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 $49$ minutes after the leakage began is.

• A. $\frac{9}{2}$
• B. $\frac{9}{7}$
• C. $\frac{7}{9}$
• D. $\frac{2}{9}$

Answer 1

The correct option is D $\frac{2}{9}$

$V=\frac{4}{3}{\pi r}^2$

After $49$ minutes volume $=4500\pi -49\left(72\pi \right)=972\pi$

$\frac{4}{3}{\pi r}^3=972\pi \Rightarrow r^3=729\Rightarrow r=9$

$V=\frac{4}{3}{\pi r}^2\Rightarrow \frac{dv}{dt}=\frac{4}{3}\pi 3r^2\frac{dr}{dt}\Rightarrow 72\pi ={4\pi r}^2\frac{dr}{dt}.$

$\Rightarrow \frac{dr}{dt}=\frac{72}{\mathrm{4.9.9}}=\frac{2}{9}$