Show that the maximum rate of change, with respect to radius, of the volume of a deflating balloon is four times the sphere's initial great circle circumference

Step-by-step explanation:

Let's say R is the initial radius of the sphere, and r is the radius at time t.

The volume of the sphere at time t is:

V = 4/3 π r³

Taking derivative with respect to radius:

dV/dr = 4π r²

This is a maximum when r is a maximum, which is when r = R.

(dV/dr) max = 4π R²

This is 4 times the sphere's initial great circle area, but not the great circle circumference. The problem statement contains an error.