The volume of the sphere is given by the formula
[tex]V=\frac{4}{3}\pi r^3[/tex]
In function notation
[tex]V(r)=\frac{4}{3}\pi r^3[/tex]
Find out dV/dt
Remember that
[tex]\begin{gathered} \frac{dV}{dt}=\frac{dV}{dr}*\frac{dr}{dt} \\ \\ \frac{dV}{dt}=400\pi\frac{ft^3}{min} \end{gathered}[/tex]
Find out dV/dr
[tex]\frac{dV}{dr}=4\pi r^2[/tex]
substitute
[tex]\begin{gathered} 400\pi=(4\pi r^2)\frac{dr}{dt} \\ \\ \frac{dr}{dt}=\frac{400\pi}{4\pi r^2} \\ \\ simplify \\ \\ \frac{dr}{dt}=\frac{100}{r^2} \end{gathered}[/tex]
Evaluate at r=5 ft
[tex]\frac{dr}{dt}=\frac{100}{5^2}=4\frac{ft}{min}[/tex]
therefore
The balloon's radius is increasing at a rate of 4 ft/min
