Question

In a room shaped like an ellipse, something whispered at 1 focus can be clearly heard at the other focus. The U.S. Capitol National Statuary Hall in Washington, D.C. is such a whispering gallery. From 1807 to 1857, the U.S. House of Representatives met in this room. According to legend, when John Quincy Adams was a representative following his presidency, he used to look as though he was asleep at his desk during debate, but he was listening to what representatives of the opposing party on the other side of the room were whispering. If the elliptical pattern of this whispering gallery is 92 feet long by 58 feet wide, how far from the whisperer at 1 focus is the listener at the other focus

Answer 1

The listener is approximately 71.414 feet far from the whisperer.

Explanation:

We must remember that two fundamental variables for every ellipse are the major semiaxis (
`a`) and the minor semiaxis (
`b`). The major semiaxis equals a half of the length of the whispering gallery, whereas the minor semiaxis equals a half of the width of the whispering gallery. These are:

`a = 46\,ft` and
`b = 29\,ft`

The distance between both whisperers (
`d`), measured in feet, is two times the distance between center and any of the foci (
`c`), measured in feet, whose valued is obtained by using this Pythagorean identity:

`d = 2\cdot c`

`c = \sqrt{a^(2)-b^(2)}`

If
`a = 46\,ft` and
`b = 29\,ft`, then:

`c = \sqrt{(46\,ft)^(2)-(29\,ft)^(2)}`

`c \approx 35.707\,ft`

`d = 2\cdot (35.707\,ft)`

`d = 71.414\,ft`

The listener is approximately 71.414 feet far from the whisperer.