(10,9.5) constant. The graph is therefore a straight line. If there is no liquid in the bottle, there is no alcohol, and hence the origin (0,0) is on the line; and if the bottle contains 10 units of liquid, there are 9.5 units of alcohol in the bottle. Hence (10, 9.5) is a second point on the line, and these two points are sufficient to determine the graph. The slope represents the rate of change of a with respect to l, Aa/Al = 0.95, which gives the percentage of alcohol in the liquid. a -8 ·6 5 3 -1 6 FIG. 31. In plotting the graph by using the point (10, 9.5), it is assumed that the unit of volume is some such unit as an ounce or cubic centimeter. If a bottleful were chosen as the unit, which is convenient in many problems, it would be better to use the point (1, 0.95) and choose the unit on the l-axis as large as convenient. For then l 1 would mean that the bottle was full. = 18. Slope of a Straight Line. "Slope of a line" (definition, Section 17) is the term used technically by mathematicians for what might be called the measure of steepness of the line. It may represent many things. Thus in the example in Section 16, As/At is the slope of the line, and hence the velocity v is represented by the slope. If the slope m is computed from two points P1 (x1, y1) and P2(x2, y2) on the line, we have, in either figure, In finding Ay and Ax it is essential that both coördinates of P1 be sub 1 tracted from those of P2, or vice versa. Theorem 1. A line runs up to the right or down to the right according as its slope is positive or negative. For two points P1 and P2 may be chosen on the line so that Pilies to the left of P2, whence Ax is positive. Then Ay will be positive or negative according as P2 is above or below P1. Hence m = Ay/Ax will be positive or negative according as Pa is above or below P1, that is, according as the line runs up to the right or down to the right. 2 The precise relation between the value of m and the direction of the line as determined by the angle the line makes with the x-axis involves a transcendental function, and will be considered in a later chapter (see also Exercises 3 and 4 below). The proof of the converse is left as an exercise. Construction. To construct a line through a given point P1 whose slope is a given positive fraction a/b, take Q b units to the right of P1, and P2 a units above Q; then P1P2 is the required line. For we have If the slope is negative, take P2 below Q. If the slope is an integer, it may be regarded as a fraction with unit denominator. If two points are close together, the line through them cannot be drawn as accurately as if they were farther apart. Hence, in this construction, it is sometimes desirable to take PiQ equal to some multiple of b and QP2 equal to the same multiple of a. EXERCISES 1. Plot the following pairs of points and the lines determined by them. Find Ax, Ay, and the slope m, and indicate the graphical significance of each of these quantities. 2. Construct the lines through the points indicated with the given slope. (a) P(2, 3), m = 1. = (b) P(3, − 1), m = 글. (c) P(0, 4), m = (d) P(3, 0), m 2. 3. Show that of two lines through the same point the one which makes the greater acute angle with the x-axis has the larger slope numerically. 4. Show that the slope of a line parallel to the x-axis is zero. 5. If a telegram containing 10 words or less can be sent to a certain place for 35¢, and a 24-word message for 63¢, construct a graph showing the cost of a message containing any number of words, and determine the charge for each additional word above 10. What represents this charge? 6. Construct a graph to show the cost of any number of eggs at 40¢ a dozen. Determine from the graph the number of eggs which can be purchased for 70¢. What does the slope represent? 7. Construct a graph to show the amount of silver in any amount of an alloy containing 25% of silver. Determine from the graph how much silver there is in 20 pounds of the alloy. What does the slope represent? 8. A pound of an alloy contains 3 parts of silver and 5 of copper. Construct a graph to show the amount of silver in any amount of the alloy. Determine from the graph how much of the alloy contains 4 pounds of silver, and how much silver there is in 10 pounds of the alloy. Interpret the slope. 9. Ten ounces of alcohol 95% pure are poured into a bottle and then 5 ounces of water are added. Construct a graph showing the amount of alcohol in the bottle during the process as a function of the amount of liquid. 10. Solve Exercise 9 if 5 ounces of 50% alcohol are added instead of 5 ounces of water. 11. Is the slope of the line joining (3, 6) to (6, 12) the same as the slope of the line joining (3, 6) to (-3, -6)? What can be said of the three points? 12. Are the points (6, 1), 2, - · 3), and (0, − 2) on a straight line? 13. Show that the points (2, 1), (3, 7), (5, 3), and (0, 5) are the vertices of a parallelogram. 14. Find the value of b if the line determined by (2, b) and (− 1, 3) is parallel to the line joining (- 4, – 5) and (7, – 2). 15. Two boys start from the same point and run in the same direction, one at the rate of 15 feet per second, the other at the rate of 20 feet per second. Construct the graphs showing the distances from the starting point to each boy at any time. What do the slopes represent? Determine from the graphs how far apart the boys are after 3 seconds. Solve the same problem if the boys run in opposite directions. 16. Water is admitted into a tank through two pipes at the rates of 3 and 5 gallons per second. On the same axes, plot graphs showing the amount which has entered through each pipe at any time, and also the total amount. What do the slopes represent? 17. If the cost of setting type for a circular is 50 cents, and if the cost of paper and printing is half a cent a copy, construct a graph showing the cost of any number of copies. From it determine the cost of 500 copies. What does the slope represent? Hint: The cost of setting the type may be regarded as the cost of zero copies. 18. If $100 is deposited in a bank to draw simple interest at 6%, construct a graph to show the amount at any time. What does the slope represent? The intercept on the "amount axis"? Determine graphically how long it would take the principal to double itself. What is the relation between the amount and the time? 19. One of the horizontal boards in a flight of stairs is called a tread, and one of the vertical boards a riser. How can the steepness of the stairs be expressed in terms of the widths of the treads and risers? 20. Show that the average rate of change of a function other than a linear function is represented by the slope of a secant line of the graph. 19. Graphical Solution of Problems Involving Functions which Change Uniformly. Graphical methods are used freely d -1 10.9 10.8 G 10.7 10.6 10.5 0:4 0.3 0.2 10.1 DI H 3 FIG. 34. C B by engineers and others because of their relative simplicity. A graphical solution of a problem is also often used as a simple means of checking the accuracy of some other solution. A method of obtaining approximate solutions of many problems by means of graphs is illustrated in the following problems: EXAMPLE 1. A druggist has a 50% solution of a certain disinfectant and also a 10% solution. How much of the former must be added to 3 pints of the latter to obtain a 20% solution? The graph representing the amount of disinfectant, d, in a quantity, 1, of the 50% solution is the straight line OA through the origin whose slope is 0.5. Similarly, the graphs for 10% and 20% solutions are respectively OB and OC whose slopes are 0.1 and 0.2. On the l-axis take OD = 3, and draw the ordinate DE to the line OB. Then DE represents the amount of disinfectant in 3 pints of the 10% solution. Through E draw EF parallel to OA. Then the ordinate of a point on EF represents the amount of disinfectant in the mixture as some of the 50% solution is added to the 3 pints of the 10% solution. If DE cuts OC at G, and if GH be drawn perpendicular to the l-axis, then in OH pints of the combined mixture there are HG pints of disinfectant; and since G lies on OC, the strength of the combined mixture is 20 %. Hence we must add DH, or 1 pint approximately, of the 50% solution. EXAMPLE 2. A freight train goes from A to B at the rate of 15 miles per hour. Four hours after it starts, an express train leaves A and moves at the rate of 45 miles per hour. The express arrives at B half an hour ahead of the freight. How far is it from A to B and how long does it take each train to go? Let abscissas denote time in hours measured from the time of departure of the freight train, and ordinates distances from A. The graph representing the distance of the freight at any time is the straight line OC d 100 -90 -801 -70 -60 50 -40 E H C G through the origin with the slope 15, and that of the express is the line DE through the point D(4, 0) with the slope 45. At B the express is half an hour ahead of the freight, and hence we must determine a point on OC half a unit to the right of DE. To do this, take F on the t-axis so that DF 1. Draw the line through F parallel to DE, and let it meet OC at If the line through G parallel to the t-axis cuts DE at H, then HG = 1. Hence, the required distance AB is represented by the ordinates IH JG 100 miles approximately. = G. = = The time of the freight is represented by OJ 6.7 hours, approximately, and that of the express by DI = 2.2 hours, approximately. The line FG may be interpreted as the graph of a train leaving half an hour later than the express and moving at the same rate of 45 miles an hour. By the conditions of the problem, such a train would overtake the freight at B. |
