A firm is manufacturing 2000 items. It is estimated that the rate of change of production $P$ with respect…

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 $\frac{d p}{d x}=100-12 \sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is
  1. $2500$
  2. $3000$
  3. $3500$
  4. $4500$

Solution

$\begin{aligned} & \frac{d p}{d x}=100-12 \sqrt{x} \Rightarrow \int d p=\int(100-12 \sqrt{x}) d x \\ & \Rightarrow p=100 x-12 \times \frac{2}{3} x^{3 / 2}+C \\ & \Rightarrow p=100 x-8 x^{3 / 2}+200[\because \text { at } x=0, p=2000] \\ & \Rightarrow p=100 \times 25-8 \times(25)^{3 / 2}+2000[\text { putting } x=25] \\ & \Rightarrow p=2500-1000+2000=3500\end{aligned}$

Asked in: MHT CET 2022 (06 Aug Shift 2)

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