The Cow And The Lighthouse

A Mathematical Brainteaser

Situation:

Somewhere in North Carolina, a flat-topped peninsula juts out into the Atlantic Ocean. Atop the peninsula is an ancient cylindrical stone lighthouse of radius R. The lighthouse is surrounded by a large lawn of tall green grass. To keep the grass mowed, the lighthouse-keeper has tethered a cow named Betsy to the south side of the lighthouse with a chain of length L. Assume that all areas accessible by the cow are flat, level, and covered with lush, green grass. Also assume that the length of the chain is less than or equal to one-half the circumference of the lighthouse.

Question:

What is the area, A, of the grass in which Betsy can graze? Express A algebraically in terms of R and L. Show your work.

Hints:

1. This problem is much more difficult than it at-first seems.
2. The southern portion of the cow's traverse is easy; the northern portion, however, is a whole different story!
3. To the best of my knowledge, this problem cannot be solved using high-school level math; you're going to have to break open a calculus textbook for this one.
4. Don't look for hints in chapter 1 or 2 of your calculus book; try chapter 15 or 20 instead.
5. The concepts of "involute" and "evolute" (in the mathematical sense) are fascinating, are they not? I always enjoy "reciprocal" relationships like that.
6. Determinants are a very useful mathematical tool.

Don't be too depressed if you can't solve this problem right away. I learned of this problem around February of 1974, but I didn't actually figure out the solution until around February of 1989, 15 years later. I have a complete solution worked out on paper. If you're really desperate, I'll email you jpg pictures of the four pages of my solution. (I bought a scanner in December of 2005, so I was able to make really nice full-color scans of my original 1989 solution, in grey graphite on yellow legal-pad paper, complete with wrinkles, tears, and smears. Rather quaint, actually.)

So go ahead and send me your solution. My contact info may be found at:

Update as of January 2006: Several people have sent me solutions over the past few years, all of them incorrect. (Though one person, who's first name is "Sam" ;-) sent me a solution that's almost correct.) So the honor of being to first person to send me the correct solution is still up for grabs. Surely there are some folks out there who can make creative use of multivariable calculus? (Or perhaps you know another method of solution?) Speak up.

Cheers,
Robbie Hatley

Written by Robbie Hatley circa 1996.
Added to MMM circa January 1998.
Last updated Monday August 3, 2020.

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