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In a whispering gallery, a person standing at one focus of the ellipse can whisper and be heard by another person standing at the other focus, because all sound waves that reach the ceiling from one focus are reflected to the other focus. A hall 120 feet in length is to be designed as a whispering gallery. Of foci are located 40 feet from the center, how high will the ceiling be at the center

Answer :

Answer:

Hence, the height of the ceiling at the center is 44.72 feet.

Explanation:

Consider that a whispering gallery has a length of 120 feet, and the foci are located 40 feet from the center.

Assume that the shape is a horizontal semi-ellipse with a center (0,0) and has the equation,

[tex]\frac{x^{2} }{ a^{2} } + \frac{y^{2} }{b^{2} } =1[/tex]  where a>b,

so,

2a = 120

a = 60

and

c = 40

Therefore, the equation of the ellipse is,

[tex]c^{2} = a^{2} -b^{2} \\\\40^{2} = 60^{2} -b^{2}\\\\b^{2} =2000\\\\b= 44.72.[/tex]

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