Question

The radius of a sphere is decreasing at a constant rate of 5 centimeters per minute. At the instant when the volume of the sphere is 892,892 cubic centimeters, what is the rate of change of the volume? The volume of a sphere can be found with the equation v = 4/3∏r^3 Round your answer to three decimal places.

Answer 1

To find the rate of change of the volume of a sphere, we can use the formula dV/dt = 4πr^2(dr/dt). Substituting the given values, we can find the rate of change of the volume at the instant when the volume is 892,892 cm^3.

Step-by-step explanation:

To find the rate of change of the volume of the sphere, we need to take the derivative of the volume equation with respect to time.

The volume of a sphere is given by the equation V = (4/3)πr^3, where r is the radius of the sphere.

By differentiating this equation with respect to time, we get dV/dt = 4πr^2(dr/dt). We are given that dr/dt = -5 cm/min, so we can substitute this value into the equation to find the rate of change of the volume.

Substituting the given values into the equation, we have dV/dt = 4π(r^2)(-5) = -20πr^2 cm^3/min. To find the rate of change of the volume at the instant when the volume is 892,892 cm^3, we need to substitute r = (3V/4π)^(1/3) into the equation and evaluate it.