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[solved] A tale of two rabbits (self.dailymaths)

submitted 1 week ago* (last edited 1 day ago) by joelthelion to c/dailymaths

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I have ten meters of mesh fence. I want to make two enclosures for my rabbits who can't stand each other (they can share a wall). What is the shape and the area of the largest enclosure (in terms of area) I can build? Each rabbit needs to have access to the same area. The shape can be arbitrary (although it would be nice if it were continuous or smooth to some extent, and each area contiguous).

Examples:

A square of 2x2m, divided by a 2m wall. Area: 4 square meters

A circle of radius 1.21m, divided by a wall. Area: ~4.58 square meters.

Is it possible to do better?

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[–] joelthelion 1 points 1 week ago (1 children)

Hi, I'm familiar with the concept of Euler-Lagrange equations, however I wouldn't know how to use them in practice to solve a problem like this one. ChatGPT isn't very helpful and suggest a single circle cut in half is the optimal solution, even when I tell it to use Euler-Lagrange.

[–] [email protected] 2 points 1 week ago

maybe instead of asking it to solve from scratch, just dump in my reply maybe that may help, i am not a expert with prompt engineering. Although, you can find similar problems in standard textbooks. I remember in one of my exams i had basically same problem, but had to minimise area instead.

for usage of euler lagrange, maybe the following may help

https://math.stackexchange.com/questions/346027/satisfy-the-euler-lagrange-equation https://math.stackexchange.com/questions/2337413/find-the-surface-of-least-area-spanned-by-a-given-contour https://www.sas.rochester.edu/mth/undergraduate/honorspaperspdfs/alfredovargas2018.pdf https://math.uchicago.edu/~may/REU2019/REUPapers/Zheng,SiqiClover.pdf https://en.wikipedia.org/wiki/Minimal_surface#History