E-RC01 · Constant volume rate, changing length rate
A spherical balloon has \(\displaystyle V=\frac43\pi r^3\) and surface area \(A=4\pi r^2\). Its volume increases at the constant rate \(48\pi\ \mathrm{cm^3\,s^{-1}}\).
Find \(\displaystyle \frac{dr}{dt}\) when \(\displaystyle r=2 \mathrm{cm}\).
Find \(\displaystyle \frac{dA}{dt}\) at the same instant.
Find the radius rate and area rate when \(\displaystyle r=4 \mathrm{cm}\), and explain whether a constant volume rate forces either of these other rates to be constant.
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