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
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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} $
