Question

To find the rate at which the radius of the balloon increases when the radius is $15$cm. A balloon which always remains spherical is being inflated by pumping in $900c{m}^3$ of gas per second.

Answer 1

$\begin{array}{c}Radiusofballon=15cm\\ Aballonissphericalbypumpingin900c{m}^3\\ RateofVolumeofsphericalballonis\left(\forall \right)\\ =\frac{4}{3}\pi r^3\\ Then,rateofradius\left[\frac{dr}{dt}\right].\\ \Rightarrow V=\frac{4}{3}\pi r^3\\ differentiatewithrespectto\left(time\right)\\ \Rightarrow \frac{dv}{dt}=\frac{4}{3}\pi \frac{d{r}^3}{dt}\\ 900=\frac{4}{3}\pi \times 3r^3\frac{dr}{dt}\\ 900=4\pi \times {\left(15\right)}^2\frac{dr}{dt}\\ \therefore Rateofradiusofballon\left(\frac{dr}{dt}\right)=\frac{900}{225\times 4\pi }\\ \frac{dr}{dt}=\frac{1}{\pi }cm/sAns.\end{array}$