A Rope, Two Telephone Poles... and Some Confounding Math

Apr 14, 2003

RAY: I promised a mathematical puzzler this week, so here it is.

TOM: Real numbers-and irrational conclusions, I bet.

RAY: Probably. There are two telephone poles. Each one is 100-feet tall. They are parallel-and an unknown distance apart.

We're going to attach a 150-foot rope from the very top of one of the poles, to the top of the other. This rope will, of course, droop down somewhat.

That drooping rope is called a "catenary," from the Latin word for chain.

TOM: Did they have chains in ancient Rome?

RAY: Of course! The lions were chained to the floor! No, that was the Christians.

So, we've got these two 100-foot poles, and a 150-foot rope. The rope is between the two poles, and it's going to droop down, making an arc.

The question is, what must be the distance between the two poles, so that the lowest point of this catenary is 25-feet above the ground?


RAY: The question is, what must the distance be between the two poles so that the lowest point of this catenary is 25 feet above the ground?

If you made the mistake that I made and you drew the picture, and then went immediately and started to look up what the equation for a catenary was, you weren't going to get the answer.

But if you thought about it for a little longer you would realize that, it's impossible for that rope to get down to 25 feet above the ground, unless the two poles are touching each other.

TOM: I'll bet you all the little nerds went right to their graphing calculators. What a dirty trick!

RAY: I'm sorry. It was a dirty trick. It wasn't really a mathematical puzzler.

Do we have a winner?

TOM: The winner is Kirsten Gilbert from Winter Park, Florida.

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