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# In a Sphere the Rate of Change of Volume is (A) π Times the Rate of Change of Radius (B) Surface Area Times the Rate of Change of Diameter (C) Surface Area Times the Rate of Change of Radius - Mathematics

#### 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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