Question

Surface Area of a Balloon The surface area S (in square meters) of a hot-air balloon is given by

S(r) = 4π r2

where r is the radius of the balloon (in meters). If the radius r is increasing with time t (in seconds) according to the formula , t ≥ 0, find the surface area S of the balloon as a function of the time t.

Answer 1

You have the formula:
\[S=4 \pi r ^{2}\]
since radius is increasing with time and S is a function of time
therefore both will be differentiated with dt
\[S=4 \pi r ^{2}\]
\[\frac{ dS }{ dt } = 4 \pi (2r) \frac{ dr }{ dt }\]
\[\frac{ dS }{ dt } = 8 \pi r \frac{ dr }{ dt }\]