Word Problems for Hyperbola

Some real-world problems, like cooling towers, follow the curve of a hyperbola. In these types of problems, you may be given measurements like the diameter of the top or the height along the axis of a hyperbola. When the hyperbola is centered at a specific point, you can use the standard form of the equation of a hyperbola to model it.

The transverse axis helps you to understand how wide the opening is, and any point that lies on the hyperbola must follow the rules of its equation. Solving these types of problems involves identifying key values and using the structure of the equation to find missing dimensions.

Example Problem

Trisha wants to build a fountain in her backyard in the shape of a hyperbola. The distance between the foci of the hyperbola is $10$ feet, and the distance between the vertices along the transverse axis is $8$ feet. If Sarah wants to place a statue at one of the vertices, how far from the center of the hyperbola should she place it?

With our recent lessons on hyperbolas, we have the tools to solve this problem! Remember that the standard equation of a hyperbola is: $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$

We are given that $a=4$ (half the distance between the vertices.

We know that the distance between the foci is $2c$. In this example, $2c=10 \rightarrow c=5$.

Now that we have $a$ and $c$, we can use the relationship between $a$, $b$, and $c$ in a hyperbola: $$\begin{align} c^2 &= a^2 + b^2 \\ 5^2 &= 4^2 + b^2 \\ 25 &= 16 + b^2 \\ b^2 &= 9 \\ b &= 3 \\ \end{align}$$

Since we know that the statue is located at one of the vertices, its distance from the center of the hyperbola is $a$, which is $4$ feet. Therefore, Trisha should place the statue $4$ feet from the center of the hyperbola.

It might occur to you that we didn't need to solve for $b$! You're right - but it's good mathematical practice to understand how to get the full equation. You never know when you'll need it!

$$\frac{x^2}{4^2} - \frac{y^2}{3^2} = 1$$

(Video) Word Problem

by Anusha Rahman

In this video by Anusha Rahman, she will walk you through a simple word problem involving hyperbolas.

Summary

Here is the word problem: An architect is designing a satellite dish with a parabolic shape. The dish is required to have a focal length of $6$ feet and a width of $10$ feet. However, due to design constraints, the dish must be in the shape of a hyperbola instead. If the focal points of the hyperbolic dish are located $8$ feet apart, find the equation of the hyperbola and determine the width of the dish.

Follow along with the video to solve the problem!

Example Problem

Sam is an aspiring artist and wants to create a cloud-shaped sculpture for a local art exhibition. He decides to model the shape of the cloud using a hyperbola. Sam knows that the cloud's shape should be such that its main body is elongated horizontally with two distinct arcs curving upwards, resembling a typical cumulus cloud.

To begin his project, Sam sketches a coordinate plane on his canvas. He plans to position the cloud's main body along the $x$-axis, and the upward-curving arcs will extend symmetrically above the $x$-axis.

After some research, Sam determines that the distance between the foci of the hyperbola should be $8$ feet, representing the horizontal width of the cloud's main body. Additionally, he measures that the distance between two points on the opposite arcs of the cloud is $4$ feet.

What are the coordinates of the center of the hyperbola where he should start drawing the cloud on his canvas?

The Solution

First, we want to denote the distance between the foci of the hyperbola as $2a$, where $a$ is the distance from the center to each focus. Since we are given the distance between the foci as $8$ feet, we know: $2a=8 \rightarrow a=4$.

Next, we can denote the distance from the center of the hyperbola to a point on either arc of the cloud as $c$. $c$ is the distance from the center to each focus. We have previously learned that $c^2 = a^2 + b^2$.

$b$ is the length of the semi-minor axis, $4$ feet.

If we substitute in $a$ and $b$,

$$\begin{align} c^2 &= 4^2 + 4^2 \\ c^2 &= 16 + 16 \\ c^2 &= 32 \\ c &\approx 5.657 \\ \end{align}$$

Now, we know the distance from the center to each focus $a$ is $4$ feet, and the distance from the center to a point on either arc of the cloud $c$ is approximately $5.657$ feet. We know that the center of the hyperbola is the midpoint between the two foci. Thus, the center of the hyperbola where Sam should start drawing the cloud on his canvas is $4$ feet to the left of the left focus and $4$ feet to the right of the right focus.

So, Sam should start drawing the cloud on his canvas at $(-4,0)$!

Hyperbola Word Problem

by Anusha Rahman

In this video by Anusha Rahman, you will be able to see how to solve a hyperbola word problem.

The Problem

Taylor Swift is planning a concert tour across the country. She wants to select a central location for her first concert that minimizes the average distance her fans will have to travel to attend. Let's assume her fans are spread out across the country in such a way that their distances from her potential concert locations form a hyperbolic shape.

Given that the foci of this hyperbolic shape are two major cities, New York City and Los Angeles, which are 2,800 miles apart, Taylor wants to determine the center of the hyperbola where she should hold her first concert.

She receives data from her fan club indicating that her fans are most densely concentrated along the major highways connecting these two cities, forming the arms of the hyperbola. Additionally, she knows that the average distance her fans are willing to travel to attend her concert is 500 miles.

Using this information, can you help Taylor find the center of the hyperbola where she should hold her first concert?