A balloon which always remains spherical,


A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing when the radius is 15 cm.


Let r be the radius and V be the volume of the spherical balloon at any time t. Then, 

$\mathrm{V}=\frac{4}{3} \pi r^{3}$

$\Rightarrow \frac{d V}{d t}=4 \pi r^{2} \frac{d r}{d t}$

$\Rightarrow \frac{d r}{d t}=\left(\frac{1}{4 \pi r^{2}}\right) \frac{d V}{d t}$


$\Rightarrow \frac{d r}{d t}=\frac{900}{4 \pi(15)^{2}}$                 $\left[\because r=15 \mathrm{~cm}\right.$ and $\left.\frac{d V}{d t}=900 \mathrm{~cm}^{3} / \mathrm{sec}\right]$

$\Rightarrow \frac{d r}{d t}=\frac{900}{900 \pi}$


$\Rightarrow \frac{d r}{d t}=\frac{1}{\pi} \mathrm{cm} / \mathrm{sec}$

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