An elliptical whispering gallery has a major axis of 28 feet an a minor axis of 22 feet. Where should two speakers position themselves so as to most easily hear each other?
A) 5.34 feet in from the ends of the minor axis
B) 8.66 feet in from the ends of the minor axis
C) 5.34 feet in from the ends of the major axis
D) 8.66 feet in from the ends of the major axis

Question

Quizplus · Accepted Answer

In an elliptical whispering gallery, the best positions for two speakers to hear each other are at the two foci of the ellipse. The distance of each focus from the center of the ellipse is given by the formula $c = \sqrt{a^2 - b^2}$, where $a$ is the semi-major axis and $b$ is the semi-minor axis. Given a major axis of 28 feet and a minor axis of 22 feet, the semi-major axis $a = 14$ feet and the semi-minor axis $b = 11$ feet. Plugging these into the formula gives $c = \sqrt{14^2 - 11^2} = \sqrt{196 - 121} = \sqrt{75} \approx 8.66$ feet. Therefore, the speakers should position themselves 8.66 feet in from the ends of the major axis.

An Elliptical Whispering Gallery Has a Major Axis of 28