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Hi I am new here and have a calculus question that came up at work.

Suppose you have a $4' \times 8'$ piece of plywood. You need 3 circular pieces all equal diameter. What is the maximum size of circles you can cut from this piece of material? I would have expected I could write a function for the area of the 3 circles in terms of $x$ and $y$, then differentiate it, find a point of maxima/minima and go from there.

My coworker did cut three $33''$ circles and that solved the real-world problem. But my passion would be to find the mathematical answer to this. I hope that my new stackexchange.com friends have the same passion, and can help me find the answer to this in general terms.

What I mean by that is someone says I have a piece of material Q units by 2Q units, what are the three circles of maximum size?... I hope you understand what I am asking. I am looking to be a friend and contributor BD

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    [Circle cutting is usually a nontrivial problem.](http://downloads.hindawi.com/journals/aor/2009/150624.pdf)2011-01-04
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    Erich's Packing Center http://www2.stetson.edu/~efriedma/packing.html has experimental results for many configurations, though I don't see circles in rectangles at a quick look.2011-01-04
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    Christian Blatter has provided an elegant proof that Isaac's configuration is the best. So your problem is solved! If you agree, you should accept one of their answers (you can't accept both of them, unfortunately).2011-01-07

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