EXERCISES With this 1. Make a paper slide rule by copying the logarithmic scale given in Fig. 144 on two paper rules of an adequate degree of stiffness. instrument calculate the following and describe the process. 4 x 6,,, 6 x 9, (5 x 7)/(8 x 9). 2. Write the equations of the lines a, b, c, g, in Fig. 150. 3. Write the equations of the lines in Fig. 152. 4. Plot the graphs of the following equations on logarithmic paper. C a e g Ah 1 2 3 4 5 6 7 8 9 1 X FIG. 152. (h) pv vš = 1. 5. Plot the graphs of the following equations on semi-logarithmic paper. (a) y = 2(102), (c) y = (10%), (b) y = 3(10-2), (d) y = e2x 6. Measure the length of the logarithmic scale in Fig. 144 and calculate the radius of a circle whose circumference is equal to this length. Fasten a circular disk of this radius to a larger disk by means of a pin through their centers so that they may rotate freely. On the circumference of the smaller disk and on the circle of equal radius of the larger copy the logarithmic scale of Fig. 144. Make the calculations of Exercise 1 with this instrument. What advantage has the circular slide rule over the slide rule of the first exercise? 7. Find the equations of the lines in Fig. 150 if the given ranges of the scales are as follows: (a) x-axis, 10 to 100, y-axis, 1 to 10. (b) 0.1 to 1, 10.0 to 100. (c) 0.01 to 0.1, 8. Find the equations of the lines in Exercise 3 if the ranges of the scales are as follows: x-axis, y-axis, (a) 10 to 20, (b) - 5 to 0, (c) 4 to 8, 100 to 1000. 90. Empirical Data Problems. If the points representing a given table of empirical data appear to lie on the graph of an equation of the form y = k10m, the constants k and m may be determined as in the EXAMPLE. Determine the law representing the table, and find the value of y if x = 2.5. Plotting the points whose coördinates are x 1, 2, 3, 4 the pairs of values in the table we get a curve y 2.37, 3.75, 5.97, 9.45 which appears to be of the form To test this assumption, we plot the table on semi-logarithmic paper. As the points obtained lie very nearly on a straight line we conclude that 验 -9 -8 7 -6 -51 4 the law may be represented by an equation of the form (1) with a fair degree of accuracy (Theorem 2, Section 89). y 1+10 To determine k and m, equate the logarithms of both sides of (1), which Equation (3) is linear in the variables x and Y, and the constants m and K can be determined by the method in Section 27, page 78, from a table of values of x and Y. We therefore look up the logarithms of the given values of y, which are the values of Y. From this table we obtain, by the method referred to, the values m = 0.200 and K = 0.175. Since K = log k 0.175, reference to the table shows that X 1, 2, = 3 .4, Y .375, .574, .776, .975 k = 1.49. Substituting the values of k and m in (1), we have the required If the points representing a given table appear to lie on the graph of an equation of the form y = kr" (page 119), a value of n may be chosen by means of the form of the graph, and the value of k determined, as in Section 44, page 127. The method is unsatisfactory in that the only way to tell which of two values of n is the better, for example, whether n = 2 or n= | is the better, is to try them both. The following procedure is preferable. If the points representing the table appear to lie on the graph of an equation of the form plot the points on logarithmic paper. If the graph is now approximately straight (Theorem 1, Section 89), we conclude that the law may be well represented by (5). As the logarithms of the two sides of (5) must be equal, we have or log y = n log x + log k, where Y = log y, X = log x, and K = log k. (6) (7) With the table of logarithms build the table of values of X and Y corresponding to the given table of values of x and y. From it the values of n and K may be found by the method in Section 27, page 78. The value of k is then found from that of K. It should be noticed that if we suspect that the law has the form (1) or (5) it is not essential that the table should be plotted on semi-logarithmic or logarithmic paper, respectively. But this plotting strengthens our feeling of having a suitable law, and approximate values of the constants may be obtained easily directly from the gra ph (see Example 3, Section 89, and Exercises 2 and 3 of the preceding set). Of course, if the new graph is not approximately straight the choice of one of these two laws does not give a very good representation of the given data. If the points representing the table, plotted on ordinary crosssection paper, appear to lie on the graph of we interchange x and y, and proceed as in the example above. The graph of (5) may be distinguished from that of (1) or (8) by the fact that it passes through the origin, if n is positive, and that it does not cut either axis, if n is negative. EXERCISES 1. Find the law of growth of the population of the United States from the following data: Year Population in millions 1880, 50, 1890, 1900, 76, 1910, 92, Let x = 0 in 1880. Assuming the law of growth does not change find the population in 1920. 2. Water flows out of a sharp-edged circular opening in the vertical side of a tank. The table gives pairs of values of the head h, the height in feet of the surface of the water above the center of the opening, and the discharge Q, the number of cubic feet of water flowing through the opening in a second. Determine Q as a function of h. 0.390, 0.709, 1.013, h 3. A rectangular weir is a rectangular notch in the wall of a reservoir or channel. The table gives pairs of values of the head h, the height in feet of the surface of the water above the bottom of the notch, and the discharge Q in cubic feet per second. Find the law. Find Q if h = 0.265. h 0.053, 0.103, 0.160, Q 0.217, 0.0422, 0.1095, 0.2065, 0.3200, 4. A triangular weir is a triangular notch in the side of a tank or channel. The table gives pairs of values of the head, h, and the discharge, Q. The angle of the notch in this case was 90°. Find the law. h 0.095, 0.169, 0.249, 0.330, 0.392, 0.00615, 0.0302, 0.0780, 0.1590, 0.2445, 5. In a chemical experiment it was found that the concentration x of a solution was connected with the amount of precipitation y of a metal as in the table. Determine the law and find y if x = 5. 6. Find the velocity, v, of a certain chemical reaction as a function of the temperature, T, from the data given in the table. Find v if T = 50°. 7. Find the law connecting the maximum speed, ", of an electric vehicle and the total weight, W, in thousand pounds from the following data. What weight will reduce the maximum to 6 miles per hour? 8. Find the law connecting the cost, C, in cents per miles for tires, repairs, battery, electricity, of electric vehicles of weight, W, in thousand pounds. What will be the cost per mile of a vehicle weighing 10,000? 9. From the table find the law connecting the resistance, R (in pounds per ton), which a passenger train encounters at a speed, v (miles per hour). Find R if v = 60. R 5.9, 5.5, 5.4, 5.5, 5.6, 6.2, 10. Airships are provided with air bags within their hulls for regulating the height of ascent. The total cubic contents of these air bags when filled is V' = mV where V is the cubic contents of the airship. Values of m for various heights, H, are given in the table. Find the law and the value of m when H = 8. H in thousand ft. 1, 2, 0.04, 0.075, 4, 10, 0.141, 0.318, 11. Find the law connecting the load, L (in pounds per square foot of wing) of an aeroplane with the area, A, of wings (in square feet). Find the safe load if A = 200. 1. The temperature of a body cooling according to Newton's law, 0 = 0oemt, fell from 125° to 94° in 8 minutes. Find the equation connecting the temperature and the time of cooling. 12. Construct a scale for the function x2 following the method used in constructing the logarithmic scale. Find by means of the scale (2.3)2 and √11. = = 3. Construct a table of values for the function y 2x2+3 and plot the graph. Let u = x2 and hence y 2u +3. Construct on the horizontal axis a uniform scale of u and a corresponding non-uniform scale of X. Plot the table of values of x and y using the non-uniform scale of x and the uniform scale of y. What is the character of the graph? |