A whispering chamber is designed following the shape of a dome. Its dimensions are 50 feet wide by 100 feet long. Assume that the chamber represents a horizontal ellipse with center at
(0. 0). What is the standard form of the equation of the ellipse representing the outline of the dome? If two performers standing at the foci of this room can hear each other whisper, how far apart are the performers? Round to the nearest foot.

### Part 1: Finding the Standard Form of the Ellipse

The standard form of the equation of an ellipse with a horizontal major axis and center at \((0, 0)\) is given by:

\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]

where:
- \(a\) is the semi-major axis (half of the length of the major axis),
- \(b\) is the semi-minor axis (half of the width of the minor axis).

From the problem:
- The length of the room (major axis) is 100 feet, so \(a = \frac{100}{2} = 50\) feet.
- The width of the room (minor axis) is 50 feet, so \(b = \frac{50}{2} = 25\) feet.

Thus, the standard form of the equation of the ellipse is:

\[
\frac{x^2}{50^2} + \frac{y^2}{25^2} = 1
\]

Simplified:

\[
\frac{x^2}{2500} + \frac{y^2}{625} = 1
\]

### Part 2: Distance Between the Foci

The distance between the foci is determined by the formula:

\[
c = \sqrt{a^2 - b^2}
\]

where \(c\) is the distance from the center to each focus. Substituting the values of \(a\) and \(b\):

\[
c = \sqrt{50^2 - 25^2} = \sqrt{2500 - 625} = \sqrt{1875} \approx 43.3 \, \text{feet}
\]

The distance between the two performers standing at the foci is \(2c\):

\[
2c = 2 \times 43.3 = 86.6 \, \text{feet}
\]

Rounding to the nearest foot, the performers are approximately **87 feet** apart.

### Summary
- The equation of the ellipse representing the outline of the dome is:
  \[
  \frac{x^2}{2500} + \frac{y^2}{625} = 1
  \]
- The performers standing at the foci of the ellipse are approximately **87 feet** apart.

Would you like further clarification on any part of this solution?

Here are five related questions for further exploration:
1. How does the distance between foci affect sound reflection properties in an elliptical room?
2. What is the significance of the semi-major and semi-minor axes in real-world ellipse applications?
3. How would the equation change if the ellipse were rotated?
4. How can you find the eccentricity of the ellipse based on the dimensions given?
5. How would you calculate the area enclosed by this elliptical room?

**Tip:** When solving problems involving ellipses, always identify the major and minor axes first, as they provide key information for the equation and properties of the ellipse.

A whispering chamber is designed following the shape of a dome. Its dimensions are 50 feet wide by 100 feet long. Assume that the chamber represents a horizontal ellipse with center at (0, 0). What is the standard form of the equation of the ellipse representing the outline of the dome? If two performers standing at the foci of this room can hear each other whisper, how far apart are the performers? Round to the nearest foot.

Discover how to find the equation of an ellipse representing a whispering chamber dome with dimensions of 50 feet wide and 100 feet long. Learn how to calculate the distance between two performers standing at the foci, approximately 87 feet apart. Suitable for high school students.

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