Chapter 2.7, Problem 45E

Revenue, cost, and profit. The price–demand equation and the cost function for the production of table saws are given, respectively, by

  $\begin{array}{lll}x=6,000-30p\hfill & \text{and}\hfill & C(x)=72,000+60x\hfill \end{array}$

where x is the number of saws that can be sold at a price of $p per saw and C(x) is the total cost (in dollars) of producing x saws.

1. (A)   Express the price p as a function of the demand x, and find the domain of this function.
2. (B)   Find the marginal cost.
3. (C)   Find the revenue function and state its domain.
4. (D)   Find the marginal revenue.
5. (E)    Find R′(1,500) and R′(4,500) and interpret these quantities.
6. (F)    Graph the cost function and the revenue function on the same coordinate system for 0 ≤ x ≤ 6,000. Find the break-even points, and indicate regions of loss and profit.
7. (G)   Find the profit function in terms of x.
8. (H)   Find the marginal profit.
9. (I)      Find P′(1,500) and P′(3,000) and interpret these quantities.