section 9.6 inverse and joint variation and other applications yielding equations with fractions

Section 9.6 Inverse and Joint Variation and Other Applications

Yielding Equations with Fractions

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

Objective 1: Distinguish between direct and inverse variation.

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

Verbally

y varies directly as x.

As the magnitude of x increases, the magnitude of y _______________ linearly.

y varies inversely as x.

As the magnitude of x increases, the magnitude of y _______________.

Algebraically Algebraically

for

Example:

ky

x 0x

12y

x

Verbally

y = kx

Example: 2y x

0k

Numerical Example Numerical Example

Comparison of Direct and Inverse Variation

2

3 6

1 2

0 0

1 2

3 6

x y x

12

1 12

2 6

3 4

4 3

5 2.4

x yx

-10 -5 5 10

-10

-5

5

10

x

y

Graphical Example Graphical Example

Comparison of Direct and Inverse Variation

-5

5

-5 5

y

x

2y x

12y

x

Joint Variation

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

Verbally Algebraically

z varies jointly as x and y.

0k

z kxy

Determine whether the relationship illustrated is direct or inverse variation.

3p t 12.5a

b

1 120

2 60

3 40

4 30

5 24

6 20

x y

1 20

2 40

3 60

4 80

5 100

6 120

x y

1. 2.

3. 4.

Objective 2: Translate statements of variation.

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

5. P varies directly as T

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

6. P varies inversely as V

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

7. P varies directly as T and inversely as V

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

8. P varies jointly as T and V

Translate each equation into a verbal statement of variation.

9.k

mn

Translate each equation into a verbal statement of variation.

10.kn

mp

Translate each equation into a verbal statement of variation.

11.m knp

Objective 3: Solve problems involving variation.

Use the given statement of variation to solve each problem.

12. 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.

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

14. z varies directly as x and inversely as y and when and Find z when and

36z 12

x 8.y 5x 12.y

15. z varies directly as x and inversely as y and when and Find z when and

36z 12

x 8.y 2x 5.y

x y

2 24

4

6

8

10

16. y varies directly as x. Use this information to complete this table.

x y

2 24

4

6

8

10

17. y varies inversely as x. Use this information to complete this table.

(a) Determine the constant of variation.

(b) Write an equation relating the weight of an object on the moon to its weight on the earth.

18. The weight of an object on the moon varies directly as its weight on the earth. An astronaut weighs 150 lbs on

earth and 25 lbs on the moon.

18. The weight of an object on the moon varies directly as its weight on the earth. An astronaut weighs 150 lbs on

earth and 25 lbs on the moon.

, weight on the earth (lbs)

, weight on the moon (lbs)

120

130

140

150

160

x y

(c) Use this equation to complete the table of values.

19. 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.

(b) Write an equation relating the number of revolutions made by a wheel to its circumference for this given distance.

(a) Determine the constant of variation.

, circumference (cm)

, number of revolutions (rev)

10

15

20

25

30

c n

(c) Use this equation to complete the table of values.

Objective 4: Solve applied problems that yield equations with fractions.

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

Step 2. Select a ______________ to represent each unknown quantity. Specify precisely what each variable represents and note any restrictions on each variable.

Step 3. If necessary, make a ____________ and translate the problem into a word equation or a system of

word equations. Then translate each word equation into an ______________ equation.

Step 4. Solve the equation or system of equations, and answer the question completely in the form of a sentence.

Step 5. Check the _____________________ of your answer.

Strategy for Solving Word Problems

20. The sum of the reciprocals of two consecutive odd integers is . Find these integers.

815

(b) Write the word equation:

The _________________ of the _______________of

the two integers is

(a) Identify the variable:

Let n = the smaller odd integer

Let_________________ = the larger odd integer

8.

15

20. The sum of the reciprocals of two consecutive odd integers is . Find these integers.

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(c) Translate the word equation into an algebraic equation:

(d) Solve this equation:

20. The sum of the reciprocals of two consecutive odd integers is . Find these integers.

815

(e) Write a sentence that answers the question:

(d) Is this answer reasonable?

21. The ratio of two readings of a pressure gauge on a boiler is

The first reading was 10 units below normal and the

second was 10 units above normal. What is the normal reading?

9.

10

.

22. 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?

23. If one painter can paint a wall by himself in 8 hours and a second painter can paint the same wall by herself in 6

hours, how long will it take them to paint the wall when working together?

24. Two employees from the Roofing Company, can put new shingles on a house in 12 hours when they work together. It takes the inexperienced roofer 7 hours longer than the more experienced roofer to put new shingles on a house when working alone. How long would it take the more experienced roofer to put shingles on the house alone?