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 =  \[\frac{0 . 5}{2}cm\] = \[\frac{1}{4}cm\]

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 =  \[\pi r^2 h = \pi \times \left( \frac{1}{4} \right)^2 \times 1000 {cm}^3\] Volume of water flowing out through the pipe in t min =  \[\pi \left( \frac{1}{4} \right)^2 \times 1000t\] 

Diameter of the conical vessel = 40 cm
Radius = 20 cm
Height or depth = 24 cm
Volume of the conical vessel =  \[\frac{1}{3} \pi R^2 H = \frac{1}{3}\pi \left( 20 \right)^2 \times 24 = 3200\pi\]

Time required to fill the vessel = \[\frac{\text { capacity of the vessel }}{\text { volume of water flowing per } \min}\]

\[t = \frac{3200\pi}{\pi \left( \frac{1}{4} \right)^2 \times 1000}\]

\[t = 51 . 2\]

So, the time required is 51.2 min = 51 min 12 sec