> (XX
10
.1421
1525
.9899
9956
.1435
1569
8431
6.9682
60
20
.1449
1612
.9894
9954
.1465
1658
8342
6.8269
40
30
.1478
1697
.9890
9952
.1495
1745
8256
6.6912
30
40
.1507
1781
.9886
9950
.1524
1831
8169
6.5606
20
50
.1536
1863
.9881
9948
.1554
1915
8085
6.4348
10
9^00'
.1564
9.1943
.9877
9.9946
.1584
9.1997
0.8003
6.3138
810 (XX
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
ANGLE
COSINES
SINES
COTANGENTS
TANGENTS
ANGLE
FOURPLACE LOGARITHMS
269
ANGLE
SINES
COSINES
TANGENTS
COTANGENTS
ANGLE
Nat.
Log.
Nat.
Log.
Nat.
Log,
Log.
Nat.
9000'
.1564
9.1943
.9877
9.9946
.1584
9.1997
0.8003
6.3138
81 GO 7
10
.1593
2022
.9872
nr\AA
JW*Tt
.1614
2078
7922
6.1970
50
20
.1622
2100
.9868
9942
.1644
2158
7842
6.0844
40
30
.1650
2176
.9863
9440
.1673
2236
77&4
5.9758
30
40
.1679
2251
.9858
9938
.1703
2313
7687
5.8708
20
50
.1708
2324
.9853
9936
.1733
2389
7611
5.7694
10
10000'
.1736
9.2397
.9848
9.9934
.1763
9.2463
0.7537
5.6713
800 00'
10
.1765
2468
.9843
9931
.1793
2536
7464
5.5764
50
20
.1794
2538
.9838
9929
.1823
2G09
7391
5.4845
40
30
.1822
2606
.9833
9927
.1853
2680
7320
5.3955
30
40
.1851
2674
.9827
9924
.1883
2750
7250
5.3093
20
50
.1880
2740
.9822
9922
.1914
2819
7181
5.2257
10
11 00'
.1908
9.2806
.9816
9.9919
.1944
9.2887
0.7113
5.1446
7900'
10
.1937
2870
.9811
9917
.1974
2953
7047
5.0658
50
20
.1965
2934
.9805
9914
.2004
3020
6980
4.9894
40
30
.1994
2997
.9799
9912
.2035
3085
69 15
4.9152
30
40
.2022
3058
.9793
9909
.2065
3149
6851
4.8430
20
50
.2051
3119
.9787
9907
.2095
3212
6788
4.7729
10
12> QO>
.2079
9.3179
.9781
9.9904
.2126
9.3275
0.6725
4.7046
78 (XX
10
.2108
3238
.9775
9901
.2156
3336
6664
4.6382
50
20
.2136
3296
.9769
9899
.2186
3397
6603
4.5736
40
30
.2164
3353
.9763
9896
.2217
3458
6542
4.5107
30
40
.2193
3410
.9757
9893
.2247
3517
6483
4.4494
20
50
.2221
3466
.9760
9890
.2278
3576
6424
4.3897
10
130 (XX
.2250
9.3521
.9744
9.9887
,2309
9.3634
0.6366
4.3315
77000'
10
.2278
3575
.9737
9884
.2339
3691
6309
4.2747
50
20
.2306
3629
.9730
9881
.2370
3748
6252
4.2193
40
30
.2334
3682
.9724
9878
.2401
3804
6196
4.1653
30
40
.2363
3734
.9717
9875
.2432
3859
6141
4.1126
20
50
.2391
3786
.9710
9872
.2462
3914
6086
4.0611
10
140 OCX
.2419
9.3837
.9703
9.9869
.2493
9.3968
0.6032
4.0108
76 00'
10
.2447
3887
.9696
9866
.2524
4021
5979
3.9617
50
20
.2476
3937
.9689
9863
.2B55
4074
5926
3.9136
40
30
.2504
3986
.9681
9859
.2586
4127
5873
3.8667
30
40
.2532
4035
.9674
9856
.2617
4178
5822
3.8208
20
50
.2560
4083
.9667
9853
.2648
4230
5770
3.7760
10
150 (XX
.2588
9.4130
.9659
9.9849
.2679
9.4281
0.5719
3.7321
75 00'
10
.2616
4177
.9652
9846
.2711
4331
5669
3.6891
50
20
.2644
4223
no A A
,tf\r*
9843
.2742
4381
5619
3.6470
40
30
.2672
4269
.9636
9839
.2773
4430
5570
3.6059
30
40
,2700
4314
.9628
9836
.2805
4479
5521
35656
20
50
.2728
4369
.9621
9832
.2836
4527
6473
35261
10
160 (XX
.2756
9.4403
.9613
9.9828
,2867
9.4575
0.5425
3.4874
74000'
10
.2784
4447
.9606
9825
.2899
4622
6378
3.4495
50
20
.2812
4491
.9596
9821
.2931
4669
5331
3.4124
40
30
.2840
4533
.9588
9817
.2962
4716
5284
3.3759
30
40
.2868
4576
.9580
9814
.2994
4762
5238
3.3402
20
50
.2896
4618
.9572
9810
.3026
4808
5192
3.3052
10
17<> (XX
.2924
9.4659
.9563
9.9806
.3057
9.4853
05147
3.2709
73000'
10
.2952
4700
.9555
9802
.3089
4898
6102
3.2371
50
20
.2979
4741
.9546
9798
.3121
4943
5057
3.2041
40
30
.3007
4781
.9537
9794
.3153
4987
5013
3.1716
30
40
.3036
4821
.9528
9790
.3185
5031
4969
3.1397
20
50
.3062
4861
.9520
9786
.3217
5075
4925
3.1084
10
180 (XX
.3090
9.4900
.9511
9.9782
.3249
9.5118
0.4882
3.0777
720 (XX
Nat.
Log.
Nat.
Log,
Nat.
Log.
Log.
Nat.
ANGLE
COSINES
SINES
COTANGENTS
TANGENTS
ANGLE
270
APPLIED MATHEMATICS
ANGLE
SINES
COSINES
TANGENTS
COTANGENTS
ANGLE
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
180 00'
.3090
9.4900
.9511
9.9782
3249
9.5118
0.4882
3.0777
72000'
10
.3118
4939
.9502
9778
3281
5161
4839
3.0475
50
20
.3145
4977
.9492
9774
.3314
5203,
4797
3.0178
40
30
.3173
6015
.9483
9770
.3346
5245
4755
2.9887
30
40
.3201
6062
.9474
9765 ,
.3378
5287
4713
2.9600
20
50
.3228
5090
.9465
9761
3411
5329
4671
2.9319
10
100 00'
.3256
9.5126
.9456
9.9757
.3443
9.5370
0.4630
2.9042
71000'
10
.3283
5163
.9446
9762
.3476
5411
4589
2.8770
50
20
.3311
6199
.9436
9748
3508
5451
4549
2.8502
40
30
.3338
5235
.9426
9743
.3541
5491
4509
2.8239
30
40
.3365
5270
.9417
9739
.3574
5531
4469
2.7980
20
60
.3393
5306
.9407
9734
3607
5571
4429
2.7725
10
200 00'
.3420
9.5341
.9397
9.9730
3640
9.5611
0.4389
2.7475
70000'
10
.3448
5375
.9387
9725
3673
5650
4350
2.7228
50
20
.3475
5409
.9377
9721
3706
5689
4311
2.6985
40
30
.3502
5443
.9367
9716
3739
5727
4273
2.6746
30
40
.3529
5477
.9356
9711
3772
5766
4234
2.6511
20
50
.3557
5510
.9346
9706
3805
5804
4196
2.6279
10
21000'
.3584
9.5543
.9336
9.9702
.3839
9 .5842
0.4158
2.6051
690 (XX
10
.3611
5676
.9325
9697
.3872
5879
4121
2.6826
50
20
.3638
5609
.9315
9692
.3906
5917
4083
2.5605
40
30
.3665
5641
.9304
9687
3939
5954
4046
2.5386
30
40
.3692
5673
.9293
9682
.3973
5991
4009
2.5172
20
50
.3719
6704
.9283
9677
.4006
6028
3972
2.4960
10
220 (XX
.3746
9.5736
.9272
9.9672
.4040
9.6064
0.3936
2.4761
68000'
10
.3773
5767
.9261
9667
.4074
6100
3900
2.4545
60
20
3800
5798
.9250
9661
.4108
6136
3864
2.4342
40
30
3827
5828
.9239
9656
.4142
6172
3828
2.4142
30
40
3854
6859
.9228
9651
.4176
6208
3792
2.3945
20
50
3881
6889
.9216
9646
.4210
6243
3757
2.3750
10
23000'
3907
9.5919
.9205
9.9640
.4245
9.6279
03721
23559
670 (XX
10
.3934
5948
.9194
9635
.4279
6314
3686
2.3369
50
20
.3961
5978
.9182
9629
.4314
6348
3652
23183
40
30
3987
6007
.9171
9624
.4348
6383
3617
2.2998
30
40
.4014
6036
.9159
9618
.4383
6417
3583
2.2817
20
50
.4041
6065
.9147
9613
.4417
6452
3548
2.2637
10
24 OCX
.4067
9.6093
.9135
9.9607
.4452
9.6486
0.3514
2.2460
66000'
10
.4094
6121
.9124
9602
.4487
6520
3480
2.2286
50
20
.4120
6149
.9112
9596
.4522
6553
3447
2.2113
40
30
.4147
6177
.9100
9590
.4557
6587
3413
2.1943
30
40
.4173
6205
.9088
9584
.4592
6620
3380
2.1775
20
50
.4200
6232
.9075
9579
.4628
6654
3346
2.1609
10
250 <xx
.4226
9.6259
.9063
9.9573
.4663
9.6687
0.3313
2.1445
650 (XX
10
.4253
6286
.9051
9567
.4699
6720
3280
2.1283
50
20
.4279
6313
.9038
9561
.4734
6752
3248
2.1123
40
30
.4905
6340
.9026
9555
.4770
6785
3215
2.0966
30
40
.4331
6366
.9013
9549
.4806
6817
3183
2.0809
20
50
.4358
6392
.9001
9543
.4841
6850
3150
2.0665
10
260 00'
.4384
9.6418
.8988
9.9537
.4877
9.6882
0.3118
2.0503
640 (XX
10
.4410
6444
.8975
9530
.4913
6914
3086
2.0353
60
20
.4436
6470
.8962
9524
.4950
6946
3054
2.0204
40
30
.4462
6495
.8949
9518
.4986
6977
3023
2.0057
30
40
.4488
6521
.8936
9512
.5022
7009
2991
1.9912
20
50
.4514
6546
.8923
9605
.5069
7040
2960
1.9768
10
27000'
.4540
9.6670
.8910
0.9499
.5095
9.7072
0.2928
1.9626
63000'
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
ANGLE
COSINES
SINES
COTANGENTS
TANGENTS
ANGLE
FOURPLACE LOGARITHMS
271
ANGLE
SINES
COSINES
TANGENTS
COTANGENTS
ANGLE
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
27 00'
.4540
9.6570
.8910
9.9499
5096
9.7072
0.2928
1.9626
630 (XX
10
.4566
6595
.8897
9492
5132
7103
2897
1.9486
50
20
.4592
6620
.8884
9486
5169
7134
2866
1.9347
40
30
.4617
6644
.8870
9479
5206
7165
2835
1.9210
30
40
.4643
6668
.8857
9473
5243
7196
2804
1.9074
20
50
.4669
6692
.8843
9466
5280
7226
2774
1.8940
10
28000'
.4695
9.6716
.8829
9.9459
5317
9.7257
0.2743
1.8807
62000'
10
.4720
6740
.8816
9453
5364
7287
2713
1.8676
50
20
.4746
6763
.8802
9446
5392
7317
2683
1.8646
40
30
.4772
6787
.8788
9439
5430
7348
2652
1.8418
30
40
.4797
6810
.8774
9432
5467
7378
2622
1.8291
20
50
.4823
6833
.8760
9425
5506
7408
2692
1.8166
10
29000'
.4848
9.6856
.8746
9.9418
5643
9.7438
0.2562
1.8040
610 (XX
10
.4874
6878
.8732
9411
5581
7467
2533
1.7917
50
20
.4899
6901
.8718
9404
5619
7497
2503
1.7796
40
30
.4924
6923
.870*
9397
5668
7526
2474
1.7675
30
40
.4950
6946
.8689
9390
5696
7556
2444
1.7566
20
50
.4975
6968
.8675
9383
5735
7586
2416
1.7437
10
30000'
.5000
9.6990
.8660
9.9375
5774
9.7614
0.2386
1.7321
600 (xx
10
.5025
7012
.8646
9368
5812
7644T
2356
1.7206
50
20
.5060
7033
.8631
9361
5861
7673
2327
1.7090
40
30
.5075
7055
.8616
9353
5890
7701
2299
1.6977
30
40
.5100
7076
.8601
9346
5930
7730
2270
1.6864
20
50
.5125
7097
.8587
9338
5969
7769
2241
1.6763
10
310 OCX
.5150
9.7118
.8572
9.9331
.6009
9.7788
0.2212
1.6643
590 (XX
10
,5175
7139
.8557
9323
.6048
7816
21&4
1.6634
50
20
.5200
7160
.8542
9315
.6088
7846
2155
1.6426
40
30
.5225
7181
.8526
9308
.6128
7873
2127
1.6319
30
40
.5250
7201
.8511
9300
.6168
7902
2098
1.62 12
20
50
.5275
7222
.8496
9292
.6208
7930
2070
1.6107
10
320 00'
.5299
9.7242
.8480
9.9284
.6249
9.7968
0.2042
1.6003
580 oo/
10
.5324
7262
.8465
9276
.6289
7986
2014
15900
60
20
.5348
7282
.8460
9268
.6330
8014
1986
15798
40
30
.6373
7302
.8434
9260
.6371
8042
1968
15697
30
40
.5398
7322
.8418
9252
.6412
8070
1930
15597
20
50
.5422
7342
.8403
9244
.6453
8097
1903
15497
10
33 OCX
5446
9.7361
.8387
9.9236
.6494
9.8126
0.1875
15399
670 (XX
10
.5471
7380
.8371
9228
.6636
8163
1847
15301
60
20
.5495
7400
.8356
9219
.6677
8180
1820
15204
40
30
.5519
7419
.8339
9211
.6619
8208
1792
15108
30
40
.5544
7438
.8323
9203
.6661
8236
1765
15013
20
50
.5668
7457
.8307
9194
.6703
8263
1737
1.4919
10
34^00'
5592
9.7476
.8290
9.9186
.6745
9.8290
0.1710
1.4826
660 (XX
10
5616
7494
.8274
9177
.6787
8317
1683
1.4733
50
20
5640
7513
.8258
9169
.6830
8344
1656
1.4641
40
30
5664
7631
.8241
9160
.6873
8371
1629
1.4660
30
40
5688
7560
.8225
9151
.0916
8398
1602
1.4460
20
50
5712
7568
.8208
9142
.6059
8426
1675
1.4370
10
35<>00'
5736
9.7586
.8192
9.9134
.7002
9.8462
0.1648
1.4281
550 (xx
10
5760
7604
.8176
9126
.7046
8479
1521
1.4193
50
20
5783
7622
.8158
9116
.7089
8506
1494
1.4106
40
30
5807
7640
.8141
9107
.7133
8633
1467
1.4019
30
40
5831
7657
.8124
9098
.7177
8669
1441
13934
20
50
5854
7676
.8107
9089
.7221
8586
1414
13848
10
360 (XX
5878
9.7692
.8090
9.9080
.7266
9.8613
0.1387
1.3764
540 (XX
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
ANGLE
COSINES
SINES
COTANGENTS
TANGENTS
ANGLE
272
APPLIED MATHEMATICS
ANGLE
SINES
COSINES
TANGENTS
COTANGFNTS
ANGLE
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
36 00'
.5878
9.7692
.8090
9.9080
.7265
9.8613
0.1387
1.3764
640 oo/
10
.5901
7710
.8073
9070
.7310
8699
1361
1.3680
50
20
.5925
7727
.8066
9061
.7356
8666
1334
1.3597
40
30
.5948
7744
.8039
9052
,7400
8692
1308
1.3514
30
40
.5972
7761
.8021
9042
'.7445
8718
1282
1.3432
20
50
.5995
7778
.8004
9033
.7490
8745
1255
1.3351
10
37000'
.6018
9.7795
.7986
9.9023
.7536
9.8771
0.1229
1.3270
530 00'
10
.6041
7811
.7969
9014
.7681
8797
1203
1.3190
50
20
.6065
7828
.7951
9004
.7627
8824
1176
1.3111
40
30
.6088
7844
.7934
8995
.7673
8850
1150
1.3032
30
40
.6111
7861
.7916
8985
.7720
8876
1124
1.2954
20
50
.6134
7877
.7898
8975
.7766
8902
1098
1.2876
10
380 OCK
.6157
9.7893
.7880
9.8965
.7813
9.8928
0.1072
1.2799
52000'
10
.6180
7910
.7862
8955
.7860
8954
1046
1.2723
50
20
.6202
7926
.7844
8945
.7907
8980
1020
1.2647
40
30
.6225
7941
.7826
8935
.7954
9006
0994
1.2572
30
40
.6248
7957
.7808
8925
.8002
9032
0968
1.2497
20
50
.6271
7973
.7790
8915
.8050
9058
0942
1.2423
10
39<>00'
.6293
9.7989
.7771
9.8905
.8098
9.9084
0.0916
1.2349
51 (XX
10
.6316
8004
.7753
8895
.8146
9110
0890
1.2276
50
20
.6338
8020
.7735
8884
.8195
9135
0866
1.2203
40
30
.6361
8035
.7716
8874
.8243
9161
0839
1.2131
30
40
.6383
8050
.7698
8864
.8292
9187
0813
1.2069
20
60
.6406
8066
.7679
8853
.8342
9212
0788
1.1988
10
40 00'
.6428
9.8081
.7660
9.8843
.8391
9.9238
0.0762
1.1918
600 (XX
10
.6450
8096
.7642
8832
.8441
9264
0736
1.1847
50
20
.6472
8111
.7623
8821
.8491
9289
0711
1.1778
40
30
.6494
8125
.7604
8810
.8641
9316
0686
1.1708
30
40
.6517
8140
.7585
8800
.8691
9341
0669
1.1640
20
50
.6539
8155
.7566
8789
.8642
9366
0634
1.1571
10
4POO'
.6561
9.8169
.7547
9.8778
.8693
9,9392
0.0608
1.1604
49000'
10
.6583
8184
.7528
8767
.8744
9417
0583
1.1436
50
20
.6604
8198
.7509
8766
.8796
9443
0557
1.1369
40
30
.6626
8213
.7490
8745
.8847
9468
0632
1.1303
30
40
.6648
8227
.7470
8733
.8899
9494
0606
1.1237
20
50
.6670
8241
.7451
8722
.8952
9519
0481
1.1171
10
420 (XK
.6691
9.8255
.7431
9.8711
.9004
9.9544
0.0466
1.1106
480 oo/
10
.6713
8269
.7412
8699
.9057
9670
0430
1.1041
50
20
.6734
8283
.7392
8688
.9110
9696
0406
1.0977
40
30
.6756
8297
.7373
8676
.9163
9621
0379
1.0913
30
40
.6777
8311
.7363
8665
.9217
9646
0364
1.0850
20
50
.6799
8324
.7333
8663
.9271
9671
0329
1.0786
10
43 00 7
.6820
D.8338
.7314
9.8641
.9325
9.9697
0.0303
1.0724
47000'
10
.6841
8351
.7294
8629
.9380
9722
0278
1.0661
50
20
.6862
8365
.7274
8618
.9436
9747
0253
1.0599
40
30
.6884
8378
.7254
8606
.9490
9772
0228
1.0538
30
40
.6905
8391
.7234
8594
.9645
9798
0202
1.0477
20
50
.6926
8405
.7214
8582
.9601
9823
0177
1.0416
10
44000'
.6947
9.8418
.7193
9.8669
.9667
9.9848
0.0152
1.0366
4eooo'
10
.6967
8431
.7173
8667
.9713
9874
0126
1.0296
50
20
.6988
8444
.7153
8646
.9770
9899
0101
1.0236
40
30
.7009
8457
.7133
8632
.9827
9924
0076
1.0176
30
40
.7030
8469
.7112
6620
.9884
9949
0051
1.0117
20
50
,7060
8482
.7092
8607
.9942
9975
0025
1.0068
10
46000'
.7071
9.8495
.7071
9.8495
1.0000
0.0000
0.0000
1.0000
460 OCX
Nat.
Log.
Nat.
Log.
Nat.
Log.
Log.
Nat.
ANGLE
COSINES
SINES
COTANGENTS
TANGENTS
ANGLE
INDEX
Algebra, geometrical exercises for, Electromotive force, 212
153
Ammeter, 217
Ampere, 214
Angle functions, 134
Angles, 64, 134
Approximate number, 2, 120
Archimedes, principle of, 47
Beams, 36
Brake, Prony, 21 ; friction, 21
British thermal unit, 202
Calipers, vernier, 9 ; micrometer, 12
Calorie, 202
Characteristic, 121
Cosine, 135
Cosines, law of, 146
Crane, 148
Density, 42
Digit, 2
Division, of approximate numbers, Horse power, 17
5 ; by logarithms, 124 ; by slide
rule, 129
Dynamos, 236; efficiency of, 237,
240
Equations, graphical solution of, 85
Errors, 1
Field magnets, 236
Foot pound, 16
Fulcrum, 28
Function, 92
Functionality, 91
Geometry, algebraic applications,
52, 97, 153 ; exercises in solid, nu
merical, 1 77 ; graphical, 186 ; alge
braic, 190
Graphs, 65, 223
Gravity, 42
Heat, 195 ; linear expansion, 199 ;
measurement of, 202 ; mechanical
equivalent of, 202 ; specific, 204 ;
latent, 209 ; generated by an elec
tric current, 231
Inequality of numbers, 92
Joule, 202
Kilowatt, 227
Efficiency, 23, 237, 240
Electricity, 212 ; units, 213 ; work Kilowatt hour, 227
and power, 227; generation of
heat, 231; wiring for light and Latent heat, 209
power, 288 ; dynamos and motors, Leverage, 28
236 Levers, 27
273
274
APPLIED MATHEMATICS
Logarithmic paper, 248
Logarithms, 120
Proportion, 110
Protractor, 54
Ratio, 109
" Ready reckoner," 69, 248
Mantissa, 120
Mass, 42
Maximum and minimum values, 98
Measurements, 4
Mechanical advantage, 28
Melting points, 198
Mil, 238
Mil foot, 233
Motors, 286 ; efficiency of, 237, 240 Slide rule, 128
Multiple circuit, 221 Squared paper, use of, 66
Multiplication, of approximate
numbers, 2 ; by logarithms, 124 ; Tangent, 135
by slide rule, 129
Scale, drawing to, 62
Series circuit, 216
Significant figures, 2
Sine, 135
Sines, law of, 144
Numbers, exact, 2 ; approximate, 2,
120 ; scale, 91
Ohm, 218
Ohm's law, 214
Parallel circuit, 221
Parallel lines, 69
Parallelogram, 69
Perpendicular, 55
Power, 17, 226
Prony brake, 21
Thermometers, 195
Triangle, of reference, 134; of
forces, 147
Variables, 62
Variation, 164 ; inverse, 166 ; joint,
167
Volt, 218
Voltmeter, 217
Watt, 226
Watt minute, 226
Weight, 42
Work, 16, 226