Application of the derivatives concept question?

Question

Suppose the radius r, height h, and volume V of a cylinder are functions of time t. How is dV/dt related to dh/dt if the radius of the cylinder is constant?

Answer 1

Volume of the cylinder V = πr²h

Differentiate with respect to t.

dV/dt = d/dt(πr²h)

= π [d/dt (r²h) + r² d/dt(h)]

dV/dt = π [2r(dr/dt)(h) + r² (dh/dt) ]

If r is constant then d/dt(r) = 0

Substitute dr/dt = 0 in dV/dt.

dV/dt = π [2r(0)(h) + r² (dh/dt) ]

dV/dt = πr² (dh/dt).