With V = (1/12) π h^3, if h' = 3 cm/s and h = 8 cm, what is dV/dt?

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Multiple Choice

With V = (1/12) π h^3, if h' = 3 cm/s and h = 8 cm, what is dV/dt?

Explanation:
Use the chain rule for related rates since volume depends on height: dV/dt = (dV/dh) · dh/dt. Differentiate V = (1/12)π h^3 with respect to h to get dV/dh = (1/12)π · 3h^2 = (1/4)π h^2. At h = 8 cm, dV/dh = (1/4)π · 64 = 16π. With dh/dt = h' = 3 cm/s, the rate of change of volume is dV/dt = (16π)(3) = 48π cm^3/s.

Use the chain rule for related rates since volume depends on height: dV/dt = (dV/dh) · dh/dt. Differentiate V = (1/12)π h^3 with respect to h to get dV/dh = (1/12)π · 3h^2 = (1/4)π h^2. At h = 8 cm, dV/dh = (1/4)π · 64 = 16π. With dh/dt = h' = 3 cm/s, the rate of change of volume is dV/dt = (16π)(3) = 48π cm^3/s.

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