Question

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

• A. $6/7$
• B. $4/9$
• C. $2/9$
• D. $Noneofthese$

Answer 1

The correct option is C $2/9$

The volume of spherical balloon is given by $V=\frac{4}{3}\pi R^3$
Initial volume $=4500\pi$
After $49$ minutes volume$=4500\pi -49\left(72\pi \right)=972\pi$
By solving these above equations, we get $r=9$ (Radius of the baloon $49$ minutes after the leakage)
$\frac{dV}{dt}=\frac{4}{3}\pi 3r^2\frac{dr}{dt}$
$\Rightarrow 72\pi =\frac{4}{3}\pi 3r^2\frac{dr}{dt}$
$\Rightarrow \frac{dr}{dt}=\frac{2}{9}$