Question

A product can be manufactured at a total cost $C\left(x\right)=\frac{x^2}{100}+100x+40$, where x is the number of units produced. The price at which each unit can be sold is given by $P=(200-\frac{x}{400})$. Determine the production level x at which the profit is maximum. What is the price per unit and total profit at the level of production?

Answer 1

Profit will be $Pro\left(x\right)=xP-C\left(x\right)$

$Pro\left(x\right)=x(200-\frac{x}{400})-(\frac{x^2}{100}+100x+40)$

We want to maximize $Pro\left(x\right)$ so differentiate it w.r.t $x$ and put it $=0$

$\frac{d\left(Pro\right(x\left)\right)}{dx}=200-\frac{2x}{400}-\frac{2x}{100}-100=0\to x=4000$

Hence, price per unit will be $P=200-\frac{4000}{400}=190$

Total Profit will be $Pro\left(x\right)=4000\times 190-(\frac{(4000{)}^2}{100}+100\times 4000+40)=32\times {10}^4-40=319960.$