Here we will look at the sofa moving problem. I assume that you voluntarily clicked on the link to this note and thus you are of adventurous nature, so some more adventurous math will not scare you. Therefore we will address the question here in full generality.
A corridor A meters wide turns at right angle into a corridor
B meters wide. We want to find the maximal length such that an object
of that length and w meters wide
We will try to follow the basic idea of the simpler problem with
Unfortunately, this is not the case, since the new position of the corner depends on the angle of the segment.
However, the idea of ignoring the inner segment and keep the outer segment at the proper position using a connecting perpendicular bar of length w is sound, since it is essentially the only reasonable way of somehow describing its position. Thus we can use this to find some structures that we can handle, namely we will try triangles.
By similarity, the four triangles in this picture share the angle a and thus we can calculate the length of the segment by the following formula:
We need to find the minimum of this function on the interval
This derivative is equal to zero for a satisfying
That is an equation that cannot be solved analytically, so we are out of luck. Not just in general, but even if we have concrete values for A, B, and w, we are in most cases unable to solve this equation. If we really need to know this a, we have to find it numerically using a suitable software (or directly as it for the approximation of minimum of l).
Are there any simpler cases that we could solve? Obviously we do know how to
solve this problem for
but that is of no help, since it is still impossible to solve directly. However,
when we look at the original equation, we can guess that with
Thus we can use the Pythagoras rule and get