Chapter 1.7, Problem 34E

To determine

To find: profit function $p\left(x\right)$ .

Answer to Problem 34E

  $p\left(x\right)=399.9x-0.2x^2-200$

Explanation of Solution

Given information:

  $\begin{array}{l}p\left(x\right)=\left(r-c\right)\left(x\right)\\ r\left(x\right)=400x-0.2x^2\\ c\left(x\right)=0.1x+200\end{array}$

Here, $x$ are items, $p\left(x\right)$ is profit function, $r\left(x\right)$ is revenue in dollars, and $c\left(x\right)$ is cost of making and selling items

Calculation:

The function $p\left(x\right)$ can be simplified as follows:

  $\begin{array}{c}p\left(x\right)=\left(r-c\right)\left(x\right)\\ =r\left(x\right)-c\left(x\right)\left[\left(f-g\right)x=f\left(x\right)-g\left(x\right)\right]\\ =\left(400x-0.2x^2\right)-\left(0.1x+200\right)\left[\text{Substituting }r\left(x\right)\text{ and }c\left(x\right)\right]\\ =400x-0.2x^2-0.1x-200\\ =399.9x-0.2x^2-200\end{array}$

Thus, the profit function is $399.9x-0.2x^2-200$ .