A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π 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 :

Engineering

Mathematics

Rate Measure

Question

A spherical balloon is filled with 4500π cubic meters of helium gas. If a leak in the balloon causes the gas to escape at the rate of 72π 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 :

A

$\frac{9}{2}$

B

$\frac{7}{9}$

C

$\frac{9}{7}$

D

$\frac{2}{9}$

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Given $\frac{- dV}{dt} = 72 π$

t = 0

Volume of gas = 4500 $π$

t = 49 minute,

Volume of gas = 72 $π$ × 49 = 3528 $π$

$∴$ After 49 minute volume of gas inside balloon = (4500 $π$ – 3528 $π$ ) = 972 $π$

$∴ \frac{4}{3} [r^{3} = 972 π$ $∴$ r = 9m

Now, $V = \frac{4}{3} {πr}^{3}$

$\frac{dV}{dt} = 4 {πr}^{2} \frac{dr}{dt} - 72 π = 4 π (9^{2}) \frac{dr}{dt}$

$\Rightarrow \frac{dr}{dt} = \frac{- 2}{9}$