Final answer:
The height of the ceiling at the center of a whispering gallery hall, which is 100 feet in length with the foci 20 feet from the center, is approximately 45.83 feet.
Step-by-step explanation:
The question involves designing a hall as a whispering gallery, which means it's a problem related to the geometry of ellipses.
For an ellipse, the distance between the center and each focus is known as the focal length (f).
The sum of the distances from each focus to any point on the ellipse is constant and equal to the major axis (2a).
Given that the hall is 100 feet in length, we can assume this is the major axis, so a = 50 feet. The foci are 20 feet from the center,
hence the focal length f is 20 feet.
To find the height of the ceiling at the center of the hall, we can use the relationship between the semi-major axis a, the semi-minor axis b, and the focal length f: a^2 = b^2 + f^2.
Solving for b, we get b = √(a^2 - f^2) = √(50^2 - 20^2) = √(2500 - 400) = √2100 feet = ≈45.83 feet (approximately).
The height of the ceiling at the center, which is the semi-minor axis b, is therefore approximately 45.83 feet.