1. If x varies as y2, and x = 2 when y=4, find x when y = 16. 2. If x varies inversely as y3, and x=2 when y = 4, find Then, when y = 2, m 2 48' m = 128. 128 x= x= 16. Result. 3. If s is the sum of two quantities, one of which varies directly as 2 and the other inversely as x, and if s = 6 when x=2, and s=2 when x 2, find s when ≈ — — - 1. = 4. The volume of a sphere varies as the cube of its diameter. If three metal spheres whose diameters are 6, 8, and 10 inches, respectively, are melted and recast into a single sphere, what is its diameter? Let V denote the volume of the required sphere, and D its diameter. Then Whence, V ∞ D3. V = mD3. Denote the volumes of the three given spheres by V1, V2, and Vз. (1) Hence, mD3=1728 m, D3 = 1728, D12. Result. That is, the diameter of the sphere obtained from the given spheres is 12 inches. Exercise 123 1. If x varies as y, and x = 10 when y = 2, find x when y = 5. 2. If x varies as y, and x=3.2 when y = 0.8, find x when y= 5.6. 3. If x + 1 varies as y -1, and x = 6 when y = 4, find x when y = 7. 4. If 2x 3 varies as 3y+2, and y = 2 when x= y when x = 1.5. 0.2, find 5. If 2 varies as y2, and x=3 when y = 2, find y when x=4. 6. If x varies inversely as y, and x=2 when y = 4, find x when y = 3. 7. If x varies inversely as y2, and x= when x = 11⁄21⁄2. = 2 when y = 1}, find y 8. If x varies jointly as y and z, and x= z=2, find x when y = 5 and z = 4. -3 when y = = 4 and 9. If x varies inversely as y2—1, and x=4 when y =5, 11. If the square of a varies as the cube of y, and if x = 6 when y = 4, find the value of y when x = = 30. 12. If s is the sum of two quantities, one of which varies as a while the other varies inversely as x; and if s= 2 when x= and s=2 when x = – 1, find the equation between s and x. 13. If w varies as the sum of x, y, and z, and w = 4 when x = 2, y = -2, and z = 5, find x if w= z=-6. — 3, y= 2, and 14. Given that s the sum of three quantities that vary as x, x2, and 3, respectively. If x = 1, s = 3; if x = 2, s = 6; and if x = 4, 8= 16. Express the value of s in terms of x. 15. The area of a circle varies as the square of its diameter. Find the diameter of a circle whose area shall be equivalent to the sum of the areas of two circles whose diameters are 6 and 8 inches respectively. 16. The intensity of light varies inversely as the square of the distance from the source. How far from a lamp is a certain much light as a point 25 feet point that receives just half as distant from the lamp? 17. The volume of a sphere varies as the cube of its diameter. If three spheres whose diameters are 3, 4, and 5 inches, respectively, are melted and recast into a single sphere, what is the diameter of the new sphere? 18. The volume of a rectangular solid varies jointly as the length, width, and height. If a cube of steel 8 inches on an edge is rolled into a bar whose width is 6 inches and depth 2 inches, what will be the length of the bar in feet? 19. If the amount earned while erecting a certain wall varies jointly as the number of men engaged and the number of days they work, how many days will it take 4 men to earn $100 when 6 men working 9 days earn $135? 20. The pressure of the wind on a plane surface varies jointly as the area of the surface and the square of the wind's velocity. The pressure on a square foot is one pound when the wind is blowing at a rate of 15 miles an hour. What will be the velocity of a wind whose pressure on a square yard is 81 pounds? SOM. EL. ALG.- - 22 CHAPTER XXV PROGRESSION ARITHMETICAL PROGRESSION 398. A series is a succession of terms formed in accordance with a fixed law. 399. An arithmetical progression is a series in which each term, after the first, is greater or less than the preceding term by a constant quantity. This constant quantity is the common difference. 400. We may regard each term of an arithmetical progression as being obtained by the addition of the common difference to the preceding term; hence, An increasing arithmetical progression results from a positive common difference, and a decreasing arithmetical progression results from a negative common difference. 1, 5, 9, 13, .. Thus: etc., is an increasing series in which the common difference is 4. 7, 4, 1, −2, . . . etc., is a decreasing series in which the common difference is -3. 401. In general, if a is the first term, and d the common difference, |