Differential Equations 3 Question 5
5. 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 $\frac{d P}{d x}=100-12 \sqrt{x}$. If the firm employees 25 more workers, then the new level of production of items is
(2013 Main)
(a) 2500
(b) 3000
(c) 3500
(d) 4500
Objective Questions II
(One or more than one correct option)
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Answer:
Correct Answer: 5. (c)
Solution:
- Given, $\frac{d P}{d x}=(100-12 \sqrt{x}) \Rightarrow d P=(100-12 \sqrt{x}) d x$
On integrating both sides, we get
$$ \begin{gathered} \int d P=\int(100-12 \sqrt{x}) d x \\ P=100 x-8 x^{3 / 2}+C \end{gathered} $$
When $x=0$, then $P=2000 \Rightarrow C=2000$
Now, when $x=25$, then is
$$ \begin{aligned} P & =100 \times 25-8 \times(25)^{3 / 2}+2000 \\ & =2500-8 \times 125+2000 \\ & =4500-1000=3500 \end{aligned} $$
