How far to bend a board to shorten its length by a desired amount Not exactly sure where to start so here is the real world application of my question...
I'm putting new decking in a utility trailer. Both ends of the trailer have a welded cap on them so I have to bend the last few boards to get them in place and slide them into the welded cap to avoid cutting and re-welding. My question is, if the board is 12 feet long and the end cap is one inch deep, how much would the middle of the board need to be bent upward in order to reduce the length by one inch so that it can clear the end cap?
In other, possibly easier to understand, words... Think of a flat 2x6 board that is 12 feet long and bending it into an arch to reduce the overall length from tip to tip by one inch.
This is more curiosity than actually applying this to the job. I would like to figure out how to calculate such a thing in the future.
Thanks,
~Mike 
 A: The precise answer will depend on how you bend the board.  If you step on it in the middle it takes a different shape than if you push it inward from the end.  But any answer will necessarily be an approximation.
Let's start with the simplest approximation: say the board bends exactly at the middle and is straight everywhere else.  It looks like a board with a hinge in the middle.  Now it forms an isosceles triangle, 11'11" wide (143 inches), and $x$ inches tall.  The Pythagorean formula tells you that $$x^2+(143/2)^2=(144/2)^2.$$
This gives you an approximate answer of 8.5 inches.
Next let's assume you somehow bend it into a perfect arc of a circle.  The circle will have a radius $r$ and the arc will cover $\theta$ radians.  The board is 144 inches so $r\theta=144$, and the shortening is defined by $2r\sin(\theta/2)=143$.  Solving the first two equations for $r$ and $\theta$ isn't straightforward, but the answer is $r\approx 352$ inches and $\theta=0.4086$ which give the amount the board will have bent away from flat as $d=r-r\cos(\theta/2)= $ 7.3 inches.
It makes sense this is smaller, because you give up some board length in the curve of the circle.
