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Question

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


Solution

Profit will be $$Pro(x)=xP-C(x)$$
$$Pro(x)=x\left(200-\dfrac{x}{400}\right)-\left ( \dfrac{x^2}{100}+100x+40\right)$$
We want to maximize $$Pro(x)$$ so differentiate it w.r.t $$x$$ and put it $$=0$$
$$\dfrac{d(Pro(x))}{dx}=200-\dfrac{2x}{400}-\dfrac{2x}{100}-100=0\rightarrow x=4000$$
Hence, price per unit will be $$P=200-\dfrac{4000}{400}=190$$
Total Profit will be $$Pro(x)=4000\times 190 - (\dfrac{(4000)^2}{100}+100\times 4000+40)=32\times 10^4-40=319960.$$

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