Question

The base of an auditorium is in the form of an eclipse 200 feet long and 100 feet wide a pin drop near one focus can clearly be heard at the other focus determine the distance between the foci to the nearest 10th of a foot.... how would I work this out??

Answer 1

Answer:

Let the coordinate of focus be $(\pm c , 0)$

As per the statement: The base of an auditorium is in the form of an eclipse 200 feet long and 100 feet wide.

⇒Length of Major axis=base of an auditorium = 200 feet and Length of a minor axis=wide of a auditorium = 100 ft

Semi-major axis (a) = 100 ft and

semi-minor axis(b) = 50 ft

Then, by an equation:

$c^2 = a^2-b^2$

Solve for c:

Substitute the given values we have;

$c^2=(100)^2-(50)^2$

Simplify:

$c^2 = 7500$

or

$c=\sqrt{7500} = 86.6025404$ ft

Distance between the foci is,  $2c = 2 \cdot 86.6025404 = 173.205081$

Therefore, the distance between the foci to the nearest 10th of a foot is, 173.2 ft