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Hopefully this is the right place for the following story problem.
A whispering gallery has an elliptical ceiling. A person standing at one focus of the ellipse can whisper and be heard
by another person standing at the other focus, because all the sound waves that reach the ceiling from one focus
are reflected to the other focus. A hall 100 feet in length is to be designed as a whispering gallery.
If the foci are located 25 feet from the center, how high will the ceiling be at the center?
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In terms of an ellipse characteristic points and characteristic elements
(see the lesson Ellipse definition, canonical equation, characteristic points and elements in this site)
we are given the semi-major axis "a" = = 50 ft and the linear eccentricity "c" = = 25 ft,
and we need to find the semi-minor axis "b" based on this data.
OK. We have = , or = .
Hence, = = ,
which gives us
b = = 43.30 ft.
Answer. The ceiling height should be 43.30 ft.