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A spherical ballon is filled with $4500 π$ cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the ratio of $72 π$ cubic meters per minute , then the rate (in meters per minute) at which the radius of the ballon decreases 49 minutes after the leakage began is

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$\frac{2}{9}$

$\frac{9}{2}$

$\frac{9}{7}$

$\frac{7}{9}$

answer is A.

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Detailed Solution

If r is the radius of the spherical baloon 49 minutes after the leakage began then

$4500 π - \frac{4 r^{2}}{3} π r^{3} = 72 π (49) = 3528 π$

$\Rightarrow \frac{4}{3} π r^{3} = 4500 π - 3528 π \Rightarrow \frac{4}{3} r^{3} = 972 \Rightarrow r = 9$

$V = \frac{4}{3} π^{3} \Rightarrow \frac{d V}{d t} = 4 π^{2} \frac{d r}{d t} \Rightarrow 72 π = 4 π (9)^{2} \frac{d r}{d t} = \frac{2}{9}$

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