Question

A spherical balloon is inflated at a rate of 32pi/3 cubic feet per minute.
a. Assuming it was empty at the beginning of this experiment, after one minute how fast is the radius increasing?
b. When is the radius increasing the fastest and why?

Answer 1

Simply what you do when you encounter problems like these is to think of the smallest possible formulas to relate the variables you're interested in. In this case it's Volume and Radius which are related by V=4/3pi*r^3

Now let's see, they give us dV/dt and we want dr/dt for part a. Unfortunately we don't have time in our equation that relates volume to radius. That's no problem, because we can kind of make up a little "cheat" for ourselves:

\[\frac{ dV }{ dt }=\frac{ dV }{ dr }*\frac{ dr }{ dt }\]

See how on the right side that the "dr" term cancels out if you were to multiply them as if they were fractions? We're allowed to do that and make these up when we need them!

The whole point of making that up is because we can take the derivative of volume with respect to radius AND we have dr/dt which is exactly what we need!

\[V=\frac{ 4\pi }{ 3 }r^3\]
differentiating gives...
\[\frac{ dV }{ dr }=4\pi*r^2\]
So now you can divide dV/dt by dV/dr to get dr/dt (just a little algebra of the thing we made, don't be scared.

\[\frac{ (\frac{ dV }{ dt }) }{ (\frac{ dV }{ dr }) }=\frac{ dr }{ dt }\]

Plug it all in and figure it out.

For part b you can see for yourself at which point the radius is increasing the fastest. Just look at the function of dr/dt and ask yourself, is this number higher or lower at r=0, r=infinity or r=what?