Q. 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 $\frac{d P}{d x}=100-12\sqrt{x}$ . If the firm employs $25$ more workers, then the new level of production of items is
NTA AbhyasNTA Abhyas 2022
Solution:
Given $\frac{d P}{d x}=100-12\sqrt{x}$
$\Rightarrow dP=\left(100 - 12 \sqrt{x}\right)dx$
Integrating both sides, we get
$\displaystyle \int dP=\displaystyle \int \left(100 - 12 \sqrt{x}\right)dx$
$\Rightarrow P=100x-12\times \frac{2}{3}\left(x\right)^{\frac{3}{2}}+c$
$\Rightarrow P=100x-8\left(x\right)^{\frac{3}{2}}+c$
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 $x=25,P=100\left(25\right)-8\left(25\right)^{\frac{3}{2}}+2000=2500-1000+2000=3500$ .
$\Rightarrow dP=\left(100 - 12 \sqrt{x}\right)dx$
Integrating both sides, we get
$\displaystyle \int dP=\displaystyle \int \left(100 - 12 \sqrt{x}\right)dx$
$\Rightarrow P=100x-12\times \frac{2}{3}\left(x\right)^{\frac{3}{2}}+c$
$\Rightarrow P=100x-8\left(x\right)^{\frac{3}{2}}+c$
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 $x=25,P=100\left(25\right)-8\left(25\right)^{\frac{3}{2}}+2000=2500-1000+2000=3500$ .