Question

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. The dimensions of Statuary Hall are 46 feet wide by 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?

Answer 1

Answer:

a) $\frac{x^{2}}{2352.25} + \frac{y^{2}}{529} = 1$, b) $c = 42.7 ft$, c) $A \approx 3504.447 ft^{2}$

Step-by-step explanation:

a) An Ellipse centered at origin is modelled by using this formula:

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

Where $a, b$ represents the lengths of horizontal and vertical axis, respectively. Let consider that horizontal axis is parallel and coincident with width of Statuary Hall. So, the measures of each axis are, respectively:

$a = 48.5 ft, b = 23 ft$  

By substituting known variables, the equation that models the hall is:

$\frac{x^{2}}{2352.25} + \frac{y^{2}}{529} = 1$

b) The distance between origin and any of the foci is:

$c = \sqrt{a^{2} - b^{2}} \\c = 42.7 ft$

c) The area of ellipse can determined by applying this formula:

$A = \pi \cdot a \cdot b$

$A \approx 3504.447 ft^{2}$