Chapter 2, Problem 88RE

Break-even analysis. The research department in a company that manufactures AM/FM clock radios established the following price-demand, cost, and revenue functions:

$\begin{array}{c}p\left(x\right)=50-1.25x\text{Price-demand}\text{function}\\ C\left(x\right)=160+10x\text{Cost}\text{function}\\ R\left(x\right)=xp\left(x\right)\\ =x\left(50-1.25x\right)\text{Revenue}\text{function}\end{array}$

where $x$ is in thousands of units, and $C\left(x\right)$ and $R\left(x\right)$ are in thousands of dollars. All three functions have domain $1\le x\le 40.$

(A) Graph the cost function and the revenue function simuta-neously in the same coordinate system.

(B) Determine algebraically when $R=C$. Then, with the aid of part (A), determine when $R<C\text{ and }R>C$ to the nearest unit.

(C) Determine algebraically the maximum revenue (to the nearest thousand dollars) and the output (to the nearest unit) that produces the maximum revenue. What is the wholesale price of the radio (to the nearest dollar) at this output?