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10.1 – Model Inverse and Joint Variation
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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 10.1 – Model Inverse and Joint Variation
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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 2
10.1 – Model Inverse and Joint Variation
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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. 3
10.1 – Model Inverse and Joint Variation
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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 Write your questions and
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10.1 – Model Inverse and Joint Variation
Practice‐ Tell whether x and y show direct, inverse or neither variation. 2. 8 3. 6 4. 8
1.
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. ,
28 5. 5,
4 6. 3,
8 7. Determine if the x and y values in the tables show direct, inverse or neither variation. 8. 9. x Y x
Y
12 132 4
21
18 198 6
14
23 253 8
10.5
29 319 8.4 10
34 374 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 5
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10.1 – Model Inverse and Joint Variation
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. 6
10.1 – Model Inverse and Joint Variation
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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.