Question

At present, a firm is manufacturing $1000$ items. It is estimated that the rate of change of production $P$ with respect to additional number of workers $x$ is given by $\frac {dP}{dx}=100-12 \sqrt x$ If the firm employees $25$ more workers, then the new level of production of items is

• A. 2500
• B. 3000
• C. 3500
• D. 4500

Answer 1

The Correct Option is C

Given, $\frac {dP}{dx}=100-12 \sqrt x$
$\Rightarrow \, \, \, \, dP=(100-12 \sqrt x) dx $
On integrating both sides, we get
$\, \, \, \, \, \int \limits dP= \int \limits \, (100-12 \sqrt x)dx$
$\, \, \, \, \, \, \, \, \, \, \, \, P=100x-8x^{3/2}+ C$
When $x=0$, then $P=2000 \Rightarrow C=2000 $
Now, when $x = 25$, then is
$\, \, \, \, \, P=100 \times 25-8 \times (25)^{3/2}+2000 $
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =2500-8 \times 125+2000 $
$\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, =4500-1000=3500 $