1. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall

2. Chapter 7 Rational Expressions

3. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 7.7 Variation and Problem Solving

4. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 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 number k is called the constant of variation or the constant of proportionality.

5. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example If y varies directly as x, find the constant of variation k and the direct variation equation, given that y = 5 when x = 30. Solution y = kx 5 = k·30 k = 1/6 The constant of variation is 1/6. The direct variation equation is

6. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Example If y varies directly as x, and y = 48 when x = 6, then find y when x = 15. Solution y = kx 48 = k·6 8 = k So the equation is y = 8x. y = 8·15 y = 120

7. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 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. Solution So our equation is

8. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 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. We substitute our given value for the elevation into the equation.

9. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 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 number k is still called the constant of variation or the constant of proportionality.

10. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 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. Solution k = 63·3 k = 189 The inverse variation equation is

11. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall Joint Variation If the ratio of a variable y to the product of two or more variables is constant, then y varies jointly as, or is jointly proportional to the other variables. If y = kxz then the number k is the constant or variation or the constant or proportionality.

12. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall 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. Solution So our equation is

13. Copyright © 2011 Pearson Education, Inc. Publishing as Prentice Hall We substitute our given value for the height of the column into the equation.