Water flows at the rate of 10 m per 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?

Radius of cylindrical pipe, r = $\dfrac{\text{diameter}}{2} = \dfrac{0.5}{2}$ = 0.25 cm

Given, water flows at the rate of 10 m per minute.

Length of the cylindrical portion, h = 10 m = 10 × 100 = 1000 cm

Height of the conical portion, H = 24 cm

Radius of conical pipe, R = $\dfrac{\text{diameter}}{2} = \dfrac{40}{2}$ = 20 cm

Volume of water that flows in 1 min = πr²h

$= \dfrac{22}{7} \times (0.25)^2 \times 1000 \\[1em] = \dfrac{22}{7} \times 0.0625 \times 1000 \\[1em] = \dfrac{1375}{7}$

Volume of the conical vessel = $\dfrac{1}{3}$ πR²H

$= \dfrac{1}{3} \times \dfrac{22}{7} \times 20^2 \times 24 \\[1em] = \dfrac{22}{7} \times 400 \times 8 \\[1em] = \dfrac{70400}{7}$

Required time = $\dfrac{\text{Volume of conical vessel}}{\text{Volume of water that flows in 1 min}}$

$= \dfrac{\dfrac{70400}{7}}{\dfrac{1375}{7}} \\[1em] = \dfrac{70400}{7} \times \dfrac{7}{1375} \\[1em] = \dfrac{70400}{1375} \\[1em] = 51.2 \text{ min.}$

= 51 min 12 sec.

Hence, time required to fill a conical vessel is 51 min 12 sec.