245 2.4 Linear Functions EXAMPLE 10 Writing Linear Cost, Revenue, and Profit Functions Assume that the cost to produce an item is a linear function and all items produced are sold. The fixed cost is $1500, the variable cost per item is $100, and the item sells for $125. Write linear functions to model each of the following. (a) cost (b) revenue (c) profit (d) How many items must be sold for the company to make a profit? SOLUTION (a) The cost function is linear, so it will have the following form. C1x2 = mx + b Cost function C1x2 = 100x + 1500 Let m= 100 and b = 1500. (b) The revenue function is defined by the product of 125 and x. R1x2 = px Revenue function R1x2 = 125x Let p = 125. (c) The profit function is found by subtracting the cost function from the revenue function. P1x2 = R1x2 - C1x2 P1x2 = 125x - 1100x + 15002 P1x2 = 125x - 100x - 1500 Distributive property P1x2 = 25x - 1500 Combine like terms. Use parentheses here. Linear Models In Example 8, we used the graph of a line to approximate real data, a process known as mathematical modeling. Points on the straight-line graph model, or approximate, the actual points that correspond to the data. A linear cost function has the form C1x2 =mx +b, Linear cost function where x represents the number of items produced, m represents the cost per item, and b represents the fixed cost. The fixed cost is constant for a particular product and does not change as more items are made. The value of mx, which increases as more items are produced, covers labor, materials, packaging, shipping, and so on. The revenue function for selling a product depends on the price per item p and the number of items sold x. It is given by the following function. R1x2 =px Revenue function Profit is found by subtracting cost from revenue and is described by the profit function. P1x2 =R1x2 −C1x2 Profit function In applications we are often interested in values of x that will assure that profit is a positive number. In such cases we solve R1x2 - C1x2 70.
RkJQdWJsaXNoZXIy NjM5ODQ=