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-circle.Find the distance sound travels as it leaves focus F and returns to focus F'.Round your answer to one decimal place, if necessary.
A)16 meters
B)16.8 meters
C)33.5 meters
D)32 meters
E)31.2 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-circle.Find the distance sound travels as it leaves focus F and returns to focus F'.Round your answer to one decimal place, if necessary. ​  ​
A)16 meters 
B)16.8 meters 
C)33.5 meters 
D)32 meters 
E)31.2 meters

Quizplus · Accepted Answer

By definition of an ellipse, we know that the length of the major axis is twice the distance between the two focii (in this case, F and F'). We also know that the semi-circle whispering gallery is a semi-ellipse, since it is just half of the full ellipse.

Using the Pythagorean theorem, we can find the length of the minor axis (the height of the semi-ellipse) as follows:

$(AC)^2 = (AB)^2 - (BC)^2$
$(AC)^2 = (8)^2 - (6)^2$
$(AC)^2 = 28$
$AC \approx 5.3$

Therefore, the length of the major axis (the diameter of the semi-circle) is twice this value, or approximately 10.6.

Since sound travels the full length of the major axis twice (once from F to F', and again from F' to F), the total distance it travels is $10.6 	imes 2 = 21.2$. However, this is the distance the sound travels along the ellipse itself. When we unfold the semi-ellipse into a straight line (as the sound waves would travel), we effectively stretch out the major axis. The stretching factor is equal to the ratio of the length of the straight line to the length of the major axis, which is the same as the ratio of the circumference of a circle to its diameter (i.e. $\pi$).

Thus, the total distance the sound travels is $\pi \cdot 21.2 \approx \boxed{33.5 	ext{ meters}}$.

Any Sound from One Focus of an Ellipse Reflects Off