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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
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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