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Drainage of a Hemispherical Tank

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A full hemispherical tank of radius drains under the influence of gravity from a circular hole of radius at the bottom of the tank. The velocity of fluid flowing from the hole is (Torricelli's law), where is the gravitational acceleration and is the height of water at time , which is shown in the tank and plotted below.

Contributed by: Enrique Zeleny (December 2012)
Open content licensed under CC BY-NC-SA

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The differential equation for the depth of fluid at time is

.

Integrating and finding the constant yields

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The time that it takes for the tank to empty is

.

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