You won't get this one!!?

Question

A 20 foot and a 30 foot ladder lean in opposite directions against buildings in an alley. The foot of each ladder touches one of the buildings. The building walls are vertical. The point where the ladders intersect is 10 feet above the ground. How wide is the alley?
No "engineering" estimates, please.

Kyle: There is enough info. Imagine a 20 foot alley. One ladder is lying on the ground. The intersect point is ) feet off the ground. Now narrow the alley and the intersect point begins to rise. When narrows to 0 feet then the intersect point is 20 feet, the top of the ladder. Thus the intersect point is a decreasing function of alley width, with a range between 0 and 20. This is a real *****, and it requires solving a 4th power polynomial.

Answer 1

This question does not have a nice answer. By the end, I'll end up with a quartic equation, which I have no intention of actually writing out the solution to. Sorry to disappoint if this was what you intended.

I believe that there is enough information to solve; however, I went through several different methods, none of which worked out too nicely. Here's the best one:

Consider a diagram of a cross-section of the alley. Let the 30ft ladder lean on the left side and the 20ft lean to the right. Let x represent the alley width, y the vertical height of the 30ft ladder, and z the height of the 20ft ladder. Let a be the distance from the left side of the alley to the point directly below the intersection of the ladders. This leaves x-a for the remaining "floor" segment. Notice the two pairs of similar, embedded triangles. These form our four equations:
x^2+y^2=30^2
x^2+z^2=20^2
(x-a)/10=x/y
a/10=x/z

Now we have four equations, four variables, and no information is doubly represented, so we should be able to solve. However, the equations are very ugly, so it becomes rather difficult. Let's begin by expanding eq.3, and substituting from eq.4:
x/10 - a/10 = x/y
x/10 - x/z = x/y
1/10 - 1/z = 1/y
since of course x cannot be zero. Now we have an equation (5) that connects y and z directly. I solved for y, giving:
y=10z/(z-10)
Plugging this into (1), and then substituting from (2),
x^2 + 100z^2/(z-10)^2 = 900
400 - z^2 + 100z^2/(z-10)^2 = 900

Here we almost have the quartic I mentioned before. If we multiply through and expand the (z-10)^2 term, we have a rather large quartic to deal with. Since I am not adept at quartics, I simply had my calculator approximate solutions for z. There were two solutions, but one was about 7, which is impossible (z must be greater than 10, referring to the diagram). The other solution is:
z=15.761287109762

Plugging this into (2) to find x, we have that the alley is
12.3118572378 ft wide.

For the fun of it, let's see if this seems alright. First, let's calculate the other variables:
a=7.81145419917
(x-a)=4.50040303862
y=27.3572325237
the angle the 30ft ladder makes with ground=65.77deg
20ft ladder's angle=52.005deg

These values all make sense to me. Sorry if this counts as an "engineering" estimate to you. I think this is far more useful than using all the roots needed to exactly solve the quartic, especially when this seems to be the only root that makes sense for the problem anyway.

Oh, and if anyone finds a way to do this that's easier than this mess, please let me know so I don't try to do one of these again someday :P

Answer 2

You are right, we won't get this one. That's because there is not enough information. We need either an angle or the height at which one of the ladders hits the wall.]


edit: Nevermind, you are right. But I don't feel like solving it!

Answer 3

The 20ft and 30ft form the hypotenuse of 2 right angled triangles, so by Pythagoras

The distance d = √(30² - 10²) + √(20² - 10² )

..................... d = √800 + √300

..................... d = 45.6 ft

Answer 4

6 feet

Answer 5

I dont feel like doing the problem but it seems to me you can just use a combination of the law of sines, law of cosines, and the pythagorean theorm to answer the question.

Answer 6

The empirical answer that I get is 12.3118565 feet, but you want an exact answer. I haven't figured that out...yet.

Answer 7

misread Q
the 2 below are misreading Q like I did

Answer 8

45.6 or 46 ft