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Q.

If the surface area of a sphere of radius $r$ is increasing uniformly at the rate $8 c m^{2} / s e c$ , then the rate of change of its volume is :

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a

proportional to $\sqrt{r}$

b

proportional to $r^{2}$

c

constant

d

proportional to $r$

answer is C.

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Detailed Solution

$S = 4 {πr}^{2}$ ; $\frac{d s}{d t} = 8 \Rightarrow \frac{d r}{d t} = \frac{1}{πr}$

Now, $v = \frac{4}{3} {πr}^{3} \Rightarrow \frac{d v}{d t} = 4 π^{2} \frac{d r}{d t} = 4 r$

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