Start with volume equation for sphere
\(V = \frac{4}{3} \pi r^3\)
Take derivative with respect to time (Implicit differentiation)
\(\frac{dV}{dt} = 4 \pi r^2 \frac{dr}{dt}\)
Sub in rate for Volume of 100 and given radius of 5, solve for dr/dt
\(\frac{dr}{dt} = \frac{100}{4 \pi 5^2} = \frac{1}{\pi}\)
Finally we have to relate this to rate diameter is changing.
D = 2r
\(\frac{dD}{dt} = 2 \frac{dr}{dt} = \frac{2}{\pi}\)
Final Answer:
Diameter is increasing at a rate of 2/pi cm/sec
