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)59.4 meters
B)29 meters
C)29.7 meters
D)30.3 meters
E)31.4 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)59.4 meters 
B)29 meters 
C)29.7 meters 
D)30.3 meters 
E)31.4 meters

Quizplus · Accepted Answer

The distance sound travels is equal to the perimeter of the ellipse, which can be approximated using the formula $P = \pi (a+b)$, where $a$ and $b$ are the lengths of the semi-major and semi-minor axes respectively. In this case, we are given that the length of the straight section is 12 meters and the distance between the foci is 20 meters. Since the ellipse is symmetrical, we can assume that the semi-major axis is equal to half of 20 meters which is 10 meters. To find the semi-minor axis, we need to use the fact that the length of the straight section plus twice the semi-minor axis equals 20 meters. Therefore, $12 + 2b = 20$, so $b = 4$. Plugging in these values into the formula gives $P = \pi (10+4) \approx 29.7$ meters. Therefore, the best choice is C.

Any Sound from One Focus of an Ellipse Reflects Off