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

Answer 1

Answer: 3500

Given $\frac{dP}{dx}=100-12\sqrt{x}$
$\Rightarrow dP=\left(100-12\sqrt{x}\right)dx$
Integrating both sides, we get
$\int dP=\int \left(100-12\sqrt{x}\right)dx$
$\Rightarrow P=100x-12\times \frac{2}{3}{\left(x\right)}^{\frac{3}{2}}+c$
$\Rightarrow P=100x-8{\left(x\right)}^{\frac{3}{2}}+c$
Also, given that at $x=0,P=2000$ .
Putting these in the above equation, we get
$2000=100\times 0-8\times {\left(0\right)}^{\frac{3}{2}}+c$
$\Rightarrow c=2000$
So, $P=100x-8{\left(x\right)}^{\frac{3}{2}}+2000$
Now, when $x=25,P=100\left(25\right)-8{\left(25\right)}^{\frac{3}{2}}+2000=2500-1000+2000=3500$ .