TITLE: maximum area of three circles
QUESTION [12 upvotes]: 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

REPLY [4 votes]: Let $Q:=[-2,2]\times[-1,1]$ be the given rectangle and let $r_0$ be the radius computed by Isaac for this rectangle. I shall prove that 3 circles of radius $r>r_0$ cannot be placed into $Q$ without overlap. The midpoints of these circles would have to lie in the smaller rectangle $Q':=[-2+r, 2-r]\times[-1+r,1-r]$. The left half $Q'_-$ of $Q'$ has a diameter $<2r_0<2r$ (by definition of $r_0$). It follows that $Q'_-$ can contain at most one center of non-overlapping circles of radius $r$, and the same is true for the right half $Q'_+$.