The radius of a sphere is increasing at a rate of 2 mm/s. How fast is the volume increasing when the diameter is 100 mm? Evaluate your answer numerically.

The rate at which the volume of a sphere is changing can be found by taking the derivative of the function that represents the volume of the sphere with respect to time.

The volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$, where $r$ is the radius of the sphere. Therefore, the rate at which the volume is changing is given by the derivative of this function with respect to time, which is $\frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt}$.

Since the diameter of the sphere is 100 mm, the radius of the sphere is 50 mm. Therefore, the rate at which the volume is changing is $\frac{dV}{dt} = 4\pi (50\text{ mm})^2 \cdot 2$ mm/s = $20,000\pi$ mm$^3$/s = 62831.85 mm$^3$/s.

Rounding this value to the nearest whole number, we get $\frac{dV}{dt} \approx 62832$ mm$^3$/s. Therefore, when the diameter of the sphere is 100 mm, the volume of the sphere is increasing at a rate of approximately 62832 mm$^3$/s.