A radio manufacturer produces x sets per week at a total cost of $₹\begin{Bmatrix} \frac{x^2}{25}+3x+100\end{Bmatrix}$. He is a monopolist and the demand of his product is $x=75-3p $, where p is the price in rupees per se t. For maximum net revenue the number of sets produced per week is :

30

The correct answer is Option (1) → 30

Demand function, $x=75-3p$

$⇒p=\frac{x-75}{3}$

∴ Revenue, $R(x)=x.p=\frac{x×(75-x)}{3}=\frac{75x-x^2}{3}$

Now,

Profit = $R(x)-C(x)$

$=\frac{75x-x^2}{3}-\left(\frac{x^2}{25}+3x+100\right)$

$=\frac{1650x-28x^2}{75}-100$

for critical point, $P'(c)=0$

$⇒1650-56x=0$

$⇒x=\frac{1650}{56}≃30$