Illustration 1
This is adapted from The Education of T.C. Mits, by Lillian Lieber
Here’s the question again: Presume that you have a steel band that fits tightly around a perfectly spherical earth at the equator. That band will be about 25,000 miles long. Now cut open the band and add 10 feet to its length, and weld it back together again. (That’s 10 feet, not miles.) That leaves some room between the earth and the band. If you center the earth in this slightly larger band, how much room will there be?
Most people’s common sense will tell them that the answer is either C or D. After all, you are adding only 10 feet to 25,000 miles, so it must be a very small gap indeed. However, most people’s common sense is wrong. The answer is B: Enough room for a cat to walk under the band.
Illustration 1 shows the situation at the start. We know from basic geometry that the circumference of a circle is 2π times the radius. Also, 25,000 miles works out to 132,000,000 feet.
| (1) | 2πr | = | 132,000,000 feet |
After adding the band, we have the situation shown in Illustration 2 (not drawn to scale, obviously). We need to find the size of the gap, labeled x in the illustration. Here are the steps to solve for x.
| Step | Equation | Reason | ||
|---|---|---|---|---|
| (2) | 2π(r + x) | = | 132,000,010 feet | Formula for circumference |
| (3) | 2πr + 2πx | = | 132,000,010 feet | Distributing the multiplication |
| (4) | 132,000,000 + 2πx | = | 132,000,010 feet | Substitute from equation (1) |
| (5) | 2πx | = | 10 feet | Subtract 132,000,000 from both sides |
Surprise! The original radius of the earth doesn’t matter at all. It has disappeared from the equation entirely. Solving for x is now a simple matter of division:
| (6) | x | = | 10 feet / (2π) |
Using 3.14159 as an approximation for π we get x = 1.59 feet. That’s certainly more than enough room for a piece of paper, not enough for a human, but plenty for a cat. (Yes, I actually got a ruler to test this; my larger cat is less than 1.5 feet tall at the shoulder.)
Many of you who chose “piece of paper” or “none of the above” are still convinced that your common sense is right. You think I have pulled some sneaky trick like those “proofs” that show that 1 = 0. If you are one of those people, I suggest you print out this page with just the math on it and take it to a math professor and ask her to tell you whether the solution is correct or not.