Background: This is a follow on from this question of how to maximise the area of two non overlapping circles of arbitrary radii packed into a rectangle of arbitrary width and height. I proposed a solution based on greedy selection of the first circle, then the second:
With a solution of: $$r_1 = (\frac{h}{2})$$ $$r_2=(-\sqrt{2}\sqrt{w}\sqrt{h}+w+\frac{h}{2})$$
Question: How do I prove that this is the optimal solution?
Its seems intuitive that when $w=h$ or $w=2h$ this will be optimal. My concern is that somewhere in between there may be a better arrangement.
I am having difficulty setting up the equations to show that for any point in between the rate of change of area of the first circle as $r_1$ decreases is greater than the rate of change as $r_2$ increases, which I hope would prove this. Help is appreciated.
