Board Cuts for Platter A round platter is being made from a board of unit width and a length, L, between 1 and 4 units.  A crosscut and a rip cut will be made to give three pieces which will be joined.  What is the maximum diameter platter that can be made (and where is the crosscut)?


You can assume that the radius of the circle is $a/2$, since you lose nothing by making $a$ as small as possible given the correct answer. Thus the second part of the question has the same answer as the first.
Using Pythagoras from the centre of the circle to the point the circle cuts the edge of the central block, we get
$$\left(\frac{1}{2}\right)^2+\lambda^2=\left(\frac{a}{2}\right)^2,$$
where $\lambda$ is the horizontal displacement of this point from the centre of the circle.
So then
$$\lambda^2=\frac{a^2-1}{4}.$$
We need to have that $\lambda\leq b/2$ (so this point is on the wood), and also that $a/2\leq 1$ (so the top of the circle also stays on the wood), with $L=a+b$.
Rearranging, we find that
$$a=\min\left(\frac{L^2+1}{2L},2\right).$$
More explicitly,
$$a=\begin{cases}2& L\geq 2+\sqrt{3},\\ \frac{L^2+1}{2L}&L<2+\sqrt{3}.\end{cases}$$
