Chapter 4.5, Problem 44ES

A firm determines that $x$ units of its product can be sold daily at $p$ dollars per unit, where $x=1000-p$ The cost of producing $x$ units per day is $C\left(x\right)=3000+20x$

(a) Find the revenue function $R\left(x\right)$ .

(b) Find the profit function $P\left(x\right)$ .

(c) Assuming that the production capacity is at most $500$ units per day, determine how many units the company must produce and sell each day to maximize the profit.

(d) Find the maximum profit.

(e) What price per unit must be charged to obtain the maximum profit?