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 $\frac{d P}{d x}=100-2 \sqrt{x}$. If the firm employs 25 more workers, then the new level of production of items is
Answer & explanation
Correct answer: option 3
We have, $\frac{d P}{d x}=100-12 \sqrt{x}$
Integrating both sides with respect to x, we get
$P =\int(100-12 \sqrt{x}) d x$
$\Rightarrow P =100 x-8 x^{3 / 2}+C$ ....(i)
Initially x = 0, P = 2000
∴ 2000 = C
Putting C = 2000 in (i), we get
$P=100 x-8 x^{3 / 2}+2000$
When x = 25, we get
$P=2500-8(25)^{3 / 2}+2000=3500$
Hence, the new level of production is 3500 items.