Neatly on looseleaf for complete credit Printing this page i

Neatly on loose-leaf for complete credit. Printing this page is not required. Profiling from Widgets the Acme Trading Company has acquired 1000 widgets. lets goal is to make as much money from selling them as possible. But the more the company charges for a widget, the fewer it will be able to sell. Acme\'s marketing director thinks that if Acme charges d dollars for each widget, the company will be able to sell 1000 - 5d widgets. the company will give the rest of the widgets to charity. Suppose Acme charges $70 for each widget. How many widgets will Acme sell? How much money will Acme make from sales of widgets? Write an equation that shows how much Acme will get in sales revenue as a function of d. What price per widget will give Acme the maximum possible revenue? How many widgets will the company sell at that price? What do you think about the marketing director\'s formula? You may complete your work for problem 3 in standard form or in vertex form, but for either method, you must show your work.

Solution

1. (a) If Acme charges $ 70 per widget, then they will be able to sell 1000-5*70 = 1000-350 = 650 widgets.

(b) Acme will make $ 650*70 = $ 45500 from the sale of widgets.

2. If Acme charges $ d per widget, they will be able to sell 1000 -5d widgets. Then their revenue is R = d(1000- 5d) = 1000d-5d².

3. We know that, for R to be maximum, d(R)/d(d) = 0 and d²(R )/d(d)² is negative. Here, d(R)/d(d) = 1000-10d and d²(R )/d(d)² = -10. Thus, if d(R)/d(d) = 0 , then 1000-10d = 0 so that d = 100. Further, d²(R )/d(d)² is always negative regardless of the value of d. Hence, the revenue will be maximum when the price of a widget is $ 100. Then the number of widgets sold is 1000-5*100 = 1000-500 = 500.

4. The marketing director has a good plan. Rather than charging a very high price ( more than $ 100) for widgets , which will reduce the revenue, it is better to charge a price which maximizes revenue and makes available 500 widgets for charity which will earn Acme a good reputation.