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The Crossed Ladders Problem March 27, 2011

I was recently reminded of the crossed ladders problem. The following simple figure should be adequate in defining the problem:

If you haven't seen the problem before, I highly recommend trying to solve it before reading on.

I had previously tried to solve it, but without success. This time I couldn't get it out of my mind. I had become a (self-inflicted) victim of nerd sniping.

The thing is that it looks simple. But it isn't. Below is my solution to the problem (using some inspiration for making it a bit more pretty).

First, some variables must be introduced. Consider the following figures:

So $a=|AF|$, $b=|BE|$, and $h=|CD|$ are given while the width $w=|AB|$ is the sought quantity. Furthermore, $p=|BF|$ and $q=|AE|$ are auxilliary variables.

Consider first the special case $a=b$. It is then clear that

$w = \sqrt{a^2-4h^2} \quad\text{for } a=b.$

Assume now, without loss of generality, that $a > b$. The Pythegorean theorem gives us

(1)
$w^2 = a^2-p^2 = b^2-q^2 \quad \Leftrightarrow \quad a^2-b^2 = p^2-q^2 = (p+q)(p-q).$

We then use that the triangle ACD is similar to AFB, just like the triangle BDC is similar to BAE, and get

$\frac{|AD|}{w} = \frac{h}{p} \quad\text{and}\quad \frac{|BD|}{w} = \frac{h}{q}.$

This gives us

$w = |AD|+|BD| = w \left( \frac{h}{p} + \frac{h}{q} \right) \quad \Leftrightarrow \quad pq = h(p+q).$

We use this equality to get

(2)
\begin{aligned} (p-q)^2 &= p^2+q^2-2 p q = p^2+q^2+2 p q-4 p q \\ &= (p+q)^2-4 h(p+q) = (p+q)(p+q-4h). \end{aligned}

We now start from (1) and use (2) to get

$(a^2-b^2)^2 = (p+q)^2 (p-q)^2 = (p+q)^3 (p+q-4h).$

We finally divide each side by $\sqrt{a^2-b^2}^4$ and obtain

(3)
$x^3 (x-c) - 1 = 0$

where

(4)
$x = \frac{p+q}{\sqrt{a^2-b^2}} \quad\text{and}\quad c = \frac{4h}{\sqrt{a^2-b^2}}.$

Let us take a closer look at (3) by considering the fourth degree polynomium $Q(x)=x^3 (x-c) - 1$. Note first that $c$ is positive. Next that $Q$ is decreasing for $x < 0$ and $0 < x < 3c/4$ and increasing for $x > 3c/4$ (seen from the derivative $Q'(x)=x^2(4x-3c)$). And since $Q(0)=-1$ we then know that $Q$ always has exactly two zeros, one negative and one positive. We are naturally interested in the positive one.

Assume now that we have found the (positive) root $x$. We then have from (4) and (1),

$p+q = x \sqrt{a^2-b^2} \quad \text{and} \quad p-q = \frac{a^2-b^2}{p+q} = \frac{1}{x} \sqrt{a^2-b^2},$

and then

$p = \tfrac{1}{2} \left( x + \frac{1}{x} \right) \sqrt{a^2-b^2},$

and finally

$w = \sqrt{a^2-p^2}.$

And that is the width we were looking for. Note, however, that if $p > a$ then no solution is possible. This can happen, for example, if the crossing height $h$ is larger than the shortest ladder.

But how do we solve (3) and find $x$? It is possible to solve a quartic equation analytically, so let us turn to Abramowitz and Stegun and solve it. By carefully inserting to retrieve the positive root we get

$x = \tfrac{1}{4} c + \tfrac{1}{4} \sqrt{c^2 + 4 u} + \tfrac{1}{2} \sqrt{\left( \tfrac{1}{2} c + \tfrac{1}{2} \sqrt{c^2 + 4 u} \right)^2 + 2 \sqrt{u^2+4} - 2 u}$

where

$u = \sqrt[3]{\sqrt{\tfrac{64}{27}+\tfrac{1}{4} c^4}-\tfrac{1}{2} c^2} - \sqrt[3]{\sqrt{\tfrac{64}{27}+\tfrac{1}{4} c^4}+\tfrac{1}{2} c^2}.$

That concludes the solution to the crossed ladders problem. Let us finish by inserting some numbers.

From the initial illustration of the problem, we have $a=12$, $b=10$ and $h=5$. This gives us

$w \simeq 4.2973280047205172448618937103219913746175.$

There are actually lengths that have an integer solution. Take, for instance, $a=119$, $b=70$, $h=30$ and $w=56$.

A simple, but tricky, problem.

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