Question

At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $P$ w.r.t. additional number of workers $x$ is given by $\frac{dP}{dx}=100-12\sqrt{x}$. If the firm employs $25$ more workers, then the new level of production of items is :

• A. $2500$
• B. $3000$
• C. $3500$
• D. $4000$

Answer 1

The correct option is C $3500$

$\frac{dP}{dx}=100-12\sqrt{x}$
$\Rightarrow \int dP=\int (100-12\sqrt{x})dx$
$\Rightarrow P=100x-12\cdot \frac{2}{3}\cdot x^{3/2}+c$
At $x=0,P=2000$
So, $c=2000$
$\Rightarrow P=100x-8\cdot x^{3/2}+2000$
At $x=25$,
$\Rightarrow P=100\left(25\right)-8\cdot {25}^{3/2}+2000$
$=3500$