Word Problems Using Ellipses

The Oval Office in the White House is known for its distinctive elliptical shape. Suppose that the Oval Office has a major axis of $35$ feet and a minor axis of $29$ feet. The President wants to place a new desk at one of the foci of the ellipse for better Feng Shui. If the distance between the foci is $20$ feet, how far from the center of the Oval Office should the desk be placed?

To find the distance from the center of the Oval Office to where the desk should be palced (the distance from the center to one of the foci of the ellipse), we need to use the properties of ellipses.

Let's first consider the information we already have. The major axis ($2a$) is $35$ feet. The minor axis ($2b$) is $29$ feet. The distance between the foci ($2c$) is $20$ feet.

First, we need to find the value of $c$, which is half the distance between the foci:

$$c = \frac{20}{2} = 10 \text{ feet}$$

Next, we will use the relationship between $a$, $b$, and $c$ in an ellipse.

$$a^2 = b^2 + c^2$$

Substitute in the values:

$$\begin{align} a^2 &= (17)^2 + (10)^2 \\ a^2 &= 289 + 100 \\ a^2 &= 389 \\ a &\approx 19.72 \text{ feet}\\ \end{align}$$

Therefore: the distance from the center of the Oval Office to where the desk should be placed (from the center to one of the foci of the ellipse) is approximately $19.72$ feet.

Sarah is designing a decorative fountain for her garden in the shape of an ellipse. The major axis of the ellipse is $8$ meters long, and the minor axis is $6$ meters long. She plans to install a statue at one of the foci of the ellipse.

If Sarah wants the statue to be exactly $2$ meters away from the edge of the ellipse along the major axis, how far away should she place it from the edge along the minor axis?

To find the distance from the center of the ellipse to the statue (which is located at one of the foci), we'll use the equation of an ellipse: $$c^2 = a^2 - b^2$$

We know that $a$ is the length of the semi-major axis, which is $4$. We also know that $b$ is the length of the semi-minor axis, which is $3$. $c$ is the distance from the center of the ellipse to either focus.

So, if we were to substitute these values into the equation: $$\begin{align} c^2 &= 4^2 - 3^2 \\ c^2 &= 16 - 9 \\ c &= \sqrt7\\ \end{align}$$

The distance from the center of the ellipse to the statue $\sqrt7$ meters.

However: that's not what the question asked for!

To find the distance from the edge of the ellipse to the statue along the minor axis, we simply subtract 2 meters (as Sarah wants the statue to be 2 meters away from the edge along the major axis) from the distance from the center to the statue.

$$\sqrt7 - 2 = \text{ distance along minor axis}$$

Using a calculator, you'll find that the answer is approximately $0.65$ meters. Therefore, Sarah should place the statue about $0.65$ meters away from the edge of the ellipse along the minor axis.

(Video) Ellipse Word Problem

by Anusha Rahman

In this video by Anusha Rahman, you will walk through an example of a word problem involving ellipses.

Solving Real-World Problems with Ellipses

We can solve world problems with ellipses.

An ellipse has a special rule:

Any time you go from one focus to the edge of the ellipse and then to the other focus, the total distance is always the same.

That distance is equal to the length of the major axis, which is the longest line across the ellipse.

In a whispering gallery shaped like an ellipse, sound travels from one focus, bounces off the curved wall, and reaches the other focus. Because of the ellipse’s shape, the total sound path is always the same. If the major axis is 50 feet, then the sound will travel exactly 50 feet, no matter where it hits the wall.

So in this problem, even though the foci are only 40 feet apart, the sound takes a curved path that is always 50 feet long.