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The problem is:

Given $x$ feet of material with one side being $y$ long, what shape gives the maximum area that can be enclosed.

My solution is having the $y$ side a straight line, and having a curve that is semi-circular as possible to the end points of the line $Y$ (the line that is $y$ long). But, $y$ has to be small enough that $x-y > y$ thus, then there is enough lenght to connect the end points of the line $Y$.

I think this is the best solutions because from my own work the circle has the best edge length to area.

I am asking here because last week, my friend gave me the problem of connecting houses to water lines on a sheet of paper, without any pipes crossing and cutting the paper. The solution was folding the paper like a doughnut, was a complete surprise to me.

That is why I am asking here, is there something I missed? I have also checked the Questions that may already have your answer and the internet.

Edit Gerry Myerson gave a great counter example, thus my solution doesn't always work. Yet, there has to exist a curve that connects the ends of the line that is $y$ long.

El Santi
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  • Is this a rectangle of material of size $x \times y$? Are you allowed to cut it? – Henry Oct 07 '14 at 07:08
  • @Henry , I will ask after break, but I think it is just like I have a pile of material that is $x$ long, and one side has to be $y$ long. But, if you have a solution for an $x \times y$ plane, that would be awesome, as well. Post your solution, and I'll upvote as long as I don't find an error. hehe – El Santi Oct 07 '14 at 07:13
  • The trouble with having $x-y$ as an arc is that it may be geometrically impossible, e.g., if $x=100$ and $y=99$ you can't make any shape, arc or no. – Gerry Myerson Oct 07 '14 at 08:13
  • @ChristianBlatter I am not trying to be rude, but could you please explain how "area of the enclosed shape" is ambiguous. – El Santi Oct 08 '14 at 06:37

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