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Applications with Rational Expressions; Variation Section 7.5 MATH 116-460 Mr. Keltner

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Teamwork! Solving problems involving two or more people working together can be made easier by thinking of the individual efforts that are put together. The equations below help illustrate this: Person’s Rate of Work Person’s Time at Work = Amount of the task completed by that person Amount completed by 1 person Amount completed by other person + = Whole task to be completed

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Example 1 If one garden hose can fill a kiddie pool in 8 minutes by itself and another garden hose can fill the same pool in 12 minutes by itself, how long will it take to fill the kiddie pool with both hoses working?

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Go, Bennett, go!! If Bennett eats one cup (approx. 8 ounces) of dog food every day, how many days will it take him to eat a 20-pound bag of dog food? Since Bennett is given the same amount of food each day, there is no change in the rate he eats. We can say that the amount of food he eats varies directly as the number of days, because as the number of days increases, so does the amount he has eaten.

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Direct Variation Bennett’s diet is an example of direct variation. We say that some variable y varies directly as x if there is some constant k such that y = kx The equation y = kx is called a direct- variation equation. The number k is called the constant of variation.

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Example 2: Bennett’s Diet Find the constant of variation for Bennett’s diet and write an equation for the amount of food he eats as a function of the number of days he has “dined.”

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Example 3 Suppose m varies directly as the square of n. If m = 24 when n = 2, find m when n = -3.

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Inverse Variation Two variables, x and y, have inverse variation if they are related as shown below: The constant k is called the constant of variation. In this situation, we would say that “y varies inversely with x.”

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More Inverse Variation Note that, as one value increases, the other value decreases. The origin is NOT a solution. x = 0 causes division by zero, which is undefined. The shape of the graph is shown here:

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iPod Example The number of songs that can be stored on an iPod varies inversely with the average size of a song. A certain MP3 player can store 2500 songs when the average size of a song is 4 megabytes.

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iPod Example Write a model that gives the number of songs n that will fit on the iPod as a function of the average song size s (in megabytes).

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MP3 Player Example Make a table showing the number of songs that will fit on the iPod if the average size of a song is 2 MB, 2.5 MB, and 5 MB. What happens to the number of songs as the average song size increases?

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Example 5 The variables x and y vary inversely, and y = 15 when x = 1 / 3. Write an equation that relates x and y. Then find y when x = -10.

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Checking for variation With inverse variation, note that k = xy, when we solve for k. Since k is constant, the product xy will be the same for any ordered pair (x, y). If the product is not constant, check to see if direct variation exists. With direct variation, note that k = y / x, when we solve for k. Since k is constant, the quotient y / x will be the same for any ordered pair (x, y). If the quotient is not constant, then there is neither inverse nor direct variation.

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Example 6 XY 1.640 232 2.525.6 416 88 XY 10130 12156 14182 16208 18234 XY 619 925 1229 1537 1843 Determine whether x and y show direct variation, inverse variation, or neither.

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Combined Variation Combined variation occurs when a quantity varies directly with the product of two or more other quantities. Example: When we say that “y varies jointly as x and w,” we would write the equation y = kxw. The graph is a three-dimensional figure that is difficult to summarize in one graph.

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Example 7 The variable z varies jointly with x and y. Also, z = 60 when x = -4 and y = 5. Find z when x = 7 and y = 2.

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Types of Variation RelationshipEquation Y varies inversely with x.y = k/x Z varies directly with x, y, and r.z = kxyr Y varies inversely with the square of x y = k / x 2 Z varies directly with y and inversely with x.z = ky/x X varies directly with t and r and inversely with s. x = ktr / s

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Example 8 Tell whether x and y show direct variation, inverse variation, or neither. y - 3x = 0 x = 3 / y x + y = 5

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Assessment Pgs. 511-515: #’s 7-56, multiples of 7; #’s 59, 60, 61

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EXAMPLE 1 Classify direct and inverse variation

EXAMPLE 1 Classify direct and inverse variation

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