Question

A cylindrical vessel of radius 0.5 m is filled with oil at the rate of 0.25 π m³/minute. The rate at which the surface of the oil is rising, is
(a) 1 m/minute
(b) 2 m/minute
(c) 5 m/minute
(d) 1.25 m/minute

Answer 1

(a) 1 m/minute

$$\begin{gathered} \text{Let}\mathit{\text{r}}\text{be the radius,}\mathit{\text{h}}\text{be the height and}\mathit{\text{V}}\text{be the volume of the cylindrical vessel at any time}\mathit{\text{t.}}\text{Then,} \\ V=\mathrm{\pi }{r}^2\mathit{\text{h}} \\ \Rightarrow \frac{dV}{dt}=\mathrm{\pi }{r}^2\frac{dh}{dt} \\ \Rightarrow \frac{dh}{dt}=\frac{1}{\mathrm{\pi }{r}^2}\frac{dV}{dt} \\ \Rightarrow \frac{dh}{dt}=\frac{0.25\mathrm{\pi }}{\mathrm{\pi }{\left(0.5\right)}^2} \\ \Rightarrow \frac{dh}{dt}=\frac{0.25}{0.25} \\ \Rightarrow \frac{dh}{dt}=1\mathrm{m}/\min \end{gathered}$$