Answer to Question #147018 in Analytic Geometry for camille

Question

A big room is constructed so that the ceiling is a dome that is semielliptical in shape. If a person stands at one focus and speaks, the sound that is made bounces off the ceiling and gets reflected to the other focus. Thus, if two people stand at the foci (ignoring their heights), they will be able to hear each other. If the room is 34 m long and 8 m high, how far from the center should each of two people stand if they would like to whisper back and forth and hear each other?

Answer 1

We have semielliptical ceiling, so we have to consider it as a half of ellipse.

General form of ellipse is; (x-h)²/a²+(y-h)²/b²=1;

a is the length of a major radius;

b is the the length of minor radius;

in our case the length of the room is 2a and equals to 34 m, a=17;

the height of the room is taken as b and it equals to 8 m;

h and k are equal to 0,

So our ellipse is x²/289+y²/64=1;

Now we need to find the distance between the center and the foci, c;

c²=a²-b² , c²= 289-64, c=15;

Answer is 15 m;

People should stand 15 m away from the center.