Question

Water flows at the rate of 10 m / minute through a cylindrical pipe 5 mm in diameter . How long would it take to fill a conical vessel whose diameter at the base is 40 cm and depth 24 cm.

Answer 1

Diameter of the pipe = 5 mm = 0.5 cm
Radius of the pipe = $\frac{0.5}{2}\mathrm{cm}$ = $\frac{1}{4}cm$
Rate of flow of water through the pipe = 10 m/min = 1000 cm/min
Volume of water that flows out through the pipe in 1 min = ${\mathrm{\pi r}}^2\mathrm{h}=\mathrm{\pi }\times {\left(\frac{1}{4}\right)}^2\times 1000{\mathrm{cm}}^3$
Volume of water flowing out through the pipe in t min = $\mathrm{\pi }{\left(\frac{1}{4}\right)}^2\times 1000t$
Diameter of the conical vessel = 40 cm
Radius = 20 cm
Height or depth = 24 cm
Volume of the conical vessel = $\frac{1}{3}{\mathrm{\pi R}}^2\mathrm{H}=\frac{1}{3}\mathrm{\pi }{\left(20\right)}^2\times 24=3200\mathrm{\pi }$
Time required to fill the vessel = $\frac{\mathrm{capacity}\mathrm{of}\mathrm{the}\mathrm{vessel}}{\mathrm{volume}\mathrm{of}\mathrm{water}\mathrm{flowing}\mathrm{per}\min }$
$$\begin{gathered} t=\frac{3200\mathrm{\pi }}{\mathrm{\pi }{\left({\displaystyle \frac{1}{4}}\right)}^2\times 1000} \\ t=51.2 \end{gathered}$$
So, the time required is 51.2 min = 51 min 12 sec