Question

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

Answer: 2.00

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