1. § 8.4 Variation and Problem Solving

2. Martin-Gay, Beginning and Intermediate Algebra, 4ed 22 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. Martin-Gay, Beginning and Intermediate Algebra, 4ed 33 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 y = 1616 x.So the direct variation equation is Direct Variation

4. Martin-Gay, Beginning and Intermediate Algebra, 4ed 44 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 Direct Variation Example:

5. Martin-Gay, Beginning and Intermediate Algebra, 4ed 55 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. Direct Variation Continued. Example:

6. Martin-Gay, Beginning and Intermediate Algebra, 4ed 66 So our equation is We substitute our given value for the elevation into the equation. Direct Variation Example continued:

7. Martin-Gay, Beginning and Intermediate Algebra, 4ed 77 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 of variation or the constant of proportionality. Inverse Variation

8. Martin-Gay, Beginning and Intermediate Algebra, 4ed 88 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 y = 189 x So the inverse variation equation is Inverse Variation Example:

9. Martin-Gay, Beginning and Intermediate Algebra, 4ed 99 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 = kx n. y varies inversely as a power of x if there is a nonzero constant k and a natural number n such that Powers of x

10. Martin-Gay, Beginning and Intermediate Algebra, 4ed 10 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. Powers of x Example:

11. Martin-Gay, Beginning and Intermediate Algebra, 4ed 11 So our equation is We substitute our given value for the height of the column into the equation. Powers of x Example continued:

12. Martin-Gay, Beginning and Intermediate Algebra, 4ed 12 1.) Understand Read and reread the problem. Continued Variation and Problem Solving 2.) Translate 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. Let T = the number of hours spent watching television. Let s = the number of hours spent studying. We are told that the amount of time watching TV varies inversely with the amount of time spent studying.

13. Martin-Gay, Beginning and Intermediate Algebra, 4ed 13 Continued Variation and Problem Solving 3.) Solve Example continued: 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.

14. Martin-Gay, Beginning and Intermediate Algebra, 4ed 14 Variation and Problem Solving 3.) Interpret Example continued: Kathy will spend approximately 0.86 hours (or 52 minutes) watching TV.