Question

Water is flowing into a vertical cylindrical tank of radius $2$ft. at the rate of $8$ cubic ft. per minute.

The water level is rising at the speed of (ft . per minute)

• A. $\frac{\mathrm{\pi }}{2}$
• B. $\frac{2}{\mathrm{\pi }}$
• C. $2$
• D. None of these

Answer 1

The correct option is B

$\frac{2}{\mathrm{\pi }}$

Explanation for the correct option

Given

• Radius of vertical cylindrical tank is $2$ft.
• Water is flowing in vertical cylindrical tank at rate of $8$ cubic ft. per minute

Volume of the vertical cylindrical tank is given by $V$

$$\begin{gathered} V=\pi r^{2}h=4\pi h \\ \frac{dV}{dt}=4\pi \left(\frac{dh}{dt}\right) \\ \left(\frac{dh}{dt}\right)=\left(\frac{1}{4\mathrm{\pi }}\right)\left(\frac{dv}{dt}\right) \\ =\left(\frac{1}{4\mathrm{\pi }}\right)\times 8=\frac{2}{\mathrm{\pi }} \end{gathered}$$

Hence, the correct option is (B) i.e. $\frac{2}{\mathrm{\pi }}$