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

According to the property of an ellipse mentioned in the question, any sound from one focus of an ellipse reflects off the ellipse directly back to the other focus. In the given semi-circle, the distance between the two foci (F and F') can be found using the formula d = 2a (where 'a' is the length of the semi-axis). Using the given diameter of 20 meters, we can find the value of 'a' to be 10 meters. Therefore, the distance between F and F' is 2 * 10 = 20 meters. Since the sound has to travel from F to a point on the circumference of the semi-circle, and then reflect off the ellipse to reach F', the total distance traveled is the circumference of the semi-circle + the distance between F and F'. The circumference of a semi-circle is (πd)/2, where 'd' is the diameter (20 meters in this case). Therefore, the total distance traveled by the sound is (πd)/2 + 20. Substituting the value of 'd', we get (π * 20)/2 + 20 which simplifies to 30.97 meters. Rounding this value to one decimal place, we get 33.5 meters. Hence, the correct answer is C.

Any Sound from One Focus of an Ellipse Reflects Off