Consider an L-shaped ditch with straight sides and both legs of the same uniform width.

Part 1: How wide a ditch can be bridged with two 10 foot planks?

Part 2: How wide a ditch can be bridged with three 10 foot planks?

Note: consider the planks to be rigid line segments - very thin yet strong.

(re Part 2)
I don't have the solution but imagine the drawing of the corner so that the inner corner is in the first quadrant with the inner corner initially 15 feet from the outer corner.
Put the first plank making a 45-45-90 triangle with the x and y axes, but each end is on rollers so it can slide either way.
Put the second plank nominally over the first plank but with its left "roller" attached to the y axis, and its right "roller" attached to the first plank.
Place the third plank with its right end attached the inside corner (on a swivel) and its left end initially at the mid point of the second plank on a roller.
Now do a gedanken experiment where the inside corner is pulled along the y=x line away from the outside corner to the point of maximum distance.

I think the right end of the first plank will side to the right along the x axis, the left end of the second plank will slide upward along the y axis.  I think the third plank will ultimately seek to maintain a right angle with the second plank when maximal distance is reached.

I tried to do this with equations of the lines.  My plan was to differentiate the value of the inner corner's x (or y) coordinate with respect to the value of the first plank's coordinate on the x axis.  But it got messy so I was not able to complete it.

Posted by Larry on 2017-08-18 12:33:44