What is the height of the ceiling above each “whispering point”?
Since the ceiling is half of an ellipse (the top half, specifically), and since the foci will be on a line between the tops of the “straight” parts of the side walls, the foci will be five feet above the floor, which sounds about right for people talking and listening: five feet high is close to face-high on most adults. I’ll center my ellipse above the origin, so (h, k) = (0, 5). The foci are thirty feet apart, so they’re 15 units to either side of the center. In particular, c = 15. Since the elliptical part of the room’s cross-section is twenty feet high above the center, and since this “shorter” direction is the semi-minor axis, then b = 20. The equation b2 = a2 c2 gives me 400 = a2 225, so a2 = 625. Then the equation for the elliptical ceiling is: I need to find the height of the ceiling above the foci. I prefer positive numbers, so I’ll look at the focus to the right of the center. The height (from the ellipse’s central line through its foci, up to the ceiling) will be the y-value
