Question

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

• A. 2500
• B. 3000
• C. 3500
• D. 4500

Answer 1

Answer: (C)

Given, $\frac{dP}{dx}=100-12\sqrt{x}$
$\Rightarrow dP=(100-12\sqrt{x})dx$
On integrating both sides, we get
$\int dP=\int (100-12\sqrt{x})dx$
$P=100x-8x^{3/2}+C$
When $x=0$, then $P=2000\Rightarrow C=2000$
Now, when $x=25$, then is
$P=100\times 25-8\times (25{)}^{3/2}+2000$
$=2500-8\times 125+2000$
$=4500-1000=3500$