Business The next exercise is suitable for group work. A graphing calculator is needed. Us...

Business The next exercise is suitable for group work. A graphing calculator is needed. Use the table feature to make a table of values for the profit function in Example 1,with x = 0, 5, 10, …, 225. Use the table to answer the following question.

(a) What is the profit or loss (= negative profit) when 20,000 DVDs are sold? when 40,000 are sold? Explain these answers.

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(b) Approximately how many DVDs must be sold in order for the company to make a profit?

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(c) What is the profit from selling 100,000 DVDs? from 150,000 DVDs? from 200,000 DVDs?

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(d) Explain why the profit amounts shown in the table for 205,000 DVDs and beyond are meaningless.

Example 1

Hot Rocks Movies sells DVDs for $8 each (wholesale) and can produce a maximum of 200,000 DVDs. The variable cose of producing x thousand DVDs is 3550x − 8x2 dollars, and the fixed costs for the manufacturing operation are $215,000. If x thousand DVDs are manufactured and sold, find expressions for the revenue, cost, and profit.

Solution

If x thousand DVDs are sold at $8 each, then

where x ≤ 200 (because only 200,000 DVDs can be made). The variable cost of making x thousand DVDs is 3550x − 8x2, so that

Therefore, the profit is given by