## Friday, May 06, 2011

### Catenaries on "Car Talk"

Here's a puzzle from the radio show, Car Talk, with the misspelling corrected:

"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.

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

I confess, I worked out the equation for a catenary to try to answer this, but that's a waste of time if you think about the problem a little.

The answer, if you need it, is here.

KeithB said...

Did working out the math give you the "right" answer?

Jeffrey Shallit said...

I gave up once I drew a picture.

Anonymous said...

I got it all by myself! Would you say that this feat alone is enough to establish I would have success as a TCS grad student?

John said...

I had to draw it, too. Then I felt stupid for not seeing it right away.

cody said...

It took a minute of thinking, but I'm sure without your hint I would have been buried in the equation.

I guess I could try to answer the question for say 50 or 75 feet, where some more work would be required.

Anonymous said...

I am certainly the densest guy in the world. I first tried the equation, when it turned into kludge, I drew a diagram - and still didn't get it. Now 24 hrs later after drawing it again, I have figured it out! I'm feeling like a fool, a darn fool.

-Truti

Colin said...

Thought about it for a bit. Realised I was missing something obvious. Forgot about it. Nearly fell off my bike when I heard the right answer.

I like this one even more
http://www.cartalk.com/content/puzzler/transcripts/201008/index.html
because a) it's a bit less of a trick question and b) I got the answer pretty quickly.

Anonymous said...

To put a different spin on this problem: given the poles are 100 feet tall, what must the length of the cable be such that the catenary be 25 feet above the ground?

Anonymous said...

Since I apparently need more coffee, let me clarify the above question: the solution is unique only if you fix the distance between the poles, of course. So fix the distance to be 100 feet as well. Ugh, sorry about that...

Curt Cameron said...

Anonymous, that's no fun - that takes like, math to figure out.

Anonymous said...

They never said the poles are on the same plane. When I drew the picture, I drew them on a nice hillside, you know, and concluded it to be impossible in some cases.

AnonymousFan said...

To supplement my other Anonymous cohort, non-trivial solutions may* exist where the poles are situated on uneven planes. In industry speak, this is an inclined span. A "hill" is not required, only two different support points.

Diagram:
http://www.acasolutions.com/resource_center/white_papers/pdfs/J_Inclined%20Span%20Sag%20Example.pdf

As per this diagram, the sag point (xd,yd) would be a critical point of interest.

*I say "may" because I haven't the figured out the solution function. It would appear that the incline would have to be pretty steep. My guess from the sketch would be that the two poles would be quite close horizontally with the support points being farther vertically.

** Yes, I know it is not the point of this type of question.