There's a problem I really like whose result always surprises me. First, imagine you have a piece of string that’s long enough to stretch all the way around a basketball (the circumference of a basketball is 30 inches). Then you realize you have an extra 24 inches of string in your pocket, which you want to add to the string. So, you cut the circle of string somewhere, add exactly 24 inches, and then smooth it out until it makes a circle all around the ball (kind of like a ring orbiting a planet). The question is to figure out how high is the string off the basketball? It's a simple geometry calculation, and you wind up getting around 3.8 inches.
Got it? Now, try the same problem, except instead of a basketball imagine that you start by wrapping a string around the equator of the earth. Then, just as before, you find an extra 24 inches of string in your pocket, which you add on to the string, and then smooth out the resulting string until it makes a circle around the earth. How high is the string off the earth?
My intuition always tells me that the gap should be minuscule - after all, what is a mere 24 inches compared with a 25,000 mile equator? But of course, every time, the answer comes out to be ... around 3.8 inches.This definitely falls under the category of "I can prove it, but I don't believe it!"
It occurred to me that there just HAD to be some calculus in this problem (is there a problem for which that couldn't be said?), and lo and behold I found it. For their weekly challenge set, I gave my students the basketball/earth problems, and also this last one: Express the radius of a sphere as a function of its circumference, and then find the derivative of this function. Why does this make sense in light of your previous answers?
The answer is that r(c)=c/(2*pi), so r'(c) is just the constant 1/(2*pi)! So, regardless of the starting circumference, a constant change in circumference will result in a constant change in the radius. Neat, huh?
So I get it. I can prove it with calculus and without calculus. Yet somehow, I still don't really understand how this can be true. And I have to admit, after years doing math calculations that "give" the answer, it's always refreshing to come across one that doesn't...