# The Unfolding of a Non-Intuitive Problem

Below is a problem that one of my calculus classes tried solving (unsuccessfully) so we banded together and walked through a solution. The problem is this (from here):

If you have two flies on a deflated spherical balloon — one on the equator and one on the north pole — and the balloon inflating at a rate of 5 cubic centimeters a second, how fast are they moving apart from each other at some time $t_o$?

What I like about the problem is that it is looks as simple as all the other related rates problems they’ve done, but it actually gets pretty complex. And it gets tricky figuring out what you’re trying to solve for, unless you keep yourself organized. What I love most is that you’re given almost nothing, but you end up with an answer I’d call beautiful because it is so ugly. You start out with practically nothing and can get something so ugly out as answer? Awesome. Welcome to math, neophyes!

So we walked through the solution together — after they had a good amount of time a couple weeks ago to try to solve it. I gently asked a few questions prodding them and kept the information organized. What you see below is how the problem unfolded on the whiteboard.

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1. That sounds like a great problem! (I’m purposely not looking at the solution, because I want time to think about it…)

2. The constant gets ugly. The rest is the same sort of idea we work with all the time. But I got almostthere, thought “what’s the big deal?” and then saw, oh, right, in terms of time… :)

Btw, I rewrote the mess as the root of a single fraction, times t to the -2/3. The extra rearranging was fun.

Jonathan