Have you ever found yourself scratching your head over a newly bought sofa which just won't fit through the corner of your corridor? Knowing whether if the size of your furniture would be permissible before buying would be very helpful indeed!
Below we are going to derive the conditions for the maximum permissible length of a rectangular objective with width w.
This question is taken from 2004 STEP 2 (Sixth Term Examination Paper, the entrance exam for Cambridge undergraduate mathematics). You can find the original problem on the STEP Database (04-S2-Q4). It's a very practically useful question, considering its taken from a STEP paper :)
To start our problem, we begin with a simpler case. Consider a curtain pole with length L which can be modelled as a straight line segment, and corridors with width a and b joining at the corner, as demonstrated on the left. We will move on to the 2d case later, which follows a similar method.
Because we want to derive the condition for the maximum length which can pass through, we consider only the cases where the rod touches the inner corner point O; since if the longest rod can pass through, shorter ones also can, and at the same angle, other rods will always be shorter than the rod just touching the corner:
Then, we consider turning this rod touching the corner over a range of angles from 0 rad to pi/2 rad, to go from the corridor a to corridor b:
From the diagram, it is conceivable that there is an angle which corresponds to a minimum length of the touching rod allowed, between the range 0 to pi/2 rad. So, to find the condition for the longest rod which can pass through the corridor, we first want to express the minimum permissible length as a function of the angle theta, and then differentiate with respect to theta to get the value of the theta for which the permissible length is at a minimum.
Let's label the diagram:
Drawing two lines from O perpendicular to the walls, we obtain two right-angled triangles. L is divided into two pieces of smaller length L1 and L2. Doing some simple trigonometry gives:
Where the value of alpha is uniquely determined by values of corridor dimension a and b through expression (1). So the maximum length can be calculated from just the dimensions a and b.
Here we have our condition for the range of curtain poles (and other 1-d objects) permissible to pass through your corridor!
Now, we are on to the sofas (2-d objects).
Analogously, we see that we only need to consider the range of sofas just touching the corridor corner, for if they pass, any other sofas will pass at the same angle:
Say we have a sofa of width w at an angle theta to corridor A, we could find a expression for the length l of the sofa such that it will just pass through, since l is the difference between the green and orange lines joined and just the green lines, that is:
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