# Solving for the Height of a Rectangular Hyperbola

In summary, the conversation discusses the construction of an arch in the shape of a rectangular hyperbola with a maximum height of 100m and a width of 300m. The individuals discuss determining the coordinates of the vertices and points on the hyperbola and how to solve for the variables h and a. They ultimately agree that a set of coordinates with the arch endpoints symmetric about the origin, such as (0,100), (-150,0), and (150,0), would be best for solving the equations.

## 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.

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.

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?

## 1. What is a rectangular hyperbola?

A rectangular hyperbola is a type of hyperbola that has its center at the origin and has equal distances from the center to the vertices.

## 2. How do you find the height of a rectangular hyperbola?

To find the height of a rectangular hyperbola, you can use the formula h = b/a, where h is the height, b is the y-intercept, and a is the distance from the center to the vertex on the y-axis.

## 3. Can the height of a rectangular hyperbola be negative?

Yes, the height of a rectangular hyperbola can be negative if the y-intercept is negative or if the vertex on the y-axis is below the origin.

## 4. What information is needed to solve for the height of a rectangular hyperbola?

To solve for the height of a rectangular hyperbola, you need to know the coordinates of either the y-intercept or the vertex on the y-axis, as well as the distance from the center to that point.

## 5. Are there any other methods for finding the height of a rectangular hyperbola?

Yes, there are other methods for finding the height of a rectangular hyperbola, such as using the focus and directrix or using the equation of the hyperbola. However, the simplest and most straightforward method is using the formula h = b/a.

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