10. Mention some function of the form mx + b arising in daily life, and discuss it graphically.

11. Plot the graphs of the equations below, using a single set of coordinate axes for each case. Each figure will contain several lines cor

responding to the values of the constants indicated.

12. Exhibit in tabular form as many as possible of the corresponding properties of a straight line and the function f(x) = mx + b of which the line is the graph (see Section 15). Indicate special properties of this function which distinguish it from all other functions.

21. Variation. We have already had a number of illustrations of the important relation y = mx, in which one variable is a constant times the other. For example, the price paid for eggs is the price per dozen times the number of dozen bought; if a man walks at a uniform rate, the distance is the rate times the time; etc. Two technical terms are used in this connection. DEFINITION. A variable is said to vary as, or to be proportional to, a second variable if the first is equal to a constant times the second. Symbolically, y varies as x, or y is propormx, m being a constant.

tional to x, if y =

This use of the word proportional is justified by the fact that two values of y are proportional to the corresponding values of x. For if y1 = mx1, and y2 = mx2 then by dividing the first

equation by the second, we obtain the proportion

Yı X1

=

Y2 X2

The phrase "variation of a function" used in Section 12, should not be confused with "variation" as a heading under which we discuss proportional variables. A more general use of the latter term will be taken up in Chapter III.

The mass of a body is the amount of matter in it. If the nature of the material in the mass varies as the volume. and V the volume, M = mV, density of the substance of which the body is composed. The value of m is the mass of a unit volume.

body is the same throughout, the That is, if M denotes the mass where m is a constant called the

EXAMPLE. If the mass of an aluminum object is 13, and its volume is 5, find the mass of an aluminum object whose volume is 8.6, and the volume of an object whose mass is 17.4.

Since

Let M denote the mass and V the volume of any aluminum object. Then since M varies as V,

Graphical solution. The graph is a straight line through the origin (Corollary 1 to Theorem 1, Section 20). It also passes through the point A (5, 13), since M 13 when V = 5. It is therefore the line OA in the figure.

=

From the graph, if V= 8.6, the value of M is CD 22.4; and if M = 17.4, the value of V is OG 7. These are the required values. }

=

The slope of the line, computed from O and A, is

=

is the mass of a unit volume of aluminum, or the density, the slope represents the density.

=

=

=

13 and V

=

5

2.6.

Algebraic solution. Substituting the given values M in (1), we obtain 13 5m, whence the density m = 13 Substituting the value of m in (1), we have the relation between the mass and volume of any aluminum object,

The graph has the advantage of exhibiting the relation to the eye, and if it is constructed carefully, it is sufficiently accurate for the purpose of a "ready reckoner." It is very useful when a number of values of either variable are desired and great accuracy is not essential, and in any case it furnishes a valuable check on the accuracy of the computation.

When we are given the fact that a variable y is proportional to a second variable x, the general form of the law connecting x and y is y = mx. The exact form of the law involves a particular value of the constant m. This value of m may be found from a pair of corresponding values of x and y, other than (0, 0), as in the algebraic solution of the example above

22. Uniform Acceleration. The motion of a body is said to be accelerated if the velocity is variable. It is said to be uniformly accelerated if the change in velocity is proportional to the change in time. The rate of change of velocity is called acceleration, and it is measured by the number of feet per second by which the velocity changes in each second.

A freely falling body moves with a uniformly accelerated motion, if the resistance of the air is neglected. The acceleration, which is called the acceleration due to gravity and which is denoted by g, is approximately 32 feet per second per second. If the positive direction is downward, g = 32, but if the positive direction is upward, g = 32.

Since the rate of change of velocity of a uniformly accelerated body is constant, v is a linear function of t, and the graph is a straight line. The slope of the line represents the acceleration, and the intercept on the v-axis represents the velocity when t = 0, which is called the initial velocity.

EXERCISES

1. Hooke's law says that the amount of stretching in a stretched elastic string is proportional to the tension. If a 2-lb. weight will stretch a string 3 feet, what tension arises when the string is stretched 1 foot? Draw the graph.

2. Prove graphically, that if y varies as x, then any two values of y are proportional to the corresponding values of x.

3. Is the definition near the end of Section 16 equivalent to the statement following? A variable y changes uniformly with respect to x if the change in y is proportional to the change in ≈ producing it.

4. The mass of a body varies as its volume. On the same axes construct the graphs showing the relation between mass and volume for bodies composed of the following substances whose densities are given: Lithium, 0.6; alcohol, 0.8; India rubber, 1.0; magnesium, 1.7; diamond, 3.5; silver, 10.5. How are the various densities represented?

5. Wihelmy's law for chemical reactions states that the amount of chemical change in a given time is proportional to the quantity of reacting substance in the system. Construct the graph.

6. The velocity acquired in t seconds by a body falling freely from rest is given by the equation v = 32t. Plot the graph and from it determine how fast the body would be falling after 4 seconds. What does the slope represent?

7. If a ball is dropped from a high building, how fast will it be moving at the end of one second? at the end of 2 seconds? at the end of 4 seconds? If a ball is thrown vertically upward, how will its velocity be affected in any second? If it is thrown up with an initial velocity of 96 feet per second, how long will it rise?

8. A train starting from rest acquires a velocity of 50 feet per second in 15 seconds. Find the average acceleration.

9. An automobile is moving at the rate of 40 feet per second, when the power is shut off and the brakes applied. If it moves thereafter with a uniform acceleration of - 5 feet per second per second, how long before it will stop?

10. Name two quantities arising in every-day experiences which are proportional, and give the value of the constant involved in the relation between them.

11. The pressure of a liquid is proportional to the depth. If the pressure per square inch on the suit of a diver is 5.2 lbs. for a depth of 12 ft. how deep can he go safely if 78 lbs. per square inch is an allowable pressure? Construct the graph.

12. For small changes in altitude, atmospheric pressure varies as the altitude. If the change in the reading of a barometer is 0.1 of a unit for each 90 ft. ascent, construct the graph of this relation and determine the difference in readings of two places with a difference in altitude of 1000 ft.

13. In a spring balance the extension of the spring is proportional to the weight. If a weight of 2 lbs. lengthens the spring 1 inch, construct a graph and determine the extension of the spring for weights of 5, 8, and 17 pounds.

14. The stock of a corporation yields an annual dividend of 5%. What is the relation between the total amount a stockholder receives in dividends and the time the stock is held? If the graph of this relation is drawn, what does the slope of the line represent? What is the value of the slope if the par value of the stock held is $10,000?

15. The rent charged to each department of a store is proportional to the percentage of the total floor space which the department occupies. If the store pays a rent of $12,000, construct a graph to show the rent charged to a department of any size, using as large a scale as convenient. From the graph, determine the rent charged to each department if the various departments occupy 17, 25, 10, 8, 5, 12, and 23 per cent of the total floor space.

16. An automobile moving at the rate of 22 feet per second (15 miles per hour) begins to coast down a hill. If its velocity after t seconds is given by v = 10t+ 22, find from the graph of v how fast it would be going after 5 seconds, and how long it would take it to acquire a velocity of 60 miles per hour. What is the acceleration?

=

32t

17. If a ball is thrown vertically upward with a velocity of 100 feet per second, its velocity after t seconds is given by the equation v +100. Plot the graph, and describe the motion. What is represented by the intercept on the v-axis? the intercept on the t-axis? the slope?

18. The ordinate of a point on one of the graphs in the figure represents the velocity v of a moving body at the time t represented by the abscissa. Describe the

The line is below the t-axis if t< 6, so that during this time v is negative and the body is moving in the opposite of a certain direction which has been assumed as positive, i.e., in the negative direction. The line is above the t-axis if t > 6, so that v is positive and the body is moving in the positive direction.

The slope of the line, computed from the points (6, 0) and (0, − 2) is the

Since the line rises to the right, the velocity constantly increases, which agrees with the fact that the acceleration is positive.

=

- 6, v =

4, and when t =

12, v = 2.

From the graph, when t Discarding the technical terms positive and negative, the motion may be described as follows: At the start the body is moving in a certain direction at the rate of 4 feet per second. During the next 12 seconds it slows down uniformly and comes to rest for an instant, and then begins to move in the opposite direction with a constantly increasing speed. After 6 seconds it has acquired a velocity of 2 feet per second. The velocity changes by of a foot per second in each second of the motion.

Notice that where the body is at any time, and in particular where it comes to rest, cannot be determined from the graph of the velocity.

19. The velocity of an automobile, in feet per second, at the time t,