2quiz Fill in the blank below: A number y varies directly with x if there exists a nonzero number k such that y=____
3Solving application problems Read the problemGather information, make a sketch, write what the variable will represent.Write an equationSolve the equationLook back and check your solutionVerify that the answer makes sense, fullfills the requirements.
4Example 1The 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.
5Example 2In 1960 only 7% of physicians were female, but the number had risen to 20% in 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.
6Break-Even problemFor 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 soldCost (C(x)) = fixed cost + item cost x # of items soldBreak-even point: R(x) = C(x)
7ExampleA 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?]
8Direct VariationWhen a situation gives rise to a linear function f(x) = kx, or y = kx, we say that we have direct variationThe 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.
9Example 1If A varies directly as B, and B = 12 when A = 4, find a linear model that relates A and B.
10Example 2If 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.
11Example 3The 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.