Using the Definition to Compute the Derivative - Exercise 5


Exercise 5. A spherical balloon is being inflated. Find the rate at which its volume V is changing with respect to the radius.

Answer. The volume of the sphere is

\begin{displaymath}V = \frac{4 \pi}{3} r^3\;.\end{displaymath}

So the desired rate of change is given by

\begin{displaymath}\frac{dV}{dr} = 4 \pi r^2\;.\end{displaymath}

In particular, when r = 5 ft, the rate of change is

$\displaystyle {\frac{dV}{dr}}$ = 100$\displaystyle \pi$  $\displaystyle {\frac{\mbox{ft}^3}{\mbox{sec}}}$,

if time is measured in seconds.


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