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Calculus/Related Rates


I was wondering if you could help me with a related rates calculus problem.
The base of a pyramid shaped tank is a square base with sides of length 12 feet, and the vertex of the pyramid is 10 feet above the base. The tank is filled to a depth of 4 feet, and water is flowing into the tank at the rate of 2 cubic feet per minute. Find the rate of change of the depth of water in the tank.

We need to calculate the surface area of the water at that point.
Since there is 4' of water out of a total 10', that is 40% full.
This makes the total width at that depth be 12(1-0.4) = 7.2.

Since we have each side as 7.2', the area at this depth is 7.2 x 7.2 = 51.84.

If the water is coming in at 2 cubic feet per minute,
we take 2/51.84 = 0.038580247 feet / minute,
or roughly 0.46 inches per minute at that moment in time.


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