A Problem with a Goat and a Rope
I heard some news the other day. A man named Ingo Ullisch, a German mathematician, has solved The Tethered Goat Problem. Does that mark a significant achievement in the advancement of science? Probably not, but Ullisch did some excellent work.
The problem seems pretty straightforward. You are presented with a goat that is tethered to a fence post on the edge of an enclosed circular space. In this case, the grazing area is precisely 1 acre. The question is: How long a rope do you need such that the goat can graze on exactly ½ of the available land? I have included a figure below.
Oddly enough, I had never heard of this problem. I decided to sit down to see if I could solve it. I used analytic geometry to come up with my answer. It didn’t take me long, I got the answer on my 6th attempt of informed estimation. Of course, there is a problem; my answer is considered an approximation; it is simply not good enough for the Mathematical Gods. They tend to be sticklers for precision.
Ullisch used Complex Analysis to get his answer. He has imaginary numbers floating around his equations in an “imaginative” way. This strategy leads him to what mathematicians call a Closed-Form solution to the problem. Now The Gods are happy, and life can get back to normal.
That is about it for this little problem. I will say this, without a computer and a piece of software written to do analytic geometry, I would never have attempted to find a solution. After all, The Collatz Conjecture is still out there, mocking me and everyone else who has heard of it.
I do not see a date on your drawing (plat)? Am I missing something?
Just an old (retired)surveyor with a question.
Mr. Nolton,
The plat was finished on 12/28/ 2021. The “Goat Problem” is a perfect one for old surveyors. My dad is 85 and he also worked on it. I was happy to have a COGO program, of course, my dad wouldn’t need one to solve it. As you know, Old School Surveyors can work magic with a pocket calculator.
RTNM