A hall 100 feet in length is to be designed as a whispering gallery. If the foci are located 21 feet from the center, how high will the ceiling be at the center?
To determine the height of the ceiling at the center of the hall, we need to understand the concept of a whispering gallery and the properties of an ellipse.
A whispering gallery is an architectural feature where sound waves are reflected off the walls and focused at a specific point called a focus. The shape that allows this phenomenon to occur is an ellipse.
In our case, the foci are located 21 feet from the center. The distance from the center of an ellipse to its farthest point is called the major axis. In this scenario, the major axis is twice the distance between the foci. So, the major axis of our ellipse is 2 * 21 feet = 42 feet.
The height of an ellipse at the center is called the minor axis. To find the minor axis, we need to use the formula for an ellipse:
(major axis/2)^2 = (minor axis/2)^2 + distance from the center to either focus^2
Substituting the known values:
(42/2)^2 = (minor axis/2)^2 + 21^2
21^2 = (minor axis/2)^2 + 21^2
0 = (minor axis/2)^2
From this, we can see that the minor axis is 0. Therefore, the height of the ceiling at the center of the hall will be 0 feet. This means that the ceiling will meet the ground at the center of the hall since it is in the shape of a flat ellipse.