Given:
The rate of change in volume = 12 cubic feet per minute.
We need to find the rate of change of radius at radius = 6 feet.
Consider the formula to find the volume of the sphere.
[tex]V=\frac{4}{3}\pi r^3[/tex]Differentiate with respect to t.
[tex]\frac{dV}{dt}=\frac{4}{3}\pi\times3r^2\times\frac{dr}{dt}[/tex][tex]\text{ Substitute }\frac{dV}{dt}=12\text{ and r=6 in the formula.}[/tex][tex]12=\frac{4}{3}\pi\times3(6)^2\times\frac{dr}{dt}[/tex][tex]12=144\pi\times\frac{dr}{dt}[/tex]Dividing both sides by 144pi, we get
[tex]\frac{12}{144\pi}=\frac{dr}{dt}[/tex][tex]\frac{dr}{dt}=\frac{1}{12\pi}[/tex][tex]\frac{dr}{dt}=\frac{1}{37.68}=0.0265\text{ feet per minute}[/tex]Hence the radius of the balloon increases by 0.03 feet per minute.
