Question

The volume of a circular cone is V = πr^2h/3, where r is the radius of the base and h is the height.
a. What is the rate of change of the volume with respect to the height if the radius is constant?
b. What is the rate of change of the volume with respect to the radius if the height is constant?

Answer 1

Differentiate both sides with respect to an independent variable:
\[\frac{\mathrm dV}{\mathrm dt}=\frac{2\pi rh}{3}\frac{\mathrm dr}{\mathrm dt}+\frac{\pi r^2}{3}\frac{\mathrm dh}{\mathrm dt}\]When either the height or radius is kept constant, you have \(\dfrac{\mathrm dh}{\mathrm dt}=0\) or \(\dfrac{\mathrm dr}{\mathrm dt}=0\), respectively.