§ 8.4

1. Variation and Problem Solving § 8.4

2. Direct Variation y varies directly as x, or y is directly proportional to x, if there is a nonzero constant k such that y = kx. The family of equations of the form y = kx are referred to as direct variation equations. The number k is called the constant of variation or the constant of proportionality.

3. Direct Variation So the direct variation equation is 1 6 y = x. If y varies directly as x, find the constant of variation k and the direct variation equation, given that y = 5 when x = 30. y = kx 5 = k·30 k = 1/6

4. Direct Variation Example: If y varies directly as x, and y = 48 when x = 6, then find y when x = 15. y = kx 48 = k·6 8 = k So the equation is y = 8x. y = 8·15 y = 120

5. Direct Variation Example: At sea, the distance to the horizon is directly proportional to the square root of the elevation of the observer.If a person who is 36 feet above water can see 7.4 miles, find how far a person 64 feet above the water can see. Round your answer to two decimal places. Continued.

6. Direct Variation Example continued: We substitute our given value for the elevation into the equation. So our equation is

7. Inverse Variation y varies inversely as x, or y is inversely proportional to x, if there is a nonzero constant k such that y = k/x. The family of equations of the form y = k/x are referred to as inverse variation equations. The number k is still called the constant ofvariation or the constant of proportionality.

8. Inverse Variation 189 x So the inverse variation equation is y = Example: If y varies inversely as x, find the constant of variation k and the inverse variation equation, given that y = 63 when x = 3. y = k/x 63 = k/3 63·3 = k 189 = k

9. Powers of x y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that y can vary directly or inversely as powers of x, as well. y varies directly as a power of x if there is a nonzero constant k and a natural number n such that y = kxn.

10. Powers of x Example: The maximum weight that a circular column can hold is inversely proportional to the square of its height. If an 8-foot column can hold 2 tons, find how much weight a 10-foot column can hold. Continued.

11. Powers of x Example continued: We substitute our given value for the height of the column into the equation. So our equation is

12. Variation and Problem Solving Example: Kathy spends 1.5 hours watching television and 8 hours studying each week. If the amount of time spent watching TV varies inversely with the amount of time spent studying, find the amount of time Kathy will spend watching TV if she studies 14 hours a week. 1.) Understand Read and reread the problem. 2.) Translate We are told that the amount of time watching TV varies inversely with the amount of time spent studying. Let T = the number of hours spent watching television. Let s = the number of hours spent studying. Continued

13. Variation and Problem Solving Example continued: 3.) Solve To find k, substitute T = 1.5 and s = 8. We now write the variation equation with k replaced by 12. Replace s by 14 and find the value of T. Continued

14. Variation and Problem Solving Example continued: 3.) Interpret Kathy will spend approximately 0.86 hours (or 52 minutes) watching TV.