Question

A balloon which always remains 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

Answer: (A)

We have, $\frac{dV}{dt}=10c{m}^3/s$
$r=15cm$
$\therefore V=\frac{4}{3}\pi r^3$
$\Rightarrow \frac{dV}{dr}=4\pi r^2=4\pi (15{)}^2=900\pi$
Now, $\frac{dV}{dt}=\frac{dV}{dr}\times \frac{dr}{dt}$
$\Rightarrow 10=900\pi \times \frac{dr}{dt}$
$\Rightarrow \frac{dr}{dt}=\frac{1}{90\pi }cm/s$