SOLUTION: Please help me solve this problem. The dome of a whispering gallery has the form of a semi-ellipse so that the two person standing at the foci will be able to hear each other.

Question 1190079: Please help me solve this problem.
The dome of a whispering gallery has the form of a semi-ellipse so that the two person standing at the foci will be able to hear each other. This is because sound waves from one focus, when they reach the semi-elliptical ceiling, bounce off to the other focus. If one such whispering gallery has a focus of 3m away from the end of the semi-ellipse, and the foci are 14m away from each other, how high is the dome at the center?
Answer by CPhill(1959) (Show Source): You can put this solution on YOUR website!
Here's how to solve this problem:
1. **Visualize:** Imagine the semi-ellipse as a horizontal slice of a full ellipse. The two people are standing at the foci (F1 and F2). The distance from a focus to the nearest end of the semi-ellipse is given as 3m. The distance between the foci (2c) is 14m.
2. **Find 'c':** The distance from the center of the ellipse to each focus is 'c'. Since the foci are 14m apart, 2c = 14m, so c = 7m.
3. **Find 'a':** The distance from the center to the end of the semi-ellipse along the major axis is 'a'. We know that the distance from a focus to the nearest end is 3m. Since the distance from the center to the focus is 7m, we have:
a = c + 3m = 7m + 3m = 10m
4. **Find 'b':** 'b' is the distance from the center to the highest point of the semi-ellipse (the height of the dome at the center). We use the relationship for ellipses:
a² = b² + c²
10² = b² + 7²
100 = b² + 49
b² = 100 - 49
b² = 51
b = √51 ≈ 7.14m
Therefore, the dome is approximately 7.14 meters high at the center.

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