Question

Water is leaking from a conical funnel at the rate of $5c{m}^5/sec$. If the radius of base of funnel is $10cm$ and the altitude is $20cm$. Find the rate at which water level is dropping when it is $10cm$ from the top.

Answer 1

The volume of Water present in the conical flask at any time is given by $V=\frac{1}{3}\pi r^{2}h\cdots \left(1\right)$

Rate in decrease in volume $\frac{dV}{dt}=5c{m}^3/sec$

when the flask is full $r=10$ and $h=20$ the ratio of $r:h$ remains same all through the cone

so, $\frac{r}{h}=\frac{1}{2}r=\frac{h}{2}$

Put it in $\left(1\right)$

$⟹V=\frac{1}{3}\pi {\left(\frac{h}{2}\right)}^{2}hV=\frac{1}{12}\pi h^3$

Differentiating with respect to $t$

$\frac{dV}{dt}=\frac{1}{12}\pi 3h^2\frac{dh}{dt}5=\frac{\pi }{4}{h}^2\frac{dh}{dt}$

when $h=10$

• $\frac{20}{\pi }={10}^2\frac{dh}{dt}⟹\frac{dh}{dt}=\frac{5}{\pi }cm/sec$