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

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9/2

9/7

7/9

2/9

answer is C.

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Detailed Solution

$\begin{matrix} Volume V = 4500 π \\ \frac{dv}{dt} = 75 π \end{matrix} \begin{matrix} 1 \min = 72 π \\ 49 \min = 49 \times 72 π \\ = 3528 π \\ remains gas \end{matrix} V = \frac{4}{3} {πr}^{3} = 4500 π - 3528 π 972 π = \frac{4}{3} π r^{3} = 972 π \Rightarrow r^{3} = 729 \Rightarrow r = 9 \frac{dv}{dt} = 4 {πr}^{2} \frac{dr}{dt} 72 π = 4 π \times 81 \frac{dr}{dt} \frac{dr}{dt} = \frac{2}{9}$