Question

Water is leaking from the corner of a vertical cone-like container at a rate of 5 $c{m}^3/min$. If $r:h=2:3$, then how fast is the water level dropping at the instant when the water is exactly 7.5 cm deep ?

• A. $\frac{1}{\pi }cm/min$
• B. $\frac{1}{5\pi }cm/min$
• C. $\frac{1}{2\pi }cm/min$
• D. $\frac{2}{3\pi }cm/min$

Answer 1

Answer: (B)

$\frac{1}{5\pi }cm/min$

Given $\frac{dV}{dt}=-5c{m}^3/min$
Also, $\frac{r}{h}=\frac{2}{3}$

$\Rightarrow r=\frac{2h}{3}$
Volume of cone $V=\frac{1}{3}\pi r^{2}h$
$V=\frac{4}{27}\pi h^3$
$\frac{dV}{dt}=\frac{4}{9}\pi h^2\frac{dh}{dt}$
$\Rightarrow \frac{dh}{dt}=-\frac{1}{5\pi }cm/min$
-ve sign indicates that water level is decreasing.