At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production P w.r.t…
At present a firm is manufacturing 1000 items. It is estimated that the rate of change of production P w.r.t. additional number of worker $x$ is given by $\frac{\mathrm{dp}}{\mathrm{d} x}=100-12 \sqrt{x}$.
If the firm employees 9 more workers, then the new level of production of items is
$1684$
$1648$
$2116$
$1116$
Solution
$\begin{array}{ll}
& \frac{\mathrm{dP}}{\mathrm{d} x}=100-12 \sqrt{x} \\
& \text { Integrating both sides, we get } \\
& \int \mathrm{dp}=\int(100-12 \sqrt{x}) \mathrm{d} x \\
\therefore \quad & \mathrm{P}=100 x-8 x \sqrt{x}+\mathrm{c} \\
& \text { Given that } \mathrm{P}=1000, \text { when } x=0 \\
\Rightarrow & 1000=100(0)-8(0)+\mathrm{c} \\
\Rightarrow \mathrm{c}=1000 & \\
\therefore \quad & \mathrm{P}=100 x-8 x \sqrt{x}+1000 \\
& \text { When } x=9, \text { we get } \\
& \mathrm{P}=900-216+1000=1684
\end{array}$
$\therefore \quad$ The new level of production of items is 1684 .