Q 63. - Calculus

Question

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

with respect to its radius r is equal to the surface

area of the sphere. Why does it make geometric sense

that the surface area would be related to this rate of

change?

Step-by-Step Solution

Answer

Rate of change of volume of the sphere$=4\pi r^2$

Step 1. Given Information

The radius of the sphere is $"r"$.

Step 2. Calculation

The volume of the sphere is 

$V=\frac{4}{3}\pi r^3$

Now, $\frac{dV}{dr}=4\pi r^2$

Which is equal to the surface area of the sphere.

Step 3. Explanation

If we paint a sphere. Then the amount of paint we used is based on the surface area and becomes part of the volume of the sphere. The outside of the paint is a new boundary of the sphere and the inside of the paint is added to the volume. That's why the derivative of the volume of a sphere is equal to its surface area.