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Fred_Kohlhepp23-Emerald I
23-Emerald I
December 25, 2020
Solved

The goat problem

  • December 25, 2020
  • 2 replies
  • 7828 views

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/

 

 

    Best answer by Werner_E

    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:

    Werner_E_1-1608909203500.png

     

    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

     

    2 replies

    W
    Werner_E25-Diamond IAnswer
    25-Diamond I
    December 25, 2020

    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:

    Werner_E_1-1608909203500.png

     

    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

     

      F
      Fred_Kohlhepp23-Emerald IAuthor
      23-Emerald I
      December 25, 2020

      Okay, where's my mistake?

      Fred_Kohlhepp_0-1608916916113.png

       

      W
      Werner_E25-Diamond I
      25-Diamond I
      December 25, 2020
      @Fred_Kohlhepp wrote:

      Okay, where's my mistake?

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

      Werner_E_0-1608921965487.png

       

      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.

       

        V
        24-Ruby IV
        ValeryOchkov24-Ruby IV
        24-Ruby IV
        December 25, 2020

        See the attach and

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

         

        W
        Werner_E25-Diamond I
        25-Diamond I
        December 25, 2020

        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:

        Werner_E_0-1608923377012.png

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

         

         

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