Gas is being pumped into a spherical balloon at the rate of 30 ft³/min. Then the rate at which the radius increases when it reaches the value 15 ft, is

$\frac{1}{30 \pi}$ ft/min

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

It is given that $\frac{d V}{d t}$ = 30 ft³/min

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

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

$\Rightarrow \left(\frac{d V}{d t}\right)_{r=15}=4 \pi \times 15^2 \times\left(\frac{d r}{d t}\right)_{r=15}$

$\Rightarrow 30=900 \pi\left(\frac{d r}{d t}\right)_{r=15}$

$\Rightarrow \left(\frac{d r}{d t}\right)_{r=15}=\frac{1}{30 \pi}$ ft/min