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Question 1185775: Yohan and Jesx want to test the acoustics of a whispering gallery. Yohan is
standing at one of the focus of the gallery, 5 feet away from the nearest wall. Jesx is standing at the other focus and is 9 feet away from Yohan.
a. What is the highest point of the ceiling from the ground?
b. How high is the ceiling from the point where Yohan and Jesx stand?
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Here's how to solve this problem:
**Understanding the Geometry**
A whispering gallery is shaped like an ellipse. The foci are the points where whispers can be heard clearly. The key property we'll use is that the *sum* of the distances from any point on the ellipse to the two foci is constant.
**a. Finding the Highest Point of the Ceiling**
1. **Distance to Foci:** Yohan is 5 feet from the nearest wall (which is also a vertex of the ellipse), and Jesx is 9 feet away from Yohan. This means the distance between the foci (2c) is 9 feet. Therefore, c = 9/2 = 4.5 feet.
2. **Distance from Focus to Vertex:** Yohan is 5 feet from the nearest wall, which is a vertex. The distance from a focus to the nearest vertex is *a - c*, where 'a' is the semi-major axis of the ellipse. So, a - c = 5.
3. **Solving for 'a':** a - 4.5 = 5 => a = 9.5 feet.
4. **Center of the Ellipse:** The center of the ellipse is halfway between the foci. Since Yohan is 5 feet from the nearest wall (vertex), and the distance between the foci is 9 feet, the distance from Yohan to the center is (9/2) + 5 - 9.5 = 5 feet.
5. **Semi-minor Axis (b):** We can find 'b' using the relationship in an ellipse: c² = a² - b².
4.5² = 9.5² - b²
b² = 9.5² - 4.5²
b² = 90 - 20.25
b²= 70.75
b = √70.75 ≈ 8.41 feet
6. **Height of the Ceiling:** The highest point of the ceiling is the distance from the center of the ellipse to the top, which is equal to the semi-minor axis *b*. Therefore, the highest point of the ceiling from the ground is approximately 8.41 feet.
**b. Height of the Ceiling Where Yohan and Jesx Stand**
1. **Horizontal Distance:** Yohan is at one focus. The foci are on the horizontal axis. So, the horizontal distance from the center of the ellipse to where Yohan stands is *c* = 4.5 feet.
2. **Equation of the Ellipse:** If we place the center of the ellipse at the origin (0,0), the equation of the ellipse is:
(x²/a²) + (y²/b²) = 1
(x²/9.5²) + (y²/70.75) = 1
3. **Substitute x = 4.5 and solve for y:** We want to find the height *y* when *x* = 4.5 (where Yohan stands).
(4.5²/9.5²) + (y²/70.75) = 1
0.225 + (y²/70.75) = 1
y²/70.75 = 0.775
y² = 54.84
y = √54.84 ≈ 7.4 feet
Therefore, the height of the ceiling where Yohan and Jesx stand is approximately 7.4 feet.
**Final Answers:**
* **a.** The highest point of the ceiling from the ground is approximately 8.41 feet.
* **b.** The height of the ceiling from the point where Yohan and Jesx stand is approximately 7.4 feet.
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