Question

Prove that the rate of change of the volume of a cylinder

with fixed height with respect to its radius r is equal to the

lateral surface area of the cylinder. Why does it make geometric

sense that the lateral surface area would be related

to this rate of change?

Short answer

The rate of change in volume of the cylinder is$=2\pi rh$

Step-by-step solution

Step 1. Given Information

The radius of the cylinder is$"r"$and the height remains constant.

Step 2. Calculation

The volume of the cylinder is$=\pi r^{2}h$

Where$"h"$is the height of the cylinder.

Let, $V=\pi r^{2}h$

Now, $\frac{dV}{dr}=2\pi rh$

Which is equal to surface area of the cylinder.

Step 3. Explanation

If we paint a cylinder then the outside of the paint is the new boundary of the cylinder and the inside of the paint is added to its volume.

This explains why the derivative of the volume is the surface area.