Question

The radius of a sphere is increasing at a rate of 3 mm/s. How fast is the volume increasing when the diameter is 80 mm?

Answer 1

\[Volume=\frac{4}{3}\pi r^3\]
In terms of the diameter (\(D=2r\)):
\[V=\frac{4}{3}\pi\left(\frac{D}{2}\right)^3\\
V=\frac{1}{6}\pi D^3\]
Implicit differentiation:
\[\frac{dV}{dt}=\frac{1}{2}\pi D^2\frac{dD}{dt}\]
The radius is increasing at 3 mm/s, so
\[\frac{dr}{dt}=3~\Rightarrow~\frac{dD}{dt}=6\]
Now plug in \(\dfrac{dD}{dt}\) and the given \(D=80~mm\):
\[\frac{dV}{dt}=\frac{1}{2}\pi (80~mm)^2\left(6\frac{mm}{s}\right)\]