Question

A balloon which always remain spherical is being inflated by pumping in $10$ cubic centimeters of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is $15$ cms.

• A. $\frac{1}{90\pi }$ cm/sec
• B. $\frac{1}{9\pi }$ cm/sec
• C. $\frac{1}{30\pi }$ cm/sec
• D. $\frac{1}{\pi }$ cm/sec

Answer 1

The correct option is A $\frac{1}{90\pi }$ cm/sec

Let $r$ be the radius of the balloon.
Then Volume, $V=\frac{4}{3}\pi r^3$
$\Rightarrow \frac{dV}{dt}=4\pi r^2\frac{dr}{dt}\cdots \left(1\right)$
$\text{Given},\frac{dV}{dt}=10$
$\Rightarrow \frac{dr}{dt}=\frac{5}{2\pi r^2}\left[\text{From}\right(1\left)\right]$
$\therefore \frac{dr}{dt}{]}_{r=15}=\frac{1}{90\pi }\text{cm/sec}$