Related Rates (Derivatives)

Question

How to show that if the volume of a balloon is decreasing at a rate proportional to its surface area, the radius of the balloon is shrinking at a constant rate.

Answer 1

Since we're given the starting point of volume, we should start with the volume formula for a sphere: V = 4/3 πr³.

We also know that it is decreasing proportional to surface area, meaning that dV/dt = -k*SA, where k is just a constant (it's negative because the problem says decreasing). So let's use the original volume formula to find dV/dt: if we take the derivative with respect to time, we get:

dV/dt = 4πr² * dr/dt, and since we know dV/dt = -k*SA, we can simplify to 4πr² * dr/dt = -k*SA. The surface area for a sphere is SA = 4πr², so we can sub that in and get:

4πr² * dr/dt = -k * 4πr².

The 4πr² s cancel, and we get dr/dt = -k, which means that the radius is changing with respect to time in a constant way. So the radius must be decreasing (because of the negative sign) at a constant rate.