Question

A right circular cone has height $9cm$ and radius of the base $5cm$. It is inverted and water is poured into it. If at any instant the water level rises at the rate of $\left(\pi /A\right)cm/sec$. Where $A$ is the area of the water surface at that instant, show that the vessel will be full in $75$ seconds.

Answer 1

Let $r$ be radius of cross section at height $h$.

$h=9cm$ and $r=5cm$....given

$\Rightarrow \frac{r}{5}=\frac{h}{9}\Rightarrow r=\frac{5h}{9}$

Given, $\frac{dh}{dt}=\frac{\pi }{A}$

$\Rightarrow \frac{dh}{dt}=\frac{\pi }{\pi r^2}$

$\Rightarrow \frac{dh}{dt}=\frac{81}{25h^2}$

$\Rightarrow \frac{25}{81}{\int }_0^9{h}^{2}dh={\int }_0^9\frac{h^3}{3}dt$

$\Rightarrow \frac{25}{81}\times \frac{729}{3}=t$

$\Rightarrow t=75seconds$.