Question

A balloon, which always remains spherical on inflation, is being inflated by pumping in $900$ cubic centimetres of gas per second. Find the rate at which the radius of the balloon increases when the radius is $15\text{cm}$.

Answer 1

Let $x$be the radius of the spherical balloon Volume of balloon is,

$V=\frac{4}{3}\pi x^3$

Given,

$\frac{dV}{dt}=900$

$\Rightarrow \frac{d}{dt}\left(\frac{4}{3}\pi x^3\right)=900$

$\Rightarrow \frac{4\pi }{3}.\frac{d}{dt}\left(x^3\right)=900$

$\Rightarrow \frac{d}{dt}\left(x^3\right)=900\times \frac{3}{4\pi }$

$\Rightarrow 3x^2\frac{dx}{dt}=900\times \frac{3}{4\pi }$

When $x=15\text{cm}$

$\Rightarrow (15{)}^2\frac{dx}{dt}=900\times \frac{1}{4\pi }$

$\Rightarrow 225\frac{dx}{dt}=\frac{225}{\pi }$

$\therefore \frac{dx}{dt}=\frac{1}{\pi }\text{cm/s}$

Hence, radius of the balloon is increasing at the rate of

$\frac{1}{\pi }\text{cm/s}$