Any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus. This property explains whispering galleries such as Statuary Hall in Washington, D.C. The whispering gallery shown in illustration has the shape of a semi-ellipse. Find the distance sound travels as it leaves focus F and returns to focus F'.
A)63.2 meters
B)33.5 meters
C)31.6 meters
D)32 meters
E)31 meters

Question

Any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus. This property explains whispering galleries such as Statuary Hall in Washington, D.C. The whispering gallery shown in illustration has the shape of a semi-ellipse. Find the distance sound travels as it leaves focus F and returns to focus F'. 
A)63.2 meters 
B)33.5 meters 
C)31.6 meters 
D)32 meters 
E)31 meters

Quizplus · Accepted Answer

We can use the formula for the distance between a point on an ellipse and its two foci:

c^2 = a^2 - b^2

where c is the distance between the foci, a is the distance from the center of the ellipse to the farthest point on the ellipse (the "major radius"), and b is the distance from the center of the ellipse to the closest point on the ellipse (the "minor radius").

In this case, we are dealing with a semi-ellipse, so we only need to use half of these values. From the diagram, we can see that the distance between the foci is 16 meters, and the major radius is 20 meters (halfway between 12 and 28). So:

c^2 = a^2 - b^2 
16^2 = 20^2 - b^2 
b^2 = 20^2 - 16^2 
b^2 = 144

Now we can find the distance by adding twice the distance from F to the closest point on the ellipse:

distance = 2(sqrt(a^2 - b^2) + c) 
distance = 2(sqrt(20^2 - 144) + 16) 
distance = 2(sqrt(256) + 16) 
distance = 2(16 + 16) 
distance = 64

However, we need to remember that the sound travels from F to the closest point on the ellipse, and then back to F'. So the final distance is half of what we calculated:

distance = 64/2 
distance = 32

Therefore, the answer is choice C, 31.6 meters.

Any Sound from One Focus of an Ellipse Reflects Off