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 dp \over dx = 100 - 12\sqrt x . If the firm employs 25 more workers, then the new level of production of items is

Differential Equations PYQ — JEE Main 2013 · Shift 0 · Q30 with Solution

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Correct answer: C

Let the production be $P(x)$ , where $x$ is the additional number of workers.
Given: $\frac{dP}{dx} = 100 - 12\sqrt{x}$ and at present, when no additional workers are employed, production is $P(0)=2000.$
We need the production level when $25$ more workers are employed, i.e. find $P(25)$ .
Integrate the rate equation: $P(25)-P(0)=\int_0^{25} \left(100-12\sqrt{x}\right)dx.$
Evaluate the integral: $\int_0^{25} 100\,dx = 100(25)=2500$

and $\int_0^{25} 12\sqrt{x}\,dx = 12\int_0^{25} x^{1/2}dx = 12\left[\frac{2}{3}x^{3/2}\right]_0^{25}.$

Now, $25^{3/2}=(\sqrt{25})^3=5^3=125,$ so $12\cdot \frac{2}{3}\cdot 125 = 8\cdot 125 = 1000.$

Thus, $P(25)-P(0)=2500-1000=1500.$

Therefore, $P(25)=2000+1500=3500.$
Comparing with the options, the correct answer is: $\boxed{3500}$ which is option C.