Problem: Given a circle that is just barely covered by a parallel collection of adjacent rectangular boards that have length at least as much as the diameter of the circle, prove that the circle cannot be covered by the same boards if they are arranged in a non parallel configuration.
This is problem 11c from the chapter 19 appendix (integration) of Spivak's calculus 4th ed. I've thought of a few ideas, but I am not sure how to proceed with them and make progress.
I was considering looking at an extreme scenario, with an arbitrarily high number of rectangles which are very long. If I can prove this scenario fails, then any other one must too, since one large piece is just multiple smaller pieces and increasing the length of a piece cannot decrease coverage.
I was also considering looking at total area each rectangle covers minus the overlapped area with another rectangle. I was thinking about working with an expression for the total covered area, and somehow showing that only a parallel arrangement can make the total equal to the circle's area.
I am stuck at the moment and any help or insights would be greatly appreciated.