The goat problem

[Fred_Kohlhepp]

Imagine a circular fence that encloses one acre of grass. If you tie a goat to the inside of the fence, how long a rope do you need to allow the animal access to exactly half an acre?

https://www.quantamagazine.org/mathematician-solves-centuries-old-grazing-goat-problem-exactly-20201209/

[Werner_E · Accepted solution]

I wasn't aware of this new exact solution to the problem - thanks for bringing this to our attention.

Obviously we cannot expect Mathcad to give us the "exact" solution, but it sure is good enough for an approximation:

By the way, the wiki entry for the goat problem has already been supplemented by Ingo Ullisch's closed form solution -> https://en.wikipedia.org/wiki/Goat_problem#Closed-form_solution

[Fred_Kohlhepp]

Okay, where's my mistake?

[Werner_E]

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@Fred_Kohlhepp wrote:

Okay, where's my mistake?

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It here, as the area should be just half of the area of the fenced circle.

BTW, your function A.g(r) only returns a correct result for the area between the two circles if r<=sqrt(2)*R.f.

If the goat circle intersects the fence circle above the x-axis, you get wrong results.

[Fred_Kohlhepp]

So I spent a fair amount of time working to fix the 

     "If the goat circle intersects the fence circle above the x-axis, you get wrong results."

to still get the wrong answers.

Then (waking from sleep) last night the mistake your arrow pointed out leaped out!

Attached is the latest (still not completely corrected) file.  

[Fred_Kohlhepp]

And the (finally) fixed file.

[Werner_E]

If you use "min", you may also use "max"

This modified function "h" works at least OK in the range which is interesting.

[Werner_E]

This addition should do the trick (you may check using the formula Valery has given in his sheet):

P6 sheet attached

EDIT: Had not seen your final version when I started my answer. Nice idea to use the min function and more elegant compared to my brute force attack 😉

[ValeryOchkov]

See the attach and

https://community.ptc.com/t5/PTC-Mathcad/NEW-CONSTANT-in-math/m-p/352621

[Werner_E]

Nice idea to make the area ratio selectable.
But the link in the thread you point to is dead and the plot in the attached worksheet fails. I guess the file is from the times of the Mathcad Application Server and you added units. If you do so, you would also have to change the plot by providing a range for l and unfixing the upper limit of the y-axis:

Unfortunately the file does not show the derivation of the formula using the capabilities of Mathcad.