Question

PLEASE HELP I'M DESPERATE!!!!A whispering gallery is designed using an elliptical ceiling. It operates on the principle that the sound projected from one focus of an ellipse reflects off the ceiling and back to the other focus. The United States Capital contains such an elliptical room. The room is 96 feet in length, 46 feet in width, and has a ceiling that is about 23 feet high.

a) Write an equation modeling the shape of the room. Assume that it is centered at the origin and the major axis is horizontal.

Answer 1

So we assume this is actually a hemi-ellipsoid. (the top half of a 3D elliptical shape). The cool thing is that the equation for such a shape is very easy to describe.
\[\frac{x^2}{a^2}+\frac{y^2}{b^2}+\frac{z^2}{c^2} = 1\] With z being positive only.

Answer 2

When you are standing in the center of the room, x and y are zero because you are at the origin and the height of the ceiling (z) is solved for with:
\[\frac{0}{a^2}+\frac{0}{b^2}+\frac{z^2}{c^2} = 1\]

Answer 3

What is the value of c if the room is 23 feet high?

Answer 4

hmm lemme think hold on

Answer 5

4.796?

Answer 6

Close. z=23 when x=0 and y=0 so for
\[\frac{z^2}{c^2}=1; \ \ c=23\]
Similarly, b will be half the width of the room and a will be half of the length.