My Favorite Algebra Problem

I have an algebra problem that my adviser, who was also my calculus professor for 2nd semester calculus, showed me one time.

Over time I've grown to like this problem more and more, and I'll explain why after I describe it.

The Problem:

Assume the earth is a perfect sphere. Now imagine a rope tied snugly against the equator. Now take some addition rope and splice in 10 additional feet to the rope and then spread out the additional rope evenly so that the rope is same distance above the surface of the earth all the way around. (Imagine when you have a very loose belt and hold it out so that it is the same distance away from you waist all the way around.)

So now the question. Would you be able to crawl under the rope around the earth then? Answer quickly - what does your intuition say?

The Answer:

The answer is, surprisingly, yes! In fact you will have ~1.6 feet of space to crawl under.

Now I don't know about you, but this makes no intuitive sense to me, 10 feet evenly spread out over such a huge object? But it's true, and here's the proof.

We know the the formula for the circumference is C=2πr where r is the radius of the sphere. So let's call the radius of the earth R. If that's the case, then:

2πR + 10 = the new circumference

Now to determine the height this gives us above the earth we'll use the variable x to represent the additional radius we get, so then we can express the circumference a 2nd way:

2π(R + x) = the new circumference

So now we can make an equation and solve.

2π(R + x)=2πR + 10
Next expand the right hand side

2πR + 2πx = 2πR + 10
Notice the 2πR on both sides - get rid of it

2πx = 10
Now just divide by 2π

x = 10/2π
And we're done, the additional height is 10/2π or ~1.6 feet

This is one of the things I love about math! That something can run completely against your intuition, and you can prove it's true. I love those kinds of math moments.

Even better, this is a cool problem you can do with any Algebra class.

But now for the things I love even more about this problem. Did you notice how the terms 2πR canceled out? Do you understand what that means? That means the size of the original object does not matter. That this experiment would have the same result whether the original sphere was the size of a marble, basketball, earth or the galaxy! Isn't that amazing? That's even more counter-intuitive! You could even have a class try this experiment with different size circles out on the playground to demonstrate this in the real world.

Next is the part that makes this problem all the richer. Once students have learned calculus, you can actually revisit this problem. Now you can look at the formula for the circumference and ask - what's the derivative?

Since C=2πR then the derivative is C'=2π

It's a constant! This shows that a change in the radius is simply multiplied by a constant to determine what kind of change you see in the circumference and the size of the original object is not part of it! Now you have a clear understanding (if you understand calculus) as to why the size of the original object didn't matter.

I hope you enjoyed this problem too. And the next time you see a kid in need of something in mathematics that might intrigue them, now you've got something you can really wow them with.