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Question
The radius of an air bubble is increasing at the rate of 0.5 cm/sec. At what rate is the volume of the bubble increasing when the radius is 1 cm?
Solution
\[\text { Let r be the radius and V be the volume of the air bubble at any time t. Then },\]
\[V=\frac{4}{3}\pi r^3 \]
\[\Rightarrow\frac{dV}{dt}=4\pi r^2 \frac{dr}{dt}\]
\[\Rightarrow\frac{dV}{dt}=4\pi \left( 1 \right)^2 \times 0.5\left( \because r = 1 \text{cm and } \frac{dr}{dt} = 0 . 5 cm/\sec \right)\]
\[\Rightarrow\frac{dV}{dt} {=2\pi cm}^3 /sec\]
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