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Target audience: Everyone, with some calculus helpful at the end

It's no secret that I'm usually pretty critical of American science journalism. Naturally, however, I must make one exception. NPR's Car Talk has some of the best math coverage out there. For those of you who are unfamiliar with the show, it's a call-in program where you can get your car problems fixed by a couple of irreverant car mechanics who graduated from MIT.

The show has a weekly puzzler component. True, some of the puzzles require specialized knowledge of cars that most people might not have, but being MIT types, the guys can't help throwing in a few math puzzles as well. I'm basing this post on the puzzle which follows:

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

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 catanary is 25-feet above the ground?

As it turns out, there's a clever trick to this problem that will keep you from having to pull out your fancy hyperbolic sine functions. Anyone should be able to solve it with a little bit of creative thought.

Of course, if you're stumped, and you have some background in differential equations, you can crank out the answer (at which point you will feel awfully silly for not having seen the answer previously). Since this is the way I solved it the first time, I'll go ahead and bring up a couple of things that go into the solution.

The shape in which a telephone line hangs is called the catenary. Gallileo was the first to really consider such a shape. He thought that the catenary curve was a parabola (e.g. y(x) = -y₀ - a x² for some positive coefficient 'a'), which is an understandable mistake as the two look very similar. Remember that was living in the Golden Age of the Polynomial, in which almost everything in physics was falling into place as the result of simple polynomial models. Also, he didn't have calculus, or the notion of the hyperbolic plane, or e, so he wouldn't have been able to derive the real result, which is what we call the hyperbolic cosine function. You probably have seen the button on your calculator labelled "cosh," and wondered what it does. Well, that's what it's for. The cosh function is closely related to the natural exponential, by the equation

cosh(x) = (e^x + e^-x)/2

The cosh function is so-named because it is the analogue of the regular cosine function from trigonometry when you're working in the hyperbolic plane, but non-Riemannian geometry is a story for another day. To a calculus student, it's interesting because it's equal to its own second derivative, a fact which isn't hard to check from the formula above.

Quick question: we know about functions which are equal to their first (e^x), second (sinh and cosh), and fourth (sin, cos) derivatives. But what about functions which are equal to, say, their third derivatives? I have some graphs that I'll post later; I'm going to wait a while so that people can think it over without the temptation of skipping right to the solution.

These days, the derivation of the catenary is a standard exercise in calculus based physics: consider string tension and gravity at each point in the string with a free-body diagram, produce a differential equation, and solve it; anyone who would understand the derivation has probably already done it, so I won't try to jam it in here.

You can check out the Cartalk puzzler archive for a bunch more interesting puzzles; if that doesn't do you, you can buy their book below. My dad got this for his birthday (?) a couple years back and seemed to enjoy it. Plus the listed prices on Amazon are now absurdly low (< $1), so there's no reason not to grab a copy. Car Talk airs on your local NPR affiliate, or you can listen online.




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