1. Volume of a sphere

For a spherical balloon, $V=\frac{4}{3}\pi r^3$

Differentiate both sides with respect to time $t$: $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$

2. Volume as a function of time

Initially, the balloon contains $V_0=4500\pi$

Gas leaks out at the rate $\frac{dV}{dt}=-72\pi\ \text{m}^3/\text{min}$

So after $t$ minutes, $V(t)=4500\pi-72\pi t$

At $t=49$ minutes, $V=4500\pi-72\pi(49)$ $=4500\pi-3528\pi$ $=972\pi$

3. Find the radius at $t=49$

Using $\frac{4}{3}\pi r^3=972\pi$

Cancel $\pi$: $\frac{4}{3}r^3=972$ $r^3=972\cdot \frac{3}{4}=729$ $r=9$

4. Find $\dfrac{dr}{dt}$ at that instant

From $\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$

Substitute $\frac{dV}{dt}=-72\pi$ and $r=9$: $-72\pi=4\pi(9^2)\frac{dr}{dt}$ $-72\pi=4\pi(81)\frac{dr}{dt}$ $-72\pi=324\pi\frac{dr}{dt}$

Thus, $\frac{dr}{dt}=\frac{-72}{324}=-\frac{2}{9}$

Since the question asks for the rate at which the radius decreases, the magnitude is $\left|\frac{dr}{dt}\right|=\frac{2}{9}\ \text{m/min}$

5. Option check

• A: $\frac{9}{7}$ — incorrect
• B: $\frac{7}{9}$ — incorrect
• C: $\frac{2}{9}$ — correct
• D: $\frac{9}{2}$ — incorrect

Therefore, the correct option is C.