Question

A cylindrical tank is 12 ft in diameter and 9 ft high. Water flows into the tank at the rate of pi/10 ft^3/sec. It has a hole of radius 1/2 in. At the bottom. When will the tank be full if initially it is empty?

Answer 1

Given :

The diameter of cylinder tank = d = 12 ft

So, Radius = r = $\dfrac{d}{2}$ = 6 ft

The rate of flow of water into tank = $\dfrac{\pi }{10}$ ft³ per sec

The radius of hole to empty the tank = 0.5 inch

To Find :

The time required to full the tank

Solution :

The Volume of cylinder tank = π × radius² × height

                                               =  π × (6 ft)² × 9 ft

                                               = 324 π  ft³

Again

radius of hole to empty the tank = 0.5 inch

∵ 1 ft = 12 inch

So, radius of hole to empty the tank = $\dfrac{0.5}{12}$ = 0.041 ft

Volume of hole = volume of sphere = $\dfrac{4}{3}$ × π × radius³

                                                           = $\dfrac{4}{3}$ × π × (0.041 ft)³

                                                           = 0.000091  π  ft³

So, Volume through which water flow =  Volume of cylinder tank - Volume of hole

                                                              = 324 π  ft³ - 0.000091  π  ft³

                                                              = 323.99 π  ft³

Again

∵  $\dfrac{\pi }{10}$ ft³ water flow through hole in 1 sec

∴ 323.99 π  ft³ water flow through hole in = $\dfrac{1}{\dfrac{\pi }{10} }$ × 323.99 π

                                                                     = 32.39 sec

Hence, The tank will full after 32.39 sec  Answer