Crossing Ladders (Solution)

~1.23 feet

Let’s call the width of the alley \(w\). Cut \(w\) into two pieces at the horizontal point at which ladders A and B meet. Call the section on the left, \(w_1\) and the section on the right \(w_2\).

Using similar triangles, we can set up the following equations:

\[w_1 + w_2 = w \\ \frac{w}{\sqrt{4 - w^2}} = \frac{w_2}{1} \\ \frac{w}{\sqrt{9 - w^2}} = \frac{w_1}{1} \\\]

So, we have 3 equations and 3 unknowns. Should be able to solve it:

\[\frac{w}{\sqrt{9 - w^2}} = w - w_2 \\ \frac{w}{\sqrt{9 - w^2}} = w - \frac{w}{\sqrt{4 - w^2}} \\ \frac{w}{\sqrt{9 - w^2}} + \frac{w}{\sqrt{4 - w^2}} = w \\ \frac{1}{\sqrt{9 - w^2}} + \frac{1}{\sqrt{4 - w^2}} = 1 \\\]

I don’t know how to actually finish the solution all in closed form, but using a computer, ~1.23 is a solution.