# Calculus

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A swimming pool is 60ft long by 25ft wide. Its depth varies uniformly from 3ft at the shallow end to 15ft at the deep end. The pool is being filled with water at a rate of 800ft3/min. There is 5ft of water at the deep end.
What is the equation for finding the rate at which the water level is rising?

• Calculus -

If the water is at depth y<12, the cross-section of the water seen from the side of the pool is a triangle. The length of the water surface is y/12 * 60.

So, the volume of water is

v = 1/2 * y/12 * 60 * y * 25 = 125/2 y^2

dv/dt = 125y dy/dt

so, at depth y=5,
800 = 125(5) dy/dt
dy/dt = 1.28 ft/min

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