Daily Maths Challenges

Nice Problem you got there. I am not really in a mood to do functional calculus, but this question is a functional calculus question. I will try to give the setup, but not going to solve it (sorry, getting sleepy).

with some clever symmetry arguments, the fence should be symmetric about a line (or plane) bisecting the 2 rabbits, and since we have to also create a wall there, we have to have a line, somewhere in this plane

now we can consider a parametric form for equation of fence.

if you are unaware of functional calculus, it is a very fun branch of calculus, where instead of maximising value of a variable, we maximise a "function" it self. for a readup, refer https://en.wikipedia.org/wiki/Action_(physics)#Hamilton's_characteristic_function https://en.wikipedia.org/wiki/Euler%E2%80%93Lagrange_equation

To solve, we need a constant (or a constraint, in this case total fence length), and a function which we want to extremise (in this case, maximise the area of enclosed by fence).

to keep the problem simple, lets consider, a single valued function (f(x) -> y) (for a general proof, we can do f(x, y) -> z), and also consider some dividing segment, now lets consider the leftmost and rightmost extremes to have values +/- alpha (assuming symmetry in this direction also, not necessary to assume this, but doing for simplicity), now we have a straight fence between some section of these extremes (lets say +/- beta) now total length of fence is constant so

10 = 2 beta + twice (line integral from - alpha to + alpha of f(x))

(line integral is not same as usual integral, you may also know it by name of path integral)

this gives us a constraint

now we can find area enclosed by

twice (integral from - alpha to + alpha f(x) dx )

now use the Euler - Lagrange equation to find a solution for f(x)

this is a very general way to find the solution, and this will also mathematically prove that our solution is the best. If you are also not interested in doing maths, you can use some llm (like chatgpt) to formally setup the problem for you, and then you can use some solver like wolfram alpha to get solution.