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Except for purposes of explanation, there is no need of writing anything but the desired result.

All the tables in Huntington's Tables are arranged so that the numbers in the body of any table increase as we read from left to right, for the reason that a uniform procedure is obtained for applying the correction. In interpolating, the desired result lies between two numbers in the same row; a correction applied to the left-hand number is always added, while a correction to be applied to the right-hand number is always subtracted.

EXERCISES

4.50 to x =

4.60, using the values of

1. Plot the graph of x2 from x = x2 given in Huntington's Tables.

2. If we plot the graph of √x for values of x from 2.30 to 3.40, using the square roots given in the Tables, why will the graph be nearly straight? 3. Find the values of the numbers given below, illustrating the interpolation graphically.

(a) 3.1722. (b) √3.478. (c) 6.493. (d) V0.02814. (e) 1/926. 4. Find the numbers following without illustrating the interpolation graphically. The tabular difference and the necessary number of tenths of that difference should be obtained mentally.

(a) 1.5342, √/0.468, 8.433, 2.46, 1/647.

(b) Squares of 2.784, 3762, 0.01388, 3846000, 0.00003728.
(c) Square roots of 0.634, 3.248, 42.7, 384.3, 279,000, 0.001876.

(d) Cubes of 3.143, 0.774, 1683, 0.00004592, 4889000.

(e) Cube roots of 0.02258, 0.226, 19.34, 0.00176, 328000.

(f) Reciprocals of 31.76, 0.00647, 35990, 0.0004325, 647.

5. Find the square of 4732, and check the result by finding its square root. Find the cube root of 3479, and check the result by cubing it. Find the reciprocal of 25.63, and check the result.

6. Find √374.33, 0.024782, 1/73.262, 1/√2438, 1/48.363.

7. Solve the following equations, using the Tables to simplify the computations:

(a) x2 + 32x + 19 = 0.

(c) 3x2 + 29x - 40 = 0.

=

(b) 2x2 47x-27 0.
(d) 5x2 + 54x – 31 = 0.

8. How long an umbrella will go into a trunk measuring 31.5 × 18.5 × 22.5 inches, inside measure, (1) if the umbrella is laid on the bottom? (2) if it is placed diagonally between opposite corners of the top and bottom?

9. A house with a gambrel roof is 28.5 feet wide, the first set of rafters has a slope of, and the top set a slope of 3. If the joint in the roof comes 7.8 feet from the side of the building measured horizontally, how long is each set of rafters?

10. If five individuals weigh 120, 124, 116, 112, 123 pounds, respectively, and five others 95, 150, 132, 105, 113 pounds, respectively, then the average weight M of either group is 119. But one group is distributed very closely around the mean M, whereas the other group exhibits marked deviations from it. A measure of the variability or tendency to deviation of measurements is given by the following formula, called the standard deviation.

2

S.D.

d12 + d22 +

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+dn2

The average M is found, and each individual measurement is subtracted algebraically from M, thus obtaining a series

of deviations d. The sum of the squares of these deviations is divided by n, the number of measurements, and the square root of the quotient is the standard deviation from the average M.

The coefficient of variability is defined as the ratio of the standard deviation from the average to the average.

Compare the coefficients of variability for the two sets of measurements. 11. The strength of grip of right hand and left hand, in hectograms, for 10 boys is given in the following table. Compute the standard deviation and coefficient of variability for each.

Right hand. 158, 200, 210, 226, 248, 270, 296, 320, 348, 403. Left hand. 138, 185, 200, 224, 244, 260, 282, 305, 336, 400.

43. Variation. DEFINITION. It is said that y varies as, or is proportional to, the nth power of x if y = kx", n being positive, while y varies inversely as, or is inversely proportional to, the nth power of x if y=k/xn. If it is desired to contrast these two forms of variation, the former is called direct variation as opposed to inverse variation.

The case of direct variation in which n 1 has already been considered in Section 21, page 61. In any case, the value of the constant k may be determined from a given pair of values of x and y, as in that section, by substituting the given values of x and y.

The graph of the relation y = kan may be obtained from one of the curves considered in Sections 38 to 40, by means of the theorem in Section 30.

The language of variation is used frequently in the applica

tions of mathematics. Thus for bodies moving with velocities near that of a rifle ball, the resistance of the air varies as the cube of the velocity. The intensity of light varies inversely as the square of the distance from the source of light, etc.

The language of variation is extended also to apply to relations involving more than two variables. One quantity is said to vary jointly as two or more other quantities, if the first is equal to a constant times the product of the others: direct and inverse variation may both be involved. For example Newton's law of gravitation states that the attraction of two bodies varies jointly as their masses and inversely as the square of the distance between them. If the masses of the bodies are m and m', and if d is the distance between them, then the attraction A is given by A = kmm'/d2.

EXERCISES

1. If the area of a rectangle is 12 square inches the altitude varies inversely as the base. Represent the relation graphically. On the figure construct several rectangles of area 12.

2. A man told a contractor that he wished to have certain work done in the next three days. The contractor replied that it would be impossible, as it would take six men three weeks to do it. Whereupon the man told him that thirty-six men would do it in three days, and to have them on the job the next morning. What relation did he assume exists between the number of men and the number of days in which they could do the work?

3. An assumption which a psychologist has made states that the lawlessness of a mob varies as the square of the number of individuals involved. Illustrate by a graph. Compare the lawlessness of two mobs of 50 men and 500 men.

4. The mean distance of the earth from the sun is 93,000,000 miles and from the moon is 240,000. If the mass of the earth is taken as 1, the masses of the sun and moon are approximately 330,000 and 1/81 respectively.

(a) Does the sun or the earth exert the greater attraction on the moon? (b) Does the sun or the moon exert the greater attraction on the earth? (c) At what distance from the earth would a particle be equally attracted by the earth and the moon? By the earth and the sun?

5. The horse power required to propel a boat varies approximately as the cube of its velocity. If a 58 H.P. engine will produce a velocity of

10 feet per second, what horse power will produce a velocity of 20 feet per second? Plot the graph of the relation and illustrate the above data on it.

6. The weight of a metal disk of given thickness and material varies as the square of the diameter. (a) If a disk whose diameter is 1 inch weighs ounce, what is the weight of a disk whose diameter is 4 inches? What is the diameter of a disk weighing 7 ounces? Sketch the graph showing the weight of any such disk. (b) If a given disk weighs 12 ounces, how would its diameter compare with one whose weight is 6 ounces?

7. The volume of a sphere varies as the cube of its diameter. How many shot of an inch in diameter can be made by melting a lead sphere whose diameter is 2 inches?

8. The intensity of light varies inversely as the square of the distance from the source of light. If an object is 20 feet from a light, by how much must it be moved to receive twice as much light? Illustrate graphically.

9. If an object is 3 feet from a light, how far should it be moved to receive one-third as much light?

10. The volume of a cylinder varies jointly as the altitude and the square of the diameter. A preserving kettle 12 inches in diameter is filled to a depth of 8 inches. How many quart jars, 3 inches in diameter and 6 inches high, will be needed to hold the preserves in the kettle?

11. In enlarging a photograph, the original negative is projected on a sensitized plate, just as a lantern slide is projected on a screen. The size (area) of the enlargement varies as the square of the distance from the source of light, and is equal to the original if the sensitized plate is placed right against the negative. If a 4 x 5 negative is placed ten inches from the source of light, where should the sensitized plate be placed to obtain an enlargement 8 x 10 inches?

44. Empirical Data Problems. Since the equation y = kxn may be written in the form y/xnk, the table of values of x and y is such that the quotients obtained by dividing the values of y by the nth powers of the corresponding values of x are equal. This property enables us to determine whether a given table of values, obtained by an experiment, may be represented approximately by y=kx", and to determine the coefficient k.

If the points whose coördinates are the pairs of values in the table are plotted, and appear to lie on the graph of ka", we will have n>1 if the graph is tangent to the x-axis at the origin, 0<n<1 if the graph is tangent to the y-axis at the origin, and n<0 if the graph approaches the axes asymptotically (Summary, Section 41).

If n>1, a simple value to choose for n is 2. By the above property, if the quotients obtained by dividing the values of y by the squares of the corresponding values of x differ by very little, the relation connecting x and y resembles y = kx2, and an approximate statement of the law may be given in this form. Each quotient, y/x2, is the value of k for which the graph of y = kx2 passes through the point (x, y). A very good value to use for k is obtained by finding the average of all the quotients. If, however, the quotients differ widely, it is well to try n = 3, or n , or some other value of n, in order to see if an approximate statement of the law can be found which is better than y kx2.

If 0<n<1, a simple value to choose for n is, but if the result is not satisfactory, other values of n should be tried.

If n<0, the values occurring most frequently are n − 1 and n 2. These cases may be distinguished, provided that the same unit is used on both axes, by the fact that the graph for n = − 1 is symmetrical with respect to the bisector of the first and third quadrants. If neither of these values of n proves satisfactory, others should be tried. If n is negative, division by xn is equivalent to multiplication by x-". For example, if n = 2, we have y/x-2= yx2. The value of k to be chosen is now the average of certain products instead of quotients.

The accuracy with which the equation finally obtained represents the law may be checked as on pages 80, 83, and 106. The use of the tables of squares, square roots, etc., is advantageous in finding k.

In a later chapter we shall obtain a more satisfactory method of determining n.

x 1, 2, 3, 4,

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EXERCISES

1. In the following table the relation between x and y is y = k/x. Construct the graph and determine k. Hint. Find the product of x and y for each pair of values and take the average of these products. Check

4, 2.1, 1.5, 0.9, the accuracy of the result.

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