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**Modeling Using Variation**

Section 3.7 Modeling Using Variation Objectives: Solve direct variation problems. Solve inverse variation problems. Solve combined variation problems.

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**Example of Direct Variation**

Because light travels faster than sound, during a thunderstorm we see lightning before we hear thunder. The formula d = 1080t describes the distance, in feet, of the storm’s center if it takes t seconds to hear thunder after seeing lightning. Thus if t = 1, d = 1080; if t = 2, d = 2160; and if t = 3, d = This distance is said to vary directly.

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Direct Variation If a situation is described by an equation in the form y = kx where k is a constant, we say that y varies directly as x. The number k is called the constant of variation. As x increases, y increases. As x decreases, y decreases.

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**Writing a Direct Variation Equation**

If v varies directly as r, then: Solution: v = kr

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**Solving Variation Problems**

Write an equation that describes the given English statement. Substitute the given pair of values into the equation in step 1 and find the value of k. Substitute the value of k into the equation in step 1. Use the equation from step 3 to answer the problem's question.

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Text Example The amount of garbage, G, varies directly as the population, P. Allegheny County, Pennsylvania, has a population of 1.3 million and creates 26 million pounds of garbage each week. Find the weekly garbage produced by New York City with a population of 7.3 million. Solution Step Write an equation We know that y varies directly as x is expressed as y = kx. By changing letters, we can write an equation that describes the following English statement: Garbage production, G, varies directly as the population, P. G = kP Step 2 Use the given values to find k. Allegheny County has a population of 1.3 million and creates 26 million pounds of garbage weekly. Substitute 26 for G and 1.3 for P in the direct variation equation. Then solve for k. 26 = 1.3k 26/1.3 = k 20 = k

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**Text Example cont. Step 3 Substitute the value of k into the equation.**

G = kP Use the equation from step 1. G = 20P Replace k, the constant of variation, with 20. Step 4 Answer the problem's question New York City has a population of 7.3 million. To find its weekly garbage production, substitute 7.3 for P in G = 20P and solve for G. G = 20P Use the equation from step 3. G = 20(7.3) Substitute 7.3 for P. G = 146 The weekly garbage produced by New York City weighs approximately 146 million pounds.

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Practice #1 The pressure, P, of water on an object below the surface varies directly as its distance, D below the surface. If a submarine experiences a pressure of 25 pounds per square inch 60 feet below the surface, how much pressure will it experience 330 feet below the surface.

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**Direct Variation with Powers**

y varies directly as the nth power of x if there exists some nonzero constant k such that y = kx n. We also say that y is directly proportional to the nth power of x.

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Practice #2 The distance, s, that a body falls from rest varies directly as the square of the time, t, of the fall. If skydivers fall 64 feet in 2 seconds, how far will they fall in 4.5 seconds?

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Inverse Variation If a situation is described by an equation in the form where k is a constant, we say that y varies inversely as x. The number k is called the constant of variation. As x increases, y decreases. As x decreases, y increases.

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**Combined Variation Example**

Describe in words the variation shown by the given equation: Solution: H varies directly as T and inversely as Q

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