Question

At present, a firm is manufacturing $2000$ items. It is estimated that the rate of change of production $\text{P}$ w.r.t. additional number of workers $\text{x}$ is given by $\frac{\text{dP}}{\text{dx}} = 1 0 0 - 1 2 \sqrt{\text{x}}$ . If the firm employs $25$ more workers, then the new level of production of items is

• A. $\text{3500}$
• B. $\text{4500}$
• C. $\text{2500}$
• D. $\text{3000}$

Answer 1

Answer: (A)

$\frac{\text{dP}}{\text{dx}} = 1 0 0 - 1 2 \sqrt{\text{x}}$
$\Rightarrow \displaystyle \int \text{dP = } \displaystyle \int \left(1 0 0 - 1 2 \sqrt{\text{x}}\right) \text{dx}$
$\Rightarrow \text{P} = 1 0 0 \text{x} - 8 \text{x}^{3 / 2} + \text{C}$
Now, $\text{x}=\text{0}$
$\Rightarrow \text{P}=2000$ $ \, ⇒ \, \text{C}=2000$
hence, $P=100x-8x^{\frac{3}{2}}+2000$
$\because \, \, x=25$
$P=2500-1000+2000$
$P=3500$