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?

Question

anonymous · Answer

$$Volume=\frac{4}{3}\pi r^3$$
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 $D=80~mm$:
$$\frac{dV}{dt}=\frac{1}{2}\pi (80~mm)^2\left(6\frac{mm}{s}\right)$$