Chapter 10, Problem 80RE

To determine

Where are the foci located in the hall?

Answer to Problem 80RE

  $\boxed{6feet}$

Explanation of Solution

Given information:

The figure shows below the specifications for an elliptical ceiling in a hall designed to be a whispering gallery. Where are the foci located in the hall?

  

Calculation:

Consider the given figure,

The height of the elliptical ceiling is $=25feet$

Width $=80feet$

Length of semi-major axis, $a=40feet$

Length of semi-minor axis, $b=25feet$ .

Foci lie at the points by coordinates $\left(ae,0\right)$ and $\left(-ae,0\right)$ ,where $e$ is the eccentricity of the given elliptical ceiling.

Assume that center of ellipse lies on the origin,

Now find eccentricity of the ellipse,

  $\begin{array}{l}e=\frac{\sqrt{a^2-b^2}}{a}\\ =\frac{\sqrt{{40}^2-{25}^2}}{40}\\ =\frac{\sqrt{975}}{40}\\ =\frac{\sqrt{39}}{8}\end{array}$

The foci of the ceiling lie at a distance of $ae=5\sqrt{39}$ from each of the edge at a height of $6feet$ assuming the center of the ellipse lies at the origin.

Hence, the center is at a height of $\boxed{6feet}$ from the ground.