Elliptical Room Problem. Please Help!

SammC. · #1 $Old$ November 8th, 2009, 09:31 AM

I cannot figure this out at all!

1. Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long.

a. Find an equation that models the shape of the room.
b. How far apart are the two foci?
c. What is the area of the floor of the room?

Opalg · #2 $Old$ November 8th, 2009, 09:53 AM

Quote:

Originally Posted by SammC. $View Post$

I cannot figure this out at all!

1. Statuary Hall is an elliptical room in the United States Capitol in Washington D.C. The room is also called the Whispering Gallery because a person standing at one focus of the room can hear even a whisper spoken by a person standing at the other focus. This occurs because any sound that is emitted from one focus of an ellipse will reflect off the side of the ellipse to the other focus. Statuary Hall is 46 feet wide and 97 feet long.

a. Find an equation that models the shape of the room.
b. How far apart are the two foci?
c. What is the area of the floor of the room?

The equation for an ellipse is $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ , where a is half the larger diameter and b is half the smaller diameter. The eccentricity e of the ellipse is given by the equation $e^2 = 1-\frac{b^2}{a^2}$ , and the distance between the foci is $2ae$ . Finally, the area of the ellipse is $\pi ab$ . That tells you all you need to know to work out the answers.

galactus · #3 $Old$ November 8th, 2009, 09:56 AM

Center the ellipse at the origin. The major axis has length 97 and the minor axis length 46.

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$

a=48.5 and b=23.

$c=\sqrt{a^{2}-b^{2}}$

The area of an ellipse can be found by the formula ${\pi}ab$