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**Inverse, Joint, and Combined Variation**

Objective: To find the constant of variation for many types of problems and to solve real world problems.

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Inverse Variation Two variables, x and y, have an inverse-variation relationship if there is a nonzero number k such that xy = k, y = k/x. The constant of variation is k.

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Example 1

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Example 1

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Example 1

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Try This The variable y varies inversely as x, and y = 120 when x = Find the constant of variation and write an equation for the relationship. Then, find y when x is 1.5, 4.5, 8, 12.5, and 14.

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Try This The variable y varies inversely as x, and y = 120 when x = Find the constant of variation and write an equation for the relationship. Then, find y when x is 1.5, 4.5, 8, 12.5, and 14.

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Try This The variable y varies inversely as x, and y = 120 when x = Find the constant of variation and write an equation for the relationship. Then, find y when x is 1.5, 4.5, 8, 12.5, and 14.

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Joint Variation If y = kxz, then y varies jointly as x and z, and the constant of variation is k.

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Example 2

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Example 2

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Squared Variation If , where k is a nonzero constant, then y varies directly as the square of x. Many geometric relationships involve this type of variation, as show in the next example.

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Example 3

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Example 3

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Example 3

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Try This Write the formula for the area A, of a circle whose radius is r. Identify the type of variation and the constant of variation. Find the area of the circle when r is 1.5, 2.5, 3.5, 4.5.

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Try This Write the formula for the area A, of a circle whose radius is r. Identify the type of variation and the constant of variation. Find the area of the circle when r is 1.5, 2.5, 3.5, 4.5. The constant of variation is

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Try This Write the formula for the area A, of a circle whose radius is r. Identify the type of variation and the constant of variation. Find the area of the circle when r is 1.5, 2.5, 3.5, 4.5. The constant of variation is

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Combined Variation

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Example 4

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Example 4

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Example 4

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Homework Page 486 13-27 odd

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