Question

A whispering gallery, often elliptical in shape, has acoustic properties such that a whisper made at one point can be

heard at other distant points. A science museum is designing a new exhibit hall that will illustrate a whispering gallery.
The hall will be 110 ft in length with the ceiling 40 ft high at the center. How far are the foci from the center of the ellipse?

Answer 1

The foci of an elliptical whispering gallery can be found using the formula c = sqrt(a^2 - b^2), where a is the semi-major axis and b is the semi-minor axis of the ellipse. In this case, the hall is 110 ft in length and the ceiling is 40 ft high at the center. Therefore, the foci of the ellipse are approximately 51.24 ft from the center.

Step-by-step explanation:

The foci of an elliptical whispering gallery can be found using the formula:

c = sqrt(a^2 - b^2)

where a is the semi-major axis and b is the semi-minor axis of the ellipse. In this case, the length of the ellipse is the semi-major axis and the height of the center is the semi-minor axis.

Given that the hall is 110 ft in length and the ceiling is 40 ft high at the center, we can calculate the semi-major axis (a) as 55 ft and the semi-minor axis (b) as 20 ft. Plugging these values into the formula, we get:
c = sqrt(55^2 - 20^2)
c = sqrt(3025 - 400)
c = sqrt(2625)
c ≈ 51.24 ft

Therefore, the foci of the ellipse are approximately 51.24 ft from the center.