# The radius of a sphere is increasing at a rate

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?

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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. $\frac{dV}{dt}=4\pi (50\ mm)^{2}\cdot(2\ mm/s)\\ \frac{dV}{dt}=68.832\ mm^{3}/s$