Step 1: Define an equation that relates the volume of a sphere to its radius.
\[V=\frac{4}{3}\cdot\pi\cdot r^{3}\]
Step 2: Take the derivative of each side with respect to time (we will define time as "t")
\[\bigg(\frac{d}{dt}\bigg) V=\bigg(\frac{d}{dt}\bigg)\bigg(\frac{4}{3}\cdot\pi\cdot r^{3}\bigg) \\
\frac{dV}{dt}=4\pi r^{2} \frac{dr}{dt}\]
Step 3: We are told in the problem statement that diameter is 100m, so therefore r = 50mm. We are also told the radius of the sphere is increasing at a rate of 2mm/s, so therefore dr/dt = 2mm/s. We are looking for how fast the volume of the sphere is increasing, or dV/dt.
\[\frac{dV}{dt}=4\pi (50\ mm)^{2}\cdot(2\ mm/s)\\
\frac{dV}{dt}=68.832\ mm^{3}/s\]