Section 8.6Inverse and Joint Variation and Other Applications

Yielding Equations with Fractions

8.6 Lecture Guide: Inverse and Joint Variation and Other Applications Yielding Equations with Fractions

Objective: Solve problems involving inverse variation.

Algebraically Verbally

y varies directly as x

Direct Variation

If x and y are real variables, k is a real constant, then:

y kx

Comparison of Direct and Inverse Variation

Numerical Example 2

3 6

1 2

0 0

1 2

3 6

x y x

Graphical Example

-5

5

-5 5

x

y

2y x

Example: 2y xAs the magnitude of x increases, the magnitude of y increases linearly.

Algebraically Verbally

y varies inversely as x.

Inverse Variation

If x and y are real variables, k is a real constant, then: Comparison of Direct and Inverse Variation

Numerical Example Graphical Example

Example: As the magnitude of x increases, the magnitude of y decreases.

ky

x for 0x

12y

x

12

1 12

2 6

3 4

4 3

5 2.4

x yx

-10

10

-10 10

x

y12

yx

Algebraically

Verbally

Joint Variation

If x, y, and z are real variables, k is a real constant, and

0k then:

z varies jointly as x and y.

z kxy

Write an equation for each statement of variation. Use k as the constant of variation.

1. P varies directly as T

Write an equation for each statement of variation. Use k as the constant of variation.

2. P varies inversely as V

Write an equation for each statement of variation. Use k as the constant of variation.

3. P varies directly as T and inversely as V

Write an equation for each statement of variation. Use k as the constant of variation.

4. P varies jointly as T and V

Translate each equation into a verbal statement of variation.

5. km

n

Translate each equation into a verbal statement of variation.

6.kn

mp

Translate each equation into a verbal statement of variation.

7. m knp

Use the given statement of variation to solve each problem.

8. y varies directly as x and y = 45 when x = 9. Find y when x = 3.

Use the given statement of variation to solve each problem.

9. y varies inversely as x and y = 5 when x = 9. Find y when x = 3.

10. The weight of an object on the moon varies directly as its weight on the earth. If an astronaut weighs 150 lbs on earth and 25 lbs on the moon, what would Ann weigh on the moon if she weighs 126 lbs on earth?

11. The number of revolutions made by a wheel rolling a given distance varies inversely as the wheel’s circumference. A wheel of circumference 20 cm makes 100 revolutions in going a certain distance. How many revolutions would be required by a wheel of circumference 25 cm in going the same distance?

Objective: Solve applied problems that yield equations with fractions.

Strategy for Solving Word Problems

Step 1. Read the problem carefully to determine what you are being asked to find.

Step 2. Select a variable to represent each unknown quantity. Specify precisely what each variable represents.

Step 3. If necessary, translate the problem into word equations. Then translate the word equations into algebraic equations.

Step 4. Solve the equation(s), and answer the question asked by the problem.

Step 5. Check the reasonableness of your answer.

12. The sum of the reciprocals of two consecutive odd

integers is 8

15. Find these integers.

13. Two boats having the same speed in still water depart simultaneously from a dock, traveling in opposite directions in a river that has a current of 6 miles per hour. After a period of time one boat is 54 miles downstream, and the other boat is 30 miles upstream. What is the speed of each boat in still water?

14. If Joe can paint a wall by himself in 8 hours and Kelly can paint the same wall by herself in 6 hours, how long will it take them to paint the wall when working together?

15. Chris and Craig, two employees from the Roofing Company, can put new shingles on a house in 12 hours when they work together. It takes Chris 7 hours longer than Craig to put new shingles on a house when working alone. How long would it take Craig to put shingles on a house alone?