Geometry Problem: Solving w/o Extra Variable?

[dragonblood]

Homework Statement

I want to know if the problem needs another variable, or if it is possible to solve as it is.

Homework Equations

See figure in image included:

You are a point x, some height h above zero-level. The two mountains have known height 500m and 400m. But you don't know how far you are from either mountain, or the distance between them.

You are, however, able to measure any angle with precision.

The Attempt at a Solution

Say I can find the angles shown in the second image

.

I'm thinking comparing triangles or something, but I also suspect the distance beteen the mountains is needed.

The point of this problem is to see if one can determine how far above sea-level you are at any time, as long as you know the height of two reference points, as well as being able to measure angles relatively accurate.

 

[verty]

Think about the triangle with these three corners: mountaintop A, mountaintop B, your position C.

Either you have enough information to determine the orientation and the scale, or you don't. Which is it? How much can you determine?

 

[dragonblood]

I can measure the angle between top A and B, but that's about it. I don't know any other length in that triangle. So I would say no. But I'm not sure...

 

[dragonblood]

I'm leaning towards there not being a solution.

However, if you perform two measurements, with the distance between measuring points known, the angles could tell you the distance to either mountain. Then it would be possible!

 

[verty]

Using the fact that you don't know the distance between the two mountains, can you think of a scenario where an observer who thinks he knows the distance between the mountains is fooled?

 

[dragonblood]

I'm sorry. You're confusing me...

 

[Femme_physics]

I'm confused. They tell you that you are "able to measure any angle with precision" - doesn't that mean that all the angles here are known? Because, I don't see that...

 

[dragonblood]

The point of the problem is this:

Let's say you outside taking a hike somewhere. You can see the summit of to mountains (which you know the height of), and you're wondering how high above sea-level you are right now. The only tools you have is some kind of instrument making you able to measure angles between any points with a satisfactory presicion.

And the question is: is this possible? Or do you have to make two measurements from two different points with a known distance between them, and triangulate the distance to the mountains?

 

[Femme_physics]

I think you've practically solved it then, in your diagram where you've drawn the line that's touching the ground. You know the length of it, right? Not, just built a triangle from that length being the hypotenuse. The vertical length will be a line from X to the ground -- the length you need. Now you got the triangle you need to solve the problem. Unless I'm misreading something, that should be pretty simple...

 

[dragonblood]

No, it's not that easy unfortunatel. I don't know the distance to the zero point. In fact, I don't know any distances at all. Only the height of the two mountains. And the point was to use angles along with the mountain heights to solve the problem...

 

[Femme_physics]

With only the height of the two mountains, you can find out everything. You can pass a two lines that represent the height of the mountains, they'll be vertical lines starting from the ground, reaching the top. You'd split each mountain to 2 right triangles that way, when you have one length known (the height)! Now you just keep solving for lengths until you get to the "magic triangle" that vertically connect X position to the ground.

Edit: In your second diagram you didn't seem to have done a good enough job-- you were almost there. You need to connect the line that starts from X and touching the ground to the corner of the other mountain, otherwise it's not a triangle. There are actually a couple of ways to solve this problem -- I'm just going with what you've started with.

 

[dragonblood]

I would very much like you to explain this in detail (with drawings). Else I'm not quite sure I understand your method :P

 

[Femme_physics]

First you use the green triangle, then the brownish, then the yellowish (it's hard to see the yellow but I think you know what I'm talking about). Does that make it clearer?

 

[dragonblood]

Use the green triangle how? I don't know the short side or the hypothenus of it. I can't know how to aim the angle to the zero point 'inside' the mountain...

 

[Femme_physics]

Not knowing the angles I need is taking more geometry than I could muster...I tried a couple of ways but I got into a jungle of triangles each time. Sorry I couldn't help...I'm sure a real expert will come along. :P

 

[dragonblood]

Thanks for making an effort! It's always rewarding even if we don't solve exactly everything :)