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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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