Exercise 106

The rate of change of the volume of a snowball that is melting is proportional to the surface area of the snowball. Suppose the snowball is perfectly spherical. The volume (in centimeters cubed) of a ball of radius $r$ centimeters is $(\nicefrac{4}{3}) \pi r^3$. The surface area is $4 \pi r^2$. Set up the differential equation for how the radius $r$ is changing. Then, suppose that at time $t=0$ minutes, the radius is 10 centimeters. After 5 minutes, the radius is 8 centimeters. At what time $t$ will the snowball be completely melted?