Question

At present, a firm manufactures 1099 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 empower 25 more workers, then the new level of production of items is

• A. $2000$
• B. $2500$
• C. $3000$
• D. $3500$

Answer 1

Answer: (B)

$2500$

Answer 2

Given, $\frac{dP}{dx}=100-12\sqrt{x}$ $\Rightarrow$ $dP=(100-12\sqrt{x})dx$ On integrating both sides, we get $P=100x-12\left( \frac{{{x}^{3/2}}}{3/2} \right)+C$ $\Rightarrow$ $P=100x-8{{x}^{3/2}}+C$ ..(i) At initially $x=0,\,\,\,\,P=1000$ Then, from E (i) we get $1000=100\times 0-0+C$ $\Rightarrow$ $C=1000$ On putting the value of C in E (i), we get $P=100x-8{{x}^{3/2}}+1000$ Now, at $x=25,$ $P=100(25)-8{{(25)}^{3/2}}+1000$ $=2500-8(125)+100$ $=3500-1000=2500$