1. What is true of the appearance of graphs that reflect a direct variation between two variables?

2. If two variables vary inversely, what will an equation representing their relationship look like?

3. Is there a limit to the number of variables that can jointly vary? Explain.

For the following exercises, write an equation describing the relationship of the given variables.

4. *y* varies directly as *x* and when *x *= 6, *y *= 12.

5. *y* varies directly as the square of *x* and when *x* = 4, *y* = 80

6. *y* varies directly as the square root of *x* and when *x *= 36, *y *= 24.

7. *y* varies directly as the cube of *x* and when *x *= 36, *y *= 24.

8. *y* varies directly as the cube root of *x* and when *x *= 27, *y *= 15.

9. *y* varies directly as the fourth power of *x* and when *x *= 1, *y *= 6.

10. *y* varies inversely as *x* and when *x *= 4, *y *= 2.

11. *y* varies inversely as the square of *x* and when *x *= 3, *y *= 2.

12. *y* varies inversely as the cube of *x* and when *x *= 2, *y *= 5.

13. *y* varies inversely as the fourth power of *x* and when *x *= 3, *y *= 1.

14. *y* varies inversely as the square root of *x* and when *x *= 25, *y *= 3.

15. *y* varies inversely as the cube root of *x* and when *x *= 64, *y *= 5.

16. *y* varies jointly with *x* and *z* and when *x *= 2 and *z *= 3, *y *= 36.

17. *y* varies jointly as *x*, *z*, and *w* and when *x *= 1, *z *= 2, *w *= 5, then *y *= 100.

18. *y* varies jointly as the square of *x* and the square of *z* and when *x *= 3 and *z *= 4, then *y* = 72.

19. *y* varies jointly as *x* and the square root of *z* and when *x *= 2 and *z *= 25, then *y *= 100.

20. *y* varies jointly as the square of *x* the cube of *z* and the square root of *w*. When *x *= 1, *z *= 2, and *w *= 36, then *y *= 48.

21. *y* varies jointly as *x *and *z* and inversely as *w*. When *x *= 3, *z *= 5, and *w *= 6, then *y *= 10.

22. *y* varies jointly as the square of *x* and the square root of *z* and inversely as the cube of *w*. When *x *= 3, *z *= 4, and *w *= 3, then *y *= 6.

23. *y* varies jointly as *x* and *z* and inversely as the square root of *w* and the square of *t*. When *x *= 3, *z *= 1, *w *= 25, and *t *= 2, then *y *= 6.

For the following exercises, use the given information to find the unknown value.

24. *y* varies directly as *x*. When *x *= 3, then *y *= 12. Find *y* when *x *= 20.

25. *y* varies directly as the square of *x*. When *x *= 2, then *y *= 16. Find *y* when *x *= 8.

26. *y* varies directly as the cube of *x*. When *x *= 3, then *y *= 5. Find *y* when *x *= 4.

27. *y* varies directly as the square root of *x*. When *x *= 16, then *y *= 4. Find *y* when *x *= 36.

28. *y* varies directly as the cube root of *x*. When *x *= 125, then *y *= 15. Find *y* when *x *= 1,000.

29. *y* varies inversely with *x*. When *x *= 3, then *y *= 2. Find *y* when *x *= 1.

30. *y* varies inversely with the square of *x*. When *x *= 4, then *y *= 3. Find *y* when *x *= 2.

31. *y* varies inversely with the cube of *x*. When *x *= 3, then *y *= 1. Find *y* when *x *= 1.

32. *y* varies inversely with the square root of *x*. When *x *= 64, then *y *= 12. Find *y* when *x *= 36.

33. *y* varies inversely with the cube root of *x*. When *x *= 27, then *y *= 5. Find *y* when *x *= 125.

34. *y* varies jointly as *x* and *z*. When *x *= 4 and *z *= 2, then *y *= 16. Find *y* when *x *= 3 and *z *= 3.

35. *y* varies jointly as *x*, *z*, and *w*. When *x *= 2, *z *= 1, and *w *= 12, then *y *= 72. Find *y* when *x *= 1, *z *= 2, and *w *= 3.

36. *y* varies jointly as *x* and the square of *z*. When *x *= 2 and *z *= 4, then *y *= 144. Find *y* when *x *= 4 and *z *= 5.

37. *y* varies jointly as the square of *x* and the square root of *z*. When *x *= 2 and *z *= 9, then *y *= 24. Find *y* when *x *= 3 and *z *= 25.

38. *y* varies jointly as *x* and *z* and inversely as *w*. When *x *= 5, *z *= 2, and *w *= 20, then *y *= 4. Find *y* when *x *= 3 and *z *= 8, and *w *= 48.

39. *y* varies jointly as the square of *x* and the cube of *z* and inversely as the square root of *w*. When *x *= 2, *z *= 2, and *w *= 64, then *y *= 12. Find *y* when *x *= 1, *z *= 3, and *w *= 4.

40. *y* varies jointly as the square of *x* and of *z* and inversely as the square root of *w* and of *t*. When *x *= 2, *z *= 3, *w *= 16, and *t *= 3, then *y *= 1. Find *y* when *x *= 3, *z *= 2, *w *= 36, and *t *= 5.

For the following exercises, use a calculator to graph the equation implied by the given variation.

41. *y* varies directly with the square of *x* and when *x *= 2, *y *= 3.

42. *y* varies directly as the cube of *x* and when *x *= 2, *y *= 4.

43. *y* varies directly as the square root of *x* and when *x *= 36, *y *= 2.

44. *y* varies inversely with *x* and when *x *= 6, *y *= 2.

45. *y* varies inversely as the square of *x* and when *x *= 1, *y *= 4.

For the following exercises, use Kepler’s Law, which states that the square of the time, *T*, required for a planet to orbit the Sun varies directly with the cube of the mean distance, *a*, that the planet is from the Sun.

46. Using the Earth’s time of 1 year and mean distance of 93 million miles, find the equation relating *T *and *a*.

47. Use the result from the previous exercise to determine the time required for Mars to orbit the Sun if its mean distance is 142 million miles.

48. Using Earth’s distance of 150 million kilometers, find the equation relating *T* and *a*.

49. Use the result from the previous exercise to determine the time required for Venus to orbit the Sun if its mean distance is 108 million kilometers.

50. Using Earth’s distance of 1 astronomical unit (A.U.), determine the time for Saturn to orbit the Sun if its mean distance is 9.54 A.U.

For the following exercises, use the given information to answer the questions.

51. The distance *s* that an object falls varies directly with the square of the time, *t*, of the fall. If an object falls 16 feet in one second, how long for it to fall 144 feet?

52. The velocity *v* of a falling object varies directly to the time, *t*, of the fall. If after 2 seconds, the velocity of the object is 64 feet per second, what is the velocity after 5 seconds?

53. The rate of vibration of a string under constant tension varies inversely with the length of the string. If a string is 24 inches long and vibrates 128 times per second, what is the length of a string that vibrates 64 times per second?

54. The volume of a gas held at constant temperature varies indirectly as the pressure of the gas. If the volume of a gas is 1200 cubic centimeters when the pressure is 200 millimeters of mercury, what is the volume when the pressure is 300 millimeters of mercury?

55. The weight of an object above the surface of the Earth varies inversely with the square of the distance from the center of the Earth. If a body weighs 50 pounds when it is 3960 miles from Earth’s center, what would it weigh it were 3970 miles from Earth’s center?

56. The intensity of light measured in foot-candles varies inversely with the square of the distance from the light source. Suppose the intensity of a light bulb is 0.08 foot-candles at a distance of 3 meters. Find the intensity level at 8 meters.

57. The current in a circuit varies inversely with its resistance measured in ohms. When the current in a circuit is 40 amperes, the resistance is 10 ohms. Find the current if the resistance is 12 ohms.

58. The force exerted by the wind on a plane surface varies jointly with the square of the velocity of the wind and with the area of the plane surface. If the area of the surface is 40 square feet surface and the wind velocity is 20 miles per hour, the resulting force is 15 pounds. Find the force on a surface of 65 square feet with a velocity of 30 miles per hour.

59. The horsepower (hp) that a shaft can safely transmit varies jointly with its speed (in revolutions per minute (rpm) and the cube of the diameter. If the shaft of a certain material 3 inches in diameter can transmit 45 hp at 100 rpm, what must the diameter be in order to transmit 60 hp at 150 rpm?

60. The kinetic energy *K* of a moving object varies jointly with its mass *m *and the square of its velocity *v*. If an object weighing 40 kilograms with a velocity of 15 meters per second has a kinetic energy of 1000 joules, find the kinetic energy if the velocity is increased to 20 meters per second.