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1.6 A PPLICATIONS OF L INEAR FUNCTIONS

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QUIZ Fill in the blank below: A number y varies directly with x if there exists a nonzero number k such that y=____

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S OLVING APPLICATION PROBLEMS Read the problem Gather information, make a sketch, write what the variable will represent. Write an equation Solve the equation Look back and check your solution Verify that the answer makes sense, fullfills the requirements.

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E XAMPLE 1 The new generation of televisions has a 16:9 aspect ratio. This means that the length of the television’s rectangular screen is 16/9 times its width. If the perimeter of the screen is 136 inches, find the length and the width of the screen.

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E XAMPLE 2 In 1960 only 7% of physicians were female, but the number had risen to 20% in 1992. Write a linear function to describe the percentage of physicians that are female in terms of the year. Find the percentage of physicians that are female in 2010.

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B REAK -E VEN PROBLEM For what number (x) of items sold will the revenue collected equal the cost of producing those items? Revenue (R(x)) = selling price x # of items sold Cost (C(x)) = fixed cost + item cost x # of items sold Break-even point: R(x) = C(x)

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E XAMPLE A student is planning to produce and sell music DVDs for $5.50 each. A computer with a DVD burner costs $2500, and each blank DVD costs $1.50. 1. Find the revenue function R(x) and the cost function C(x). 2. Find the break-even point. 3. Show the solution graphically. [What portion of the graph shows the student is making a profit?]

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D IRECT V ARIATION When a situation gives rise to a linear function f(x) = kx, or y = kx, we say that we have direct variation The following three statements are equivalent. 1. y varies directly as x. 2. y is in direct proportion to x. 3. y = kx for some nonzero constant k. The number k is the constant of variation.

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E XAMPLE 1 If A varies directly as B, and B = 12 when A = 4, find a linear model that relates A and B.

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E XAMPLE 2 If w is proportional to t, and t = 18 when w = 10, (a) find a linear model that relates w and t. (b) use the model to find the value of w when t = 639.

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E XAMPLE 3 The cost of renting office space in a downtown office building varies directly with the size of the office. A 600-square-foot office rents for $2550 per month. Use this information to write a model that gives the rent in terms of the size of the office. Then use the model to find the rent on a 960 square-foot office.

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H OMEWORK PG. 70: 31, 34, 35, 38, 41, 44, 53, 54, 55, 58, 60 KEY: 31, 34, 41, 60 Reading: 2.1 Basic Functions

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