A balloon in the shape of a perfect sphere is slowly expanding. At time hours the radius is 32 feet and is increasing at an instantaneous rate of 3 .

Question

A balloon in the shape of a perfect sphere is slowly expanding. At time hours the radius is 32 feet and is increasing at an instantaneous rate of 3. Use the chain rule for the sequence of variables -- to determine the instantaneous rate of change of the volume at this moment.

Answer 1

Since we know the volume of a sphere we can use this to determine the rate of change of volume with respect to time.

V = 4/3 * π * r³

dV/dt = 4/3 * π * d/dt (r³) = 4/3 * π * (3 * r² * dr/dt) = 4 * π * r² * dr/dt

At a specific time, the radius is 32 feet and is expanding at 3 ft/hour (assumed same time unit since not given).

Therefore, the change in volume is

dV/dt = 4 * π * (32)² * 3 ft/hr = 12,288 * π ft/hr