Question

Water flows at the rate of 10 meters per minute through a cylindrical pipe having its diameter 20 mm. how much time will it take to fill a conical vessel of diameter 40 cm and depth 24 cm? 

Answer 1

For the cylindrical pipe: 

diameter = 20 mm

∴ radius = ${\displaystyle \frac{\text{diameter}}{2}=\frac{20}{2}}$ = 10 mm = 1cm

Rate of flow of water through the pipe  = 10m/minute

                                                                = 10 × 100 cm /minute

                                                                = 1000 cm/minute 

∴ Water flow's through a distance (h) of  1000 cm in a minute 

volume of water flowing through the pipe in 1 minute = ${\displaystyle \pi r^{2}h}$

                                                                                     = π × 1× 1× 1000

                                                                                      = 1000π cm³ 

For the conical vessel:

diameter = 40 cm 

∴ radius =${\displaystyle \frac{\text{diameter}}{2}=\frac{40}{2}=20}$ cm 

depth (h) = 24cm 

Volume of conical vessel = ${\displaystyle \frac{1}{3}\pi r^{2}h}$

                                       ${\displaystyle =(\frac{1}{3}\times \pi \times 20\times 20\times 24)}$ cm³ 

Time is taken to fill the conical vessel  =${\displaystyle \frac{\text{volume of conical vessel}}{\text{volume of water flowing through a pipe in 1 min}}}$

                                                        ${\displaystyle =\frac{\frac{1}{3}\times \pi \times 20\times 20\times 24}{1000\pi }}$

                                                       ${\displaystyle =\frac{\pi \times 20\times 20\times 24}{3\times \pi \times 1000}}$

                                                      ${\displaystyle =\frac{3200}{1000}=}$3.2 minutes

∴ Time taken to fill the conical veseel is 3.2 minutes.