Question

The court room in the capitol building is elliptical in shape. This creates a problem because private conversations, even whispers, can be heard at certain points in the room. These points are located on the foci of the ellipse. The court room width is 60 feet and the length of the room is 100 feet.

a. Write the equation of the ellipse if the ellipse were overlaid on a coordinate plane with its center at the origin.
b. Describe the graph of the ellipse, state the center and vertices. What is the defining feature of the equation that tells us it is an ellipse?
c. If you were a lawyer trying to hear a private conversation before court how far from the center would you and those having the private conversation have to be standing?
d. How far apart are you standing from each other?
e. If you stand at the center of the room, how far away is the nearest wall?

Answer 1

Explanation:

a.

The length of the major axis = 2a = 100' => a =50'.

The length of the minor axis = 2b = 60' => b = 30'.

Let's say that the X-axis is along the length of the centre of the court room. And, the Y-axis is along the width in the centre. Let the centre point of the court room be called the origin denoted by O with the coordinates (0,0).

The equation of the ellipse:

x²/50² + y²/30² = 1.

b. e = eccentricity of the ellipse.

e² = 1 - b²/a² = 1 - 900/2500 = 16/25

e = 4/5 .

The focii F1 and F2 are at a distance a × e = 32' from the origin 0 on either side, on the major axis.

Vertices A (-a, 0) = (-50 ', 0), : B = (a,0) = (50' ,0) .

C ( -b,0) = (0, -30) ; D(b,0) = (0,30).

The defining feature of an ellipse is that for any point P (x, y) on the ellipse, the sum of the distances PF1 and PF2 to the focii is a constant.

c.

We would stand at the Focii. They're at a distance= a×e = 32' away from the centre.

d. The Focii are 64 feet away from each other.

e. From the centre O, the nearest wall is at the distance= b = 30 '.