Solving for the Height of a Rectangular Hyperbola

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Homework Help Overview

The problem involves determining the height of a rectangular hyperbola representing an arch, given its width and maximum height. The original poster seeks to find the height at a specific distance from the ends of the arch.

Discussion Character

  • Exploratory, Assumption checking, Problem interpretation

Approaches and Questions Raised

  • Participants discuss the placement of vertices and points on the hyperbola, questioning the validity of certain coordinates. There is an exploration of how to set up the hyperbola equation without horizontal shifts and the implications of vertical translations.

Discussion Status

The discussion is ongoing, with participants providing different interpretations of the coordinates and the setup of the hyperbola. Some guidance has been offered regarding the placement of points and vertices, but no consensus has been reached on the correct approach or equations.

Contextual Notes

There are indications of confusion regarding the placement of the arch's coordinates, particularly concerning negative values and their physical interpretation. The original poster notes the need for vertical translation to ensure the arch's center is above ground level.

zaddyzad
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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.
 
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zaddyzad said:
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?
 
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.
 
zaddyzad said:
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?
 

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