Question

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

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

Answer 1

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

Let radius of the balloon be $r$,
Volume of sphere$\left(V\right)=\frac{4}{3}$$\pi$$r^3$
$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\cdots \left(1\right)$

Volume after $49$ minutes$\left(V_{49}\right)=4500\pi -72\cdot 49\pi$ cubic metres.
$\frac{4}{3}\pi (r_{49}{)}^3=972\pi$
$r_{49}=9\text{m}$

Now, from equation $\left(1\right)$
$\frac{dV}{dt}=4\pi r^2\frac{dr}{dt}$
$72\pi =4\pi (r_{49}{)}^2\frac{d{r}_{49}}{dt}$
$\frac{d{r}_{49}}{dt}=\frac{18}{9.9}$
$\Rightarrow \frac{d{r}_{49}}{dt}$ = $\frac{2}{9}\text{m}$ per minute