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Full text of "Elements Of Applied Mathematics"

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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 common-sense 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 

FOUR-PLACE 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 100-foot 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- (3-142)'. 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 safety-valve 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 safety-valve 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 cast-iron 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 12-in. 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 1-lb. 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 200-lb. 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 
I-beam 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 side-wheel steamer the engine has a 6-ft. 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 
6-in. 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 


High-speed . 
Medium-speed 


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 500-g. 
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 500-g. 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 6-lb. 
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 4-lb. 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 7-lb. 
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 5-g. and a 50-g. 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 200-g. 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 200-g. 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 mid-point 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 50-g. 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 2-lb. 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 8-ft. 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 yellow-pine 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 200-g, weight 
at the center. Read each balance. 




FIG. 13 

2, With the meter stick as in Exercise 1, place a 200-g. 
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 500-g. 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 yellow-pine 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 arc-light carbon. 

PROBLEMS 

1. How much will a brass 50-g. 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. Nickel-aluminum 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 four-sided 
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 Pi-obleml31et^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 2x-5 
(g) x x-f- 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 y-axis. 
The point is called the origin. 

Since the number of degrees is always greater than 70, we 
may call the a-axis 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' Pocket-Book," 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 9-in. gong ? of anll-in. 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 x-axis at the top of the sheet. On the x-axis 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. Straight-line 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 
24-lb. 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 

yellow-pine 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 yellow-pine 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 3-hr, 
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 x-axis let a large square = 20 mi. Let St. Louis be at 
the lower left-hand 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 
mid-point 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 ar-axis 
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 20-mL 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 aluminum-zinc 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 ,r-axis and y-axis 
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 
y-axis 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 y-b 

- 

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 x-axis, 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 : 

1-5-2 + 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 




T-IP* n\00 

CO 1-1 
\ 


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 x-axis. 

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 or-axis 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 -6x-S. 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 oj-axis 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 jr-axis. 

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 rr-axis 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 z-axis 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 -3z-9 = (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. Straight-line 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 arc-light 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 straight-line 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 x-axis 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 open-top 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 1000-mi. 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. 



7-X 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 alternate-interior 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 
x-axis and the left border line be the y-axis, 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 mid-points 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 ,r-axis, the left border line be 
the y-axis, 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 open-top 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 open-top 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 open-top 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 open-top 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 mid-points 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 mid-point 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 four-place 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 four-figure 
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 3-7-16 = '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 
1-C 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 

= | (1-732) 
= .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 = 



= \ (1-732) 



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 good-sized 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 lamp-post. If 
his shadow is 17 ft. long, what is the height of the lamp-post ? 

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 M-to 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 + 25-2 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 + 81-49 
"" 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 8-lb. 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 8-lb. 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 

8-lb. 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 
king-post 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 star-shaped 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 quarter-mile 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 
6-in. 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 mid-points 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 
mid-points 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 500-g. weight 6 cm. from the fulcrum. Sus- 
pend on the other side a 100-g. 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 2-in. 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 half-inch pipe, how many 
gallons per minute will flow through a 3-in. pipe ? 

17. The safe load on a wrought-iron 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 1-in. 
plate, what size of rivets is required for a f -in. plate ? What 
thickness of plate can be riveted with J-in. 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 T-rail 2 ft. long and 8| in. in cross-sectional 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 cross-sectional area. What is the resistance of a column of 
mercury 3 m. long and 4 sq. mm. in cross-sectional area, the 
resistance varying directly as the length and inversely as the 
cross-sectional 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 cross-sectional 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 4-in. 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 12-ft. 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' Pocket-Book" 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 1-in. 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 single-acting pump is 10 in. in 
diameter, has a 10-in. 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 40-in. 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 2-in. 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 

2-77TA. 

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 45-51 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 left-hand 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 wrought-iron 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, hot-water 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 wrought-iron 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 plate-glass 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 cast-iron 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 red-hot (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 red-hot (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 80-lb. 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 incandescent-lamp 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 110-volt 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 220-volt circuit has a 
resistance of 330 ohms, and a 16 candle power lamp for a 
110-volt 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 arc-light 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 electric-light 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 110-volt 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 
110-volt mains with a man having a hand-to-hand 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 x-k 
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 = j-j = - 

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 = 



OM-XN 
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. 

20.30-40 



2O ohms 



SOLUTION. R = 



20.30 + 20.40 
= 9.2 ohms. 



30-40 




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 110-volt 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 110-volt 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, 1736-1819, 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 
W--g. (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 = V-A = 45 x 10 = 450 Watts. 



2. A 16 candle power incandescent lamp is on a 110-volt 
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 150-kw. 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 10-kw. 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 electric-lighting 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 10-kw. 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 550-volt 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 5-kw. 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 arc-light 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 6-in.-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 6-lb. 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 110-volt 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 500-volt 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 inil-ft. 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 field-magnet coils. 

1. Series-wound dynamos. The field-magnet 
coils are connected to the armature so that 
the whole current generated passes through 

them. FIG. 97 

2. Shunt-wound dynamos. The field-mag- 
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. Compound-wound 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 series-wound 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 series-wound 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 shunt-wound 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. 
V-A^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 50-kw., 125-volt, compound-wound dynamo has a shunt 
resistance of 62.5 ohms, a series-coil 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 110-volt mains. If the armature resistance is .09 ohm, the 
series-coil resistance .078 ohm, and the shunt-coil 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 110-volt 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 dynamo-electric 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 220-volt 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 3-kw. 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 550-volt 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 x-axis and the y-axis 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 mid-point of YZ. Tack the sheet of paper on a drawing 
board so that the T-square, in position, lies on O and P. Set 
the T-square on A (making OA = 3.14) and draw AB. Set the 
T-square 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 right-hand 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 mid-points 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 T-square, in 
position, lies on and the mid-point of YZ. Move the T-square 
up to 2 on OF and draw a line from 2 to YZ. Move the 
T-square 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 = 4o5-i (g) y = .006 ar". 

y = 8a-i (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 T-square, in 
position, lies on M and N. Move the T-square to A and draw 
AB. Move the T-square 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 wrought-iron 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 V-notch, 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 yellow-pine 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 yellow-pine 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 cast-iron 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 
slow-moving 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. 



BIBLIOGRAPHY 



PERRY. The Teaching of Mathematics, pp. 101. 1902. The Macmillan 

Company. 2s. 
YOUNG. The Teaching of Mathematics, pp. 851. 1907. Longmans, Green 

& Co. $1.60. 

KENT. Mechanical Engineers 1 Pocket-Book, pp. 1129. 1906. John Wiley 
& Sons. $5.00. 

SEAVER. Mathematical Handbook, pp. 365. McGraw-Hill Book Com- 
pany. $2.60. 

SUPLEE. The Mechanical Engineer's Reference Book, pp. 859. 1905. 
J. B. Lippincott Company. $5.00. 

TRAUTWINE. Engineers' Pocket-Book, pp. 1267. 1909. John Wiley & 
Sons. $5.00. 

BRIGGS. First Stage Mathematics, pp. 312. 1910. Clive. 2s. 
GODFREY AND BELL. Experimental Mathematics, pp. 64. 1905. Arnold. 

2s. 
LEIGHTON. Elementary Mathematics, Algebra and Geometry, pp. 296. 

1907. Blackie. 2s. 
LODGE. Easy Mathematics, Chiefly Arithmetic, pp. 486. 1906. The 

Macmillan Company. 4s. 6d. 
MAIR. A School Course of Mathematics, pp. 388. 1907. Clarendon 

Press. 8s. 6d. 
MYERS. First-Year Mathematics, pp. 365. 1909. The University of 

Chicago Press. $1.00. 
MYERS. Second- Year Mathematics, pp. 282. 1910. The University of 

Chicago Press. $1.50. 
SHORT AND ELSON. Secondary School Mathematics, Book I, pp. 182. 

1910. $1.00. Book II, pp. 192. 1911. $1.00, D. C. Heath & Co. 
STONE-MILLIS. A Secondary Arithmetic, pp. 221. 1908, Benj. H* 

Sanborn & Co. 75 cents. 

BREGKENRIDGE, MERSEREAU, AND MOORE. Shop Problems in Mathe- 
matics, pp. 280. 1910. Ginn and Company. $1.00. 

261 



262 APPLIED MATHEMATICS 

CASTLE. Workshop Mathematics, pp. 331. 1900. The Macmillan Com- 
pany. 65 cents. 

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 Taylor-Holden 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. 
STONE-MI 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. McGraw-Hill Book 

Company. $2.00. 
WHITTAKER. Arithmetic of Electrical Engineering, pp. 159. Whittaker. 

25 cents. 

BHREND8EN-G5TTiNG. 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. Aufgaben-Sammlung, 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. FOUR-PLACE LOGARITHMS OF THREE-FIGURE NUMBERS 

II. THE NATURAL SINES, COSINES, TANGENTS, AND COTAN- 
GENTS OF ANGLES DIFFERING BY TEN MINUTES, AND THEIR 
FOUR-PLACE 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 



FOUR-PLACE 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 



FOUR-PLACE 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 



FOUR-PLACE 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