Question

Water is dripping out from a conical funnel of semi-vertical angle $\frac{\pi}{4}$ at the uniform rate of $2\, cm ^{3} / \sec$ in its surface area through a tiny hole at the vertex in the bottom. When the slant height of the water is $4\, cm$, if the rate of decrease of the slant height of the water, is $\frac{\sqrt{ k }}{4 \pi} cm / \sec$ then find $k$.

Answer 1

$V =\frac{1}{3} \pi r ^{3}$
$\Rightarrow \frac{ dV }{ dt }=\pi r ^{2} \frac{ dr }{ dt }$
$\Rightarrow 2=\pi r ^{2} \frac{ dr }{ dt }$
$\ell=\sqrt{2} r$
$\frac{ d \ell}{ dt }=\sqrt{2} \frac{ dr }{ dt }$
$=\sqrt{2} \frac{2}{\pi(2 \sqrt{2})^{2}}=\frac{\sqrt{2}}{4 \pi} cm / \sec .$