Question

A company manufacturers and sells a electric drills per month. The monthly cost and price-demand equations areC(x) = 50000 + 40x,P = 170 - x/30,0 < x < 5000.(A) Find the production level that results in the maximum profit.Production Level =

Answer 1

To find the maximum profit, we have to find the derivative of the cost. So:

$C^{\prime}(x)=\text{ 40}$

Then, we need to find the revenue that is equal to:

$\begin{gathered} R(x)\text{ = x \lparen p\lparen x\rparen\rparen} \\ R(x)=\text{ x\lparen170 - }\frac{x}{30}) \\ R(x)\text{ = 170x -}\frac{\text{ x}^2}{30} \\ \end{gathered}$

Profit = Revenue - Cost

Profit=

$\begin{gathered} P(x)=(170x\text{ - }\frac{x^2}{30})\text{ - 40} \\ P(x)=\frac{5100x\text{ - x}^2}{30}\text{ - 40} \\ P(x)=\frac{5100x\text{ - x}^2\text{ - 1200}}{30} \\ P(5000)=\frac{5100(5000)\text{ - \lparen5000\rparen}^2\text{ - 1200}}{30} \\ P(5000)=\frac{25,500,000-25,000,000-1200}{30} \\ P(5000)=\frac{498800}{30} \\ P(5000)=16,626.666 \\ P(5000)\approx16,627 \end{gathered}$

The production level is 16,627 units to get the maximum profit