352. THEOREM 1. If y∞ox, and x ∞ z, then y ∞ 2. 353. THEOREM 2. If yox, and zoox, then (y + z) ∞ x. y = mx and z = nx. For .. y ± z = (m ± n) x ; since mn is constant, y varies as x. 354. THEOREM 3. If yox when z is constant, and y ∞2 when x is constant, then y∞ xz when x and z are both variable. Let x', y', z', and x", y", z" be corresponding values of the variables. Let x change from x' to x", z remaining constant, and let the corresponding value of y be Y. or or Now let z change from ' to z", x remaining constant. (1) (2) (§ 342) (§ 339, V) .. the ratio y xz is constant, and y varies as xz. In like manner it may be shown that if y varies as each of any number of quantities x, z, u, etc., when the rest are unchanged, then when they all change, y oo xzu, etc. Thus, the volume of a rectangular solid varies as the length when the width and thickness remain constant; as the width when the length and thickness remain constant; as the thickness when the length and width remain constant; but as the product of length, breadth, and thickness when all three vary. 1. If y varies inversely as x, and when y = 2 the corresponding value of x is 36, find the corresponding value of x when y = 9. If 9 and 72 are substituted for y and m, respectively, the result is 9= 72 х or 9x 72. ..x=8. 2. The weight of a sphere of given material varies as its volume, and its volume varies as the cube of its diameter. If a sphere 4 inches in diameter weighs 20 pounds, find the weight of a sphere 5 inches in diameter. then, since 20 and 4 are corresponding values of W and D, (§ 352) 2. If W varies inversely as P, and W is 4 when P is 15, find W when P is 12. 3. If xy and y∞, show that xz ∞ y2. 5. If x varies inversely as y2 – 1, and is equal to 24 2 1 -, find the relation between x and z if 8. The area of a circle varies as the square of its radius, and the area of a circle whose radius is 1 foot is 3.1416 square feet. Find the area of a circle whose radius is 20 feet. 9. The volume of a sphere varies as the cube of its radius, and the volume of a sphere whose radius is 1 foot is 4.1888 cubic feet. Find the volume of a sphere whose radius is 2 feet. 10. If a sphere of given material 3 inches in diameter weighs 24 pounds, how much will a sphere of the same material weigh if its diameter is 5 inches? 11. The velocity of a falling body varies as the time during which it has fallen from rest. If the velocity of a falling body at the end of 2 seconds is 64 feet, what is its velocity at the end of 8 seconds? 12. The distance a body falls from rest varies as the square of the time it is falling. If a body falls through 144 feet in 3 seconds, how far will it fall in 5 seconds? The volume of a right circular cone varies jointly as its height and the square of the radius of its base. 13. Compare the volume of two cones, one of which is twice as high as the other, but with one half its diameter. If the volume of a cone 7 feet high with a base whose radius is 3 feet is 66 cubic feet: 14. Find the volume of a cone 9 feet high with a base whose radius is 3 feet. 15. Find the volume of a cone 7 feet high with a base whose radius is 4 feet. 16. Find the volume of a cone 9 feet high with a base whose radius is 4 feet. 17. The volume of a sphere varies as the cube of its radius. If the volume is 1793 cubic feet when the radius is 3 feet, find the volume when the radius is 1 foot 6 inches. 18. Find the radius of a sphere whose volume is the sum of the volumes of two spheres with radii 31⁄2 feet and 6 feet, respectively. 19. The distance of the offing at sea varies as the square root of the height of the eye above the sea level, and the distance is 3 miles when the height is 6 feet. Find the distance when the height is 24 feet. CHAPTER XXII. PROGRESSIONS. 355. A succession of numbers that proceed according to some fixed law is called a series; the successive numbers are called the terms of the series. A series that ends at some particular term is a finite series ; a series that continues without end is an infinite series. Arithmetical Progression. 356. A series is called an arithmetical series or an arithmetical progression when each term after the first is obtained by adding to the preceding term a constant difference. The general representative of such a series is in which a is the first term and d the common difference; the series is increasing or decreasing according as d is positive or negative. 357. The nth Term. Since each succeeding term of the series is obtained by adding d to the preceding term, the coefficient of d is always one less than the number of the term, so that the nth term is a + (n − 1) d. If the nth term is represented by l, we have l = a + (n − 1)d. |