The radius r of a sphere is increasing at a rate of 3 inches per minute. (a) Find the rates of change of the volume when r = 9 inches and r = 36 inches. (b) Explain why the rate of change of the volume of the sphere is not constant even though dr/dt is constant.

Consider the image shown below. The square is completely inscribed in the circle. The circle's radius is increasing at a rate of 0.8 cm/sec. (a) Find the area of the circle at the instant when the sides of the square are each 4 cm. (b) What is the rate of change of the circle's area at the instant the sides of the square are each 4 cm? (c) At the instant the radius of the circle is 5/2 , how fast is the area of the shaded region changing with respect to time?

The radius of a spherical balloon is increasing at a rate of 4 centimeters per minute. How fast is the surface area changing when the radius is 14 centimeters? Hint: The surface area is S = 4Tr2. Rate of change of surface area =

An ant colony explores an area enclosed by a circle centered at its nest. This area (and the circle containing it) expands as they have time to explore. Due to the need for ants to return to the colony periodically, as the radius expands, the change in the radius of the area explored with respect to time slows down. Specifically, if r is the radius of the area the ants have explored in meters, then = 3 m/day. Find the rate of change of the area explored by the ants 4 in m2/day when the radius of the area explored is 3m. dt