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A spherical balloon is expanding. If the radius is increasing at the rate of 5 inch per minute, the rate at which the volume increases (in cubic inches per minute) when the radius is 10 inch is

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$100 π$

1000 $π$

$2000 π$

$500 π$

answer is C.

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Detailed Solution

The volume $V = \frac{4}{3} π r^{3}, r$ being the radius. It is given that $\frac{d r}{d t} = 5$ inch/min and we have to find $\frac{d V}{d t}$ when $r = 5$ inches.

$\frac{d V}{d t} = 4 π r^{2} \frac{d r}{d t}$

${\Rightarrow \frac{d V}{d t}|}_{r = 10} = 4 π (100) 5 = 2000 π$

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