328 CHAPTER 6 Algebra, Graphs, and Functions 18. The time it takes an ice cube to melt in water and the temperature of the water Inverse 19. C varies directly as r. Write the variation equation given k 2 .π = C r 2π = 20. I varies inversely as R. Write the variation equation given k 120. = I R 120 = 21. A varies jointly as b and h. Write the variation equation given k . 1 2 = A bh 1 2 = 22. R varies directly as L and inversely as A. Write the variation equation given k 0.05. = R L A 0.05 = In Exercises 23–30, (a) write the variation equation and (b) determine the quantity indicated. 23. y varies directly as x. Determine y when x 12 = and k 9. = a) y kx = b) 108 24. x varies inversely as y. Determine x when y 15 = and k 45. = a) x k y = b) 3 25. m varies inversely as the square of n. Determine m when n 10 = and k 20. = a) m k n2 = b) 0.20 26. r varies directly as the square of s. Determine r when s 2 = and k 13. = a) r ks2 = b) 52 27. A varies directly as B and inversely as C. Determine A when B C 5, 10, = = and k 5. = a) A kB C = b) 2.5 28. M varies directly as J and inversely as C. Determine M when J C 12, 16, = = and k 4. = a) M kJ C = b) 3 29. F varies jointly as D and E. Determine F when D E 3, 10, = = and k 7. = a) F kDE = b) 210 30. A varies jointly as R1 and R2 and inversely as the square of L. Determine A when R R L 120, 8, 5, 1 2 = = = and k . 3 2 = a) A kR R L 1 2 2 = b) 57.6 In Exercises 31– 40, (a) write the variation equation, and (b) solve for k, and then determine the quantity indicated. 31. H varies directly as L. If H 12 = when L 40, = determine H when L 10. = a) H kL = b) 3 32. C varies inversely as J. If C 7 = when J 0.7, = determine C when J 14. = a) C k J = b) 0.35 33. q varies inversely as the square of w. If q 9 = when w 5, = determine q when w 3. = a) q k w2 = b) 25 34. A varies directly as the square of B. If A 245 = when B 7, = determine A when B 12. = a) A kB2 = b) 720 35. J varies directly as the square root of b. If J 2 = when b 36, = determine J when b 81. = a) J k b = b) 3 36. C varies inversely as the square root of m. If C 8 = when m 9, = determine C when m 100. = a) C k m = b) 2.4 37. Z varies jointly as W and Y. If Z 12 = when W 9 = and Y 4, = determine Z when W 50 = and Y 6. = a) Z kWY = b) 100 38. S varies jointly as I and the square of T. If S 4 = when I 10 = and T 4, = determine S when I 4 = and T 6. = a) S kIT2 = b) 3.6 39. F varies jointly as M1 and M2 and inversely as the square of d. If F 20 = when M M 5, 10, 1 2 = = and d 0.2, = determine F when M M 10, 20, 1 2 = = and d 0.4. = a) F kM M d 1 2 2 = b) 20 40. F varies jointly as q1 and q2 and inversely as the square of d. If F 80 = when q q 4, 16, 1 2 = = and d 0.4, = determine F when q q 12, 20, 1 2 = = and d 0.2. = a) F kq q d 1 2 2 = b) 1200 Problem Solving In Exercises 41–54, (a) write the variation equation and (b) determine the quantity indicated. 41. Profit The profit, p, from selling bicycles is directly proportional to n, the number of bicycles sold. If the profit from selling 8 bicycles is $450, determine the profit from selling 18 bicycles. a) p kn = b) $1012.50 42. Property Tax The property tax, t, on a home is directly proportional to the assessed value, v, of the home. If the property tax on a home with an assessed value of $200,000 is $2400, what is the property tax on a home with an assessed value of $225,000? a) t kv = b) $2700 43. Melting an Ice Cube The time, t, for an ice cube to melt is inversely proportional to the temperature, T, of the water in which the ice cube is placed. If it takes an ice cube 2 minutes to melt in 75°F water, how long will it take an ice cube of the same size to melt in 80°F water? a) t k T = b) 1.875 minutes 44. Pressure and Volume The volume of a gas, V, varies inversely as its pressure, P. If the volume of a gas is 6400 cubic centimeters (cc) when the pressure is 25 millimeters (mm) of mercury, determine the volume when the pressure is 200 mm of mercury. a) V k P = b) 800 cc $ $ Belchonock/123RF
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