Problem:
A ladder is to be carried down a hallway p feet wide. Unfortunately at the end of the hallway there is a right-angled turn into a hallway q feet wide. What is the length of the longest ladder that can be carried horizontally around the corner?

Visualization:

Above you can see a crude picture of the situation. The red bar represents the ladder to be carried. Click on Play to see the animation. You can change the values of p and q as well as the length of the ladder.

Now let's solve the problem. First instead of trying to find the maximal length let's see how to determine if a ladder of a given length l can be carried safely around the corner.

On the picture above we can see the critical moment. The ladder is represented by a red line segment AB of length l. It is easy to find the coordinates of B in terms of a: . So the equation of a line passing through AB is

The most important point on the picture is R. Its coordinates are (p,q). The ladder touches R if

Otherwise we have

So if

then we can carry the ladder without getting stuck if and only if

If we find the minimum of on (0,l) and test if it is greater or equal zero then we are done. Now:

The second derivative tells us that is concave up, so its critical value is a minimum. Let's solve

for a by multiplying both sides by :

If we have the values of p, q, l given, we can compute a0 and see if .

If , then our ladder has the biggest possible length. So in order to find the length of the longest ladder plug a0 expressed in terms of p, q, l into

and solve it for l. You should obtain