Single Variable Calculus, Chapter 1, 1.3, Section 1.3, Problem 54

Assume that a spherical balloon is being inflated and the radius of the balloon is increasing at a rate of 2 cm/s.

(a) We need to express the radius $r$ of the balloon as a function of the time $t$ (in seconds).


$\begin{equation} \begin{aligned} r(t) =& 2t\\ \end{aligned} \end{equation}$


(b) find $V \circ r$ and explain, if $V$ is the volume of the balloon as a function of the radius.


$\begin{equation} \begin{aligned} Volume =& \displaystyle \frac{4}{3} \pi r^3; \text{ but } r=2t\\ Volume =& \displaystyle \frac{4}{3} \pi (2t)^3\\ Volume =& \displaystyle \frac{32 \pi}{3} t^3, \text{ volume of the sphere as a function of time.} \end{aligned} \end{equation}$