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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 3 points 1 week ago

The best I was personally able to obtain is the circle divided by a wall.

I initially asked this question on the mathematics stack exchange: https://math.stackexchange.com/questions/5035579/area-maximization. Of course, it got closed for no reason, but before it did, someone offered the following solution: join two circular arc (subtended at angle 240∘ with respect to their center) at their end points, add a line segment between the end points, you can achieve an area ∼4.9457788. Recomputing the area of their solution, I get ~4.49 instead, but I'm not sure I got it right...