Example 7 from Algebra and Trigonometry, 12.1 The Ellipse

Locating the Foci of a Whispering Chamber

The Statuary Hall in the Capitol Building in Washington, D.C. is a whispering chamber. Its dimensions are 46 feet wide by 96 feet long as shown in Figure 13.

What is the standard form of the equation of the ellipse representing the outline of the room? Hint: assume a horizontal ellipse, and let the center of the room be the point $(0, 0) .$
If two senators standing at the foci of this room can hear each other whisper, how far apart are the senators? Round to the nearest foot.

A horizontal ellipse with rays originating at each focus and going through the other focus.

Figure 13

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. We are assuming a horizontal ellipse with center (0,0), so we need to find an equation of the form x2a2+y2b2=1, where a>b. We know that the length of the major axis, 2a, is longer than the length of the minor axis, 2b. So the length of the room, 96, is represented by the major axis, and the width of the room, 46, is represented by the minor axis.
  - Solving for $a,$ we have $2 a = 96,$ so $a = 48,$ and $a^{2} = 2304.$
  - Solving for $b,$ we have $2 b = 46,$ so $b = 23,$ and $b^{2} = 529.$
  Therefore, the equation of the ellipse is $\frac{x^{2}}{2304} + \frac{y^{2}}{529} = 1.$
2. To find the distance between the senators, we must find the distance between the foci, $(\pm c, 0),$ where $c^{2} = a^{2} - b^{2} .$ Solving for $c,$ we have:
  $\begin{array}{l} c^{2} = a^{2} - b^{2} \\ c^{2} = 2304 - 529 & \begin{matrix} \end{matrix} Substitute using the values found in part (a) . \\ c = \pm \sqrt{2304 - 529} & \begin{matrix} \end{matrix} Take the square root of both sides . \\ c = \pm \sqrt{1775} & \begin{matrix} \end{matrix} Subtract . \\ c \approx \pm 42 & \begin{matrix} \end{matrix} Round to the nearest foot . \end{array}$
  The points $(\pm 42, 0)$ represent the foci. Thus, the distance between the senators is $2 (42) = 84$ feet.

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Suppose a whispering chamber is 480 feet long and 320 feet wide.

ⓐ What is the standard form of the equation of the ellipse representing the room? Hint: assume a horizontal ellipse, and let the center of the room be the point $(0, 0) .$
ⓑ If two people are standing at the foci of this room and can hear each other whisper, how far apart are the people? Round to the nearest foot.

How did it go?

Your turn

The book's Try It right after this example: same idea, new numbers. Only the answer is given.

Suppose a whispering chamber is 480 feet long and 320 feet wide.

Solution

Your turn

Solution

Your turn