Something for a Friday Afternoon

Everyone knows the once-counterintuitive examples using exponentiation. Here’s one:

  • Would you rather have a million dollars, or a penny doubled each day for thirty days?

This has been a yawn for quite a while now, so don’t we need a replacement for it?

Mathematics professor Lillian Lieber included a form of the exponentiation example in one of her books. I can’t remember whether it was The Education of T. C. Mits [The Celebrated Man In The Street] or Mits, Wits, and Logic [Wits = Woman In The Street].

As it happens, she also gave another counterintuitive example.

Here’s how it went:

  • Suppose we wrap a belt around the earth, then let out 10 meters of slack, and distribute the belt evenly around the earth so that its “altitude” is everywhere the same. What is the altitude of the belt?

Okay. To begin with, let c and r be the circumference and radius of the earth, and let C and R be the new, enlarged circumference and radius of the encircling belt.

The altitude we’re trying to find will be the difference of the two radii:

R – r

From geometry, we know that

R = C / 2π
r = c / 2π

So

R – r = C / 2π – c / 2π = (C – c) / 2π

From the beginning, we’ve known the difference of the two circumferences

C – c = 10 meters

So the altitude

C – c = 10 meters / 2π ~= 1.6 meters

I thought it surprising that the altitude would be so large. (I expected it to be tiny.) But what I found really surprising is that we did not need to know the circumference or diameter of the earth!

That means that if we’d wrapped a basketball, or even something the size of the sun, the altitude would still be the same!

Counterintuitive?

I thought so!

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