Answer
Explanation
Given:
A spherical balloon is inflated with gas at the rate of 800 cubic centimeters per minute means
[tex]\frac{dV}{dt}=800\text{ }cm^3\text{/}min[/tex](a) The rates of change of the radius when r = 30 centimeters and r = 85 centimeters is calculated as follows:
[tex]\begin{gathered} V=\frac{4}{3}\pi r^3 \\ \\ \frac{dV}{dr}=\frac{4}{3}\times3\pi r^{3-1} \\ \\ \frac{dV}{dr}=4\pi r^2 \\ \\ But\frac{\text{ }dV}{dr}=\frac{dV}{dt}\div\frac{dr}{dt} \end{gathered}[/tex]So when r = 30, we have
[tex]\begin{gathered} \frac{dV}{dr}=4\pi(30)^2 \\ \\ \frac{dV}{dr}=4\times\pi\times900 \\ \\ \frac{dV}{dr}=3600\pi \\ \\ From\text{ }\frac{dV}{dr}=\frac{dV}{dt}\div\frac{dr}{dt} \\ \\ Putting\text{ }\frac{dV}{dt}=800,\text{ }we\text{ }have \\ \\ 3600\pi=800\div\frac{dr}{dt} \\ \\ \frac{dr}{dt}=\frac{800}{3600\pi}=\frac{800}{3600\times3.14} \\ \\ \frac{dr}{dt}=0.071\text{ }cm\text{/}min \end{gathered}[/tex]Therefore, the rate of change of the radius when r = 30 is dr/dt = 0.071 cm/min.
For when r = 25 cm, the rate of change is:
[tex][/tex]