Question

Wonder women, standing at one focus of a whispering gallery, is 10 feet from the nearest wall. Her friend superman is standing at the other focus 80 feet away. What is the length of the whispering gallery? How high is its elliptical ceiling at the center?

Answer 1

Answer:

• length 100 ft
• height 30 ft

Step-by-step explanation:

The center is 40 ft from the focus, so the wall is 40+10 = 50 ft from the center. From wall to wall must be 2×50 ft = 100 ft.

The length is 100 ft.

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Assuming the gallery has an elliptical vertical cross section, the distance from the focus to the center ceiling is the same as half the length of the gallery, so we can use the Pythagorean theorem to find the height:

  50² = 40² + h²

  h = √(2500 -1600) = √900 = 30

The height of the ceiling at the center is 30 feet.

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Comment on the geometry

We expect the gallery to have an elliptical floor plan, rather than an elliptical vertical cross section. In the latter case, the focus would have to be above the floor for it to work, so the ceiling at center would have to be higher than 30 ft. by the distance that the focus is above the floor.