Question

help!!!! A hall feet in length 100 is to be designed as a whispering gallery. If the foci are located 20 feet from the​ center, how high will the ceiling be at the​ center?

Answer 1

The height of the ceiling at the center of the whispering gallery is approximately 45.8 feet.

Step-by-step explanation:

To find the height of the ceiling at the center of the whispering gallery, we can use the properties of an ellipse. In an ellipse, the distance between the center and each focus is constant.

In this case, the length of the hall, which is the major axis of the ellipse, is 100 feet. The distance between the center and each focus is given as 20 feet.

We can use the formula to find the height of the ceiling at the center:

Height = sqrt((a^2 - b^2))

Where a is half the length of the major axis (w/2) and b is half the length of the minor axis (ceiling height at the center).

Plugging in the values, we have:

a = 100/2 = 50 feet

b = ?

Using the formula, we have:

Height = sqrt((50^2 - 20^2))

Height = sqrt(2500 - 400) = sqrt(2100) = 45.8 feet (rounded to one decimal place)

Therefore, the height of the ceiling at the center of the whispering gallery is approximately 45.8 feet.