Question 1289: Jake, standing at one focus of a whispering gallery, is 6 feet from the nearest wall. His friend is standing at the other focus, 100 feet away. What is the length of this whispering gallery? How high is its elliptical ceiling at the center?
Answer by khwang(438)   (Show Source):
You can put this solution on YOUR website! Sol:
Consider the diagram below:
F1, F2 are two focii, FA = 6ft, F1F2 = 100 ft
B-----F2----------------F1----A
(-a,0) (-c,0) (c,0) (a,0)
Assume theequation of the ellipse as:
x^2/a^2 + y^2/b^2 =1.
We have a^2 = b^2 + c^2.
Now, a-c = AF1= 6, 2c=F1F2 =100, so c= 50, a =56
and b^2 = 56^2 - 50^2 = 636, b= 25.21
Hence,the length of this whispering gallery is 2a = 112 ft
and it height is 2b = 50.42 ft
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