The fixed cost of a new product is ₹3,000. The variable cost is estimated to be 20% of the total revenue generated on selling x units with the demand $P(x)=\left(15-\frac{5x}{36}\right)$ . The number of units(x) at which profit in maximum.

54

The correct answer is Option (3) → 54

Revenue $(R(x))$ is,

$R(x)=P(x).x$

$=x\left(15-\frac{5x}{36}\right)=15x-\frac{5x^2}{36}$

for cost function,

Fixed cost = Rs. 3,000

Variable cost = 20% of revenue

$VC(x)=0.2R(x)=0.2\left(15x-\frac{5x^2}{36}\right)$

$=3x-\frac{x^2}{36}$

$∴C(x)=3000+3x-\frac{x^2}{36}$

Profit, $P=\left(15x-\frac{5x^2}{36}\right)-\left(3000+3x-\frac{x^2}{36}\right)$

$=12x-\frac{x^2}{9}-3000$

for max. profit, $f'(c)=0$

$\left.\frac{dP}{dx}\right|_{x=c}=12-\frac{2c}{9}$

$⇒c=\frac{12×9}{2}=54$