Geometry problem

Galileo

Consider an alley with two ladders placed crosswise.
Kinda like this sucky diagram below:

[tex]
\begin{array}{cccccccc}
|&* & & & & & &| \\
|& &* & & & & &| \\
|& & &* & & &&/| \\
|& & & &* &/& &| \\
|& & &/& &* & &| \\
|&/& & & & &* &|
\end{array}
[/tex]
( Ugh, I hope you get the idea)

One ladder has a length of 2 m. The other is 3 m long.
They cross each other at a height of 1 m.
What is the width of the alley?

It's not hard to see there's a unique solution to this question.
Good luck

Gokul43201

[tex]\sqrt{9-w^2} + \sqrt{4-w^2} = \sqrt{(9-w^2)(4-w^2)} [/tex]

This might look pretty, but it's both messy and inelegant...there's got to be a nicer way.

PS : That gives me w = about 1.231m

ceptimus

The solution to these 'crossed ladder' problems, except in trivial cases, is a quartic. They are almost always messy:

spoiler - highlight or Ctrl-A to view

w^8 - 22w^6 + 163w^4 - 454w^2 + 385 = 0

w is about 1.2312 m.

Best found (IMO) by successive approximation, ie. guessing.

Galileo

You're right. (so is Ceptimus), but how did you figure that?
On the left side I see the height of the 3m ladder plus the height of the 2m ladder and on the right side the product. Why are these equal?

NateTG

I don't know how they did things, but there are lots of similar triangles to pick from. It seems like that should be enough to work things out.

relativelyslow

could someone explain this one for me? this intrigues me

Galileo

Me too. C'mon Gokul43201!

TenaliRaman

Galileo,
this is sort of a hint and i am not sure whether Gokul did it this way...

Let the ladders be L1 and L2.
Let H_L1 be the height at which ladder L1 meets the wall
Let H_L2 be the height at which ladder L2 meets the wall

can u show that,
1/H_L1 + 1/H_L2 = 1

-- AI

GCT

Here's a hint: highlight to see

Use trigonometry (cos, sin, etc.......). It's not a difficult problem as you will see, the solution is quite "simple."

Gokul43201

That's what I did. But if you want a complete solution, I'll post it a little later...no time now.

alleycat2

Eureka !! I have discovered a truly unremarkable Al Gore Rythym to solve this teaser. *.102m Unfortunatley, it will not fit in this small margin. Get the idea? Thanx
*encrypted so not to spoil it for others>12 clock arithmetic.