Question

The radius of a sphere is increasing at a constant rate of 10 inches per second. At the instant when the volume of the sphere is 531 cubic inches, what is the rate of change of the volume? Round to 3 decimal places, work not needed.​

Answer 1

The rate of change of the volume of the sphere is 400π cubic inches per second.

Step-by-step explanation:

The volume of a sphere can be given by the formula V = (4/3)πr³, where V is the volume and r is the radius. In this problem, the radius is increasing at a constant rate of 10 inches per second. To find the rate of change of the volume, we need to take the derivative of the volume with respect to time.

Using the power rule for differentiation, we have dV/dt = 4πr²(dr/dt). Substituting the given values, dV/dt = 4π(10)² = 400π cubic inches per second.

Therefore, the rate of change of the volume is 400π cubic inches per second.