Solving for the Height of a Rectangular Hyperbola

[zaddyzad]

Homework Statement

An arch is the shape of a hyperbola. IF it s 300m wide at its base and has a maximum height of 100m, how high is the arch 30m from the end ?

Note: this is a rectangular hyperbola.

Homework Equations

(y-h)^2 - x^2 = a

The Attempt at a Solution

I determined the verticies is (0,0) and there are two points (-150,-100), (150,100) I also know that there must be a vertical translation for the centre to be higher than (0,0).

But what I can figure out is how to solve for h and a using two different coordinates. If someone could help me with the algebra that'd be awesome.

 

[haruspex]

I determined the verticies is (0,0) and there are two points (-150,-100), (150,100) I also know that there must be a vertical translation for the centre to be higher than (0,0).

How do you get (-150,-100)? That would be underground, no? What about y when x = 300?

 

[zaddyzad]

I'm trying to set it as easily as I can without a horizontal shift. However maybe a vertice of (0,100) and points (-150,0) and (150,0) would be better.

 

[haruspex]

I'm trying to set it as easily as I can without a horizontal shift. However maybe a vertice of (0,100) and points (-150,0) and (150,0) would be better.

Either way is fine, but I think you had the coordinates wrong in your first way. It looked like you had the origin on the ground at one end of the arch, right? So the y coord should never have been negative.
The set you propose now, with the arch endpoints symmetric about the origin, looks right. So, what equations do you get?