2; find s when t = 42. (a) s varies as t2, s = 64 when t = 2 and y = 3; find z (c) v varies directly as s and inversely as t, v = 4 when s = t = 3; find v when s = 7 and t = 5. 23 and (d) s varies jointly as h and r2 and inversely as t, s = 1 when r = 2, = = h 3, and t = 2. 3. The volume of a cylinder varies directly as its height when the radius is constant. Express this statement in the form of an equation. 4. The volume of a cylinder varies directly as the square of its radius when its height is constant. Express this by means of an equation. 36; find t when s = 2, r = s, and h 5. The interest due on P dollars let out at simple interest varies jointly as the rate and the time. Express this by means of an equation. 6. The number of feet a body falls varies directly as the square of the number of seconds occupied in falling. If the body falls 16 feet the first second, how many feet will it fall in 5 seconds? in 10 seconds? Ans. 400 ft.; 1600 ft. 7. The weight of an object above the surface of the earth varies inversely as the square of its distance from the center of the earth. If an object weighs 50 lbs. at the sea level, what would be its weight on top of a mountain a mile high? Assume that the radius of the earth is 4000 miles. Ans. 49.9875 lbs. 8. When an object is taken below the surface of the earth its weight varies directly as its distance from the center of the earth. If an object weighs 200 lbs. at the surface, how much would it weigh midway between the center and the surface? How much would it weigh at the center? Ans. 100 lbs.; 0 lbs. 9. The time of vibration of a simple pendulum varies directly as the square root of its length. If the time of vibration of a pendulum 39.14 inches long is one second, how long must a pendulum be in order that its time of vibration shall be two seconds? Ans. 13 ft. 0.56 in. 10. The amount of light received on a page from a given source varies directly as the size of the page and inversely as the square of its distance from the source of light. One page is twice as large as another one and twice as far from the source of light. Which page receives the more light? 11. The pressure of wind on a sail varies jointly as the area of the sail and the square of the wind's velocity. When the wind is 15 miles per hour, the pressure on a square foot is one pound. What is the speed of the wind when the pressure on a square yard is 25 pounds? 12. A paper disk is placed midway between two sources of light which are 12 ft. apart. If the amount of light falling on the disk varies inversely as the square of the distance from the source of light, show that if the disk is moved parallel to itself a distance of 2√√3 ft., the whole amount of light falling on the disk is trebled. 13. How far must the disk in Ex. 12. be moved from a point midway between the two lights in order that the total amount of light on the disk shall be doubled? 14. Kepler's third law states that the square of the number of years it takes a planet to revolve about the sun varies directly as the cube of the distance of the planet from the sun. Let the distance from the earth to the sun be 1. How long would it take a planet whose distance from the sun is 100 to complete one revolution? Ans. 1,000 yrs. 15. The safe load of a horizontal beam supported at both ends varies jointly as the breadth and square of the depth, and inversely as the length between supports. If a 2 x 6 inches white pine joist, 10 feet long between supports, safely holds up 800 lbs., what is the safe load of a 4 X 8 beam of the same material 15 feet long? Ans. 1,896 lbs. 16. A beam 15 feet long, 3 inches wide, and 6 inches deep when supported at each end can bear safely a maximum load of 1,800 lbs. What is the greatest weight that can safely be placed on a beam of the same material 18 feet long, 4 inches wide, and 4 inches deep? Ans. 889 lbs. 17. The bend b of a rod supported at both ends varies directly as the weight m hung at its middle-point, directly as the cube of the length l of the rod between supports, inversely as the width w of the rod, and inversely as the depth d. A plank 10 feet long, 10 inches wide, and 2 inches thick is supported at both ends. A weight of 180 lbs. hung at the middle makes it bend 3 inches. How much will the plank bend if placed on edge? Ans. 0.12 in. 18. The area of a triangle varies jointly as the length of the base b and the altitude a. = 4 inches. Write the law if when a = 6 inches, b 19. Similar figures vary in area as the squares of their like dimensions. A new grindstone is 48 inches in diameter. How large is it in diameter when one-fourth of it is ground away? 20. A circular silo has a diameter of a feet. What must be the diameter of a circular silo of the same height to hold 4 times as much? 21. What is the effect on the area of a regular hexagon if the length of each side of the hexagon is doubled. 22. Similar solids vary in volume as the cubes of their like dimensions. A water pail that is 10 inches across the top holds 12 quarts. Find the volume of a similar pail that is 12 inches across the top. 23. Using the reetangular pack, 432 apples 2 inches in diameter can be put in a box 12 X 12 X 24. How many 3 inch apples can be packed in the same box? How many 4 inch apples? Ans. 128; 54. 24. If a lever with a weight at each end is balanced on a fulcrum, the distances of the two weights from the fulcrum are inversely proportional to the weights. If 2 men of weights 160 lbs. and 190 lbs. respectively are balanced on the ends of a 10 foot stick, what is the length from the fulcrum to each end? Ans. 45 ft.; 54 ft. Ans. 1,406 lbs. 25. A wire rope 1 inch in diameter will lift 10,000 lbs. What will one inches in diameter lift? 26. Two persons of the same build are similar in shape; their weights should vary as the cube of their heights. A man 5 ft. tall weighs 150 lbs. Find the weight of a man of the same build and 6 feet tall. Ans. 194.74 lbs. 27. A man 5 ft. 5 in. tall weighs 140 lbs., and one 6 ft. 2 in. tall weighs 216 lbs. Which is of the stouter build? 28. The size of a stone carried by a swiftly flowing stream varies as the 6th power of the speed of the water. If the speed of a stream is doubled, what effect does it have on its carrying power? What effect if trebled? CHAPTER XI EMPIRICAL EQUATIONS 173. Empirical Formulas. In practice, the relations between quantities are usually not known in advance, but are to be found, if possible, from pairs of numerical values of the quantities discovered from experiment. In order to determine the relation between these quantities it is useful to first plot the corresponding pairs of values upon cross-section paper, and draw a smooth curve through the plotted points. If the curve so drawn resembles closely one of the following types of curves: we assume that the relation connecting the quantities is the corresponding equation of the above set and it remains to determine the constants of the equation. If the plotted data does not fit any of the type curves mentioned above, a general method of procedure is to assume an equation of the type to at least small errors, it is not to be expected that the caĺculated values of the coefficients in the assumed equations will be absolutely accurate, nor that the points that represent the pairs of values of x and y will all lie absolutely on the curve represented by the final formula. 174. Coefficients Determined by the Method of Least Squares. In case the plotted points appear to be upon a straight line, a parabola, or a curve of the nth degree, the corresponding equation is assumed and we proceed to determine the coefficients by a method which is illustrated in the following example. EXAMPLE 1. Let the observed values of x and y be Plotting this data, the points will be seen to lie roughly on a straight line. Hence we assume a relation of the form Upon substituting the four pairs of values given in the table there |