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Question
In a sphere the rate of change of volume is
Options
π times the rate of change of radius
surface area times the rate of change of diameter
surface area times the rate of change of radius
none of these
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Solution
surface area times the rate of change of radius
\[\text { Let r be the radius andVbe the volume of sphere at any time t.Then },\]
\[V = \frac{4}{3}\pi r^3 \]
\[ \Rightarrow \frac{dV}{dt} = \frac{4}{3}\left( 3\pi r^2 \right)\left( \frac{dr}{dt} \right)\]
\[ \Rightarrow \frac{dV}{dt} = 4\pi r^2 \left( \frac{dr}{dt} \right)\]
\[\text { Thus, the rate of change of volume is surface area times the rate of change of the radius }.\]
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