The vertical cross section of a dome is semielliptical in sh

The vertical cross section of a dome is semi-elliptical in shape. The cross section is 220 feet long with a maximum height of 70 feet. Where should two people stand in order to maximize the whispering point?

The people should stand on opposite sides of the room each being located at one of the foci located _____ feet from the wall.

Solution

The people should stand on opposite sides of the room each being located at one of the foci

Now Place the center of the room at (0, 0).
The ends of the room are at (±110, 0), the middle of the room\'s ceiling is at (0, 70)

a = distance from the center to the vertices along the major axis
b = distance from the center to the co-vertices along the minor axis
c = distance from the center to the foci along the major axis
The relationship between a, b, and c is
a² - b² = c²

110^2 - 70^2 = c^2

c = +/- sqrt(7200) = +/- 84.85 feet

The best places to listen are at ( 84.85, 0) and (-84.85 ,0)

i.e. 25.15 feet from the walls at foci