A fence q feet tall runs parallel to a building at a distance of p feet from the building. What is the length of the shortest ladder that
can reach from the ground over the fence to the wall of the building?
Below you can experiment with different values of p, q and with different
ladders. To see if the ladder can be used to reach the building, press
Play. You can also find the shortest ladder ( for the given p and
q ) that will reach the wall ( just click on Optimize ).
Now let's try to solve this problem.
On the picture above you can see a ladder reaching from the ground ( point
A ) to the wall ( point D ). The ladder touches the fence ( point
B ), so clearly it is the shortest ladder that can reach to the wall from
A. Let us express the length of this ladder in terms of a.
Using Pythagorean Theorem we have
It's easy to see that
In order to find the length |ED| we can use the similarity of triangles
Now we can find the minimum of l(a) for a>0 ( that will give us the
shortest ladder ). In order to simplify computations we will find the minimum
a<p1/3q2/3, then a3<pq2 and
If in turn
a>p1/3q2/3, then a3>pq2 and
Thus it is clear
has a minimum at a0 and of course
has a minimum at a0 too. We can now obtain the length of the shortest ladder
by plugging a0 into l(a):
This page including the Java applet was written by Marek Szapiel.