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AndersCarlos
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Homework Statement
Apostol, Vol 1: Section 4.12 Problem 26
Water flows into a hemispherical tank of radius 10 feet (flat side up). At any instant, let h denote the depth of the water, measured from the bottom, r the radius of the surface of the water, and V the volume of the water in the tank. Compute dV/dh at the instant when h=5 feet. If the water flows in at a constant rate of 5√3 cubic feet per second, compute dr/dt, the rate at which r is changing, at the instant t when h=5 feet.
Homework Equations
Hemispherical Volume: [2π(r^3)]/3
The Attempt at a Solution
I tried to rewrite the volume in terms of h. If we consider the circle equation, its possible to say that the radius of the water volume is: r' = √[100 - (h^2)], where 100 is the radius of the tank squared and h is the height. So, we can rearrange the volume formula to: V = {2π{[100 - (h^2)]^3/2}}/3, if we derive this with respect to h, we can find: dV/dh = -2πh√[100 - (h^2)], if we change h for 5, it gives that dV/dh = -10π√75. However, according to the answers at the end, the correct answer would be 75. Sorry if I made any mistakes. I would really appreciate any tip, correction or solution. Thank you.