I propose there is no solution to this problem with the length poles called out.

Short (and best) explanation:

1. The midpoints of the two poles are ALWAYS vertically aligned (basic geometry).
2. The intersection height is ALWAYS midway between the two midpoint heights - draw it out and see.
3. Since the midpoint height of the longer of the two poles is 2.5 meters, even as it is unfathomably close to vertical, the intersection height can NEVER even reach 2.5 meters.

Another and longer way of looking at it:

Picture the poles in the two extremes:

1. 4 meter pole lying flat (intersection height = 0)
2. alley width (x) very nearly 0 (as close as you can imagine - look at the trend as x goes to 0.)

Keeping option 2 in mind, picture this: BOTH poles will ALWAYS have their midpoints vertically aligned (the 5 meter pole above the 4 meter) as you stretch or shrink the alley width. Visualize the alley width = 4 (the 4 meter pole lying flat) The vertical 'delta' of the midpoints will be 1.5. (the poles and one of the walls make up a 3- 4-5 triangle) As you shrink x (the alley width), the pole's midpoints will start to rise (the 4 meter pole's midpoint NEVER rising above 2, and the 5 meter pole's midpoint NEVER rising above 2.5) The trend for 'delta' will start at 1.5 as seen above, and reach a minimum of .5 when x = 0 (both poles vertical and sharing the same space) If you sketch out the two poles crossing each other (or ANY combination of poles), you will see that the intersection height ALWAYS lies between the two midpoints. Therefore, even as the poles get as close to vertical as your mind can imagine, the intersection height can never actually even reach 2.5 (the height of the midpoint of the 5 meter pole, or depending on how you look at it, the height of the midpoint of the 4 meter pole plus the delta at that extreme: .5). This is in no way a proof and it needs to be worked out, but this case presents two pole lengths that make it a bit obvious there is no solution. Especially since certain combinations of 'delta' and the height of the 4 meter midpoint are actually > 3 (even though the intersection is actually a little less than that sum), but it is obvious upon sketching that in this case it is not even close.

Let me know if you think I'm way off base somewhere, I did this quickly during work!
Christopher A. Pellegrino
Mechanical Engineer
Rodgers Instruments LLC