  Question

A hemispherical tank full of water is emptied by a pipe at the rate of $$\cfrac{25}{7}$$ litres per second. How much time will it take to half-empty the tank, if the tank is $$3$$ metres in diameter?

Solution

Given,The diameter of hemispherical tank $$D=3\ m$$Hence, the radius of hemispherical tank $$R=\dfrac{D}{2}=1.5\ m$$The total volume of water in the hemispherical tank $$V=\dfrac{2}{3}\pi R^{3}$$$$V=\dfrac{2}{3}\times \dfrac{22}{7}\times (1.5)^{3}\ m^{3}$$$$V= \dfrac{44}{21}\times 3.375\ m^{3}$$$$V= \dfrac{148.5}{21}\ m^{3}$$$$V= \dfrac{148500}{21}\ liters$$          $$(\because 1\ m^{3}=1000\ liter)$$Hence, the total volume of water in the hemispherical tank is $$\dfrac{148500}{21}\ liters$$   Volume of half-empty tank $$=\dfrac{V}{2}=\dfrac{74250}{21}\ liters$$Given that time required to empty the tank by $$\dfrac{25}{7}\ liters$$ of water $$=1\ second$$Hence, the time required to empty the tank by $$1\ liter$$ of water $$=\dfrac{7}{25}\ second$$Now,  time  taken to half-empty the tank $$\dfrac{74250}{21}\ liters$$ of water $$=\dfrac{7}{25}\times \dfrac{74250}{21}\ seconds$$                                                                                                                                                                                                                       $$=990\ seconds$$                                                                                                           .                                                                                                             $$=16.5\ minutes$$                $$(\because 1\ minute=60\ seconds)$$Hence, time taken to half-empty the tank is  $$16.5\ minutes$$Mathematics

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