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Question

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


Solution

Radius of the pipe $$=\dfrac{5}{2}\ mm=\dfrac{5}{2}\times\dfrac{1}{10}\ cm=\dfrac{1}{4}\ cm$$
Speed of water $$=10 m/min=1000\ cm/min$$
Volume of water that flows in 1 minute $$=\pi r^2h$$
                                                                 $$=\dfrac{22}{7}\times\dfrac{1}{4}\times\dfrac{1}{4}\times1000$$
                                                                 $$=\dfrac{1375}{7}\ cm^3$$
Radius of conical vessel $$=\dfrac{40}{2}=20\ cn$$
Depth $$=24\ cm$$
So,
Capacity of the vessel $$=\dfrac{1}{3}\times\pi r^2h$$
                                      $$=\dfrac{1}{3}\times\dfrac{22}{7}\times20\times20\times24$$
                                      $$=\dfrac{70400}{7}\ cm^3$$
Therefore,
Time required to fill vessel $$=\dfrac{Capacity\ of\ the \ vessel}{Volume\ of\ water\ flowing\ per\ minute}$$ 
                                             $$=\dfrac{70400/7}{1375/7}$$
                                             $$=\dfrac{265}{5}\ minutes$$
                                             $$=51\ min\ 12\ sec$$

Mathematics

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