The Cow And The Lighthouse

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:

This problem is much more difficult than it at-first seems.
The southern portion of the cow's traverse is easy; the northern portion, however, is a whole different story!
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.
Don't look for hints in chapter 1 or 2 of your calculus book; try chapter 15 or 20 instead.
The concepts of "involute" and "evolute" (in the mathematical sense) are fascinating, are they not? I always enjoy "reciprocal" relationships like that.
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. Never-the-less, I strongly recommend trying to solve this problem for yoruself. To check your solution against mine (or, if you just want to see how I solved this), see here:

Solution

Cheers,
Robbie Hatley