10.1 Model Inverse Joint Variation 1

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10.1 ? Model Inverse and Joint Variation 1

Lesson goal ? Write and use inverse and joint variation models; useful in real life to model music frequencies.

Recall that you have learned about direct variation where x and y relate in the equation

That is if x increases so does y.

Inverse variation ?

Two variable show inverse variation if they are related as follows:

,

0

a is the constant of variation, and y is said to vary inversely with x

That is if the value of x increases, the value of y decreases.

You can also solve for a and get the equation

Note here that you multiply x and y and get a constant number, it can be a fraction or a decimal but will always be the same for all points in the inverse variation example.

Examples

Tell if x and y show direct, inverse or neither variation in the following:

3

7

Examples Write and equation for the given relation: x varies inversely with y and x=2 when y=12

You try ? Write inverse eq as above when

a) x=1.5 y=6

b) Find y using the eq from part a when

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10.1 ? Model Inverse and Joint Variation 2

Determine if x and y in the following table shows direct, inverse of neither variation.

You Try ? Determine if x and y in the following table shows direct, inverse of neither variation.

Example ? z varies jointly with x and y. Given: z=8, x=4, and y=10

Write eq relating x, y and z Then find z when x=5 and y=7

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10.1 ? Model Inverse and Joint Variation 3

Step 1 - Write

Step 2 ? Use

Step 3 ? Rewrite

Step 4 ? Calculate

You Try ?

The variable z varies jointly with x and y. Also, z= - 75 when x= 3 and y= - 5. Write an eq that relates x, y, and z. Then find z when x= 2 and y= 6.

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10.1 ? Model Inverse and Joint Variation 4

Comparing different types of variation ?

Note: When the relationship is said to vary jointly, the variable goes in the numerator and when it varies inversely, it goes in the denominator. Also the constant a ALWAYS is in the numerator.

You try ? Write eq for given relationship:

- x varies inversely with y and directly with w

- p varies jointly with q and r and inversely with the square root of s

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10.1 ? Model Inverse and Joint Variation 5

Practice-

Tell whether x and y show direct, inverse or neither variation.

1.

2. 8 3.

6 4. 8

The variables x and y vary inversely. Use the given values to write an eq relating x and y. Then find y when x=3.

5. 5,

4 6.

3, 8 7.

, 28

Determine if the x and y values in the tables show direct, inverse or neither

variation.

8.

9.

x Y 12 132 18 198 23 253 29 319 34 374

xY 4 21 6 14 8 10.5 8.4 10 12 7

Write an eq relating x, y, and z given that z varies jointly with x and y. Then

find z when

.

10.

2,

6, 24

11.

8, 6, 12

12. 9,

2, 6

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10.1 ? Model Inverse and Joint Variation 6

Write and eq for the given relationship. 13. x varies directly with y and inversely with z.

14. y varies jointly with x and the square of z.

15. w varies inversely with y and jointly with x and z.

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10.1 ? Model Inverse and Joint Variation 7

Application problems

1.

Digital Cameras The number n of photos your digital camera can store varies inversely with the average size s (in megapixels) of the photos. Your digital camera can store 54 photos when the average photo size is 1.92 megapixels. Write a model that gives n as a function of s. How many photos can your camera store when the average photo size is 3.87 megapixels?

2.

Snowshoes When you stand on snow, the average pressure P (in pounds per square inch) that you exert on the snow varies inversely with the total area A (in square inches) of the soles of your footwear. Suppose the pressure is 0.43 pounds per square inch when you wear the snowshoes shown. Write an equation that gives P as a function of A. then find the pressure if you wear the boots shown.

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