Final answer:
The designed whispering gallery is an elliptical shape 100 feet in length with foci 20 feet from the center.
To find the height at the center, we calculate the semi-minor axis (b) using the ellipse equation, resulting in an approximate height of 45.83 feet.
Step-by-step explanation:
The question is about the design of a whispering gallery, which involves the mathematical concept of an ellipse. Whispering galleries are architectural features that utilize the acoustic properties of ellipses.
The foci (plural of focus) of an ellipse are two points such that the sum of the distances from any point on the ellipse to each focus is constant.
In this scenario, the hall is 100 feet in length, which means the major axis of the ellipse is 100 feet long.
Since the foci are located 20 feet from the center, the distance between the two foci is 40 feet.
The focal distance (distance from the center to a focus) is therefore 20 feet.
We can use the standard ellipse equation √(a^2 - b^2) = c, where 'a' is the semi-major axis, 'b' is the semi-minor axis (half the height of the ceiling at the center), and 'c' is the focal distance.
To find the height of the ceiling at the center ('b'), we first find 'a':
½ length of hall = a => a = 50 feet
Next, we calculate 'c':
c = focal distance from center = 20 feet
Finally, solve for 'b' using the ellipse equation:
√(a^2 - c^2) = b
√(50^2 - 20^2) = b
√(2500 - 400) = b
√2100 = b
b ≈ 45.83 feet
So, the height of the ceiling at the center of the whispering gallery would be approximately 45.83 feet.