Question

Whispering Gallery : Jim, standing at one focus of a whispering gallery, is $6$feet from the nearest wall. His friend is standing at the other focus, $100$feet away. What is the length of this whispering gallery? How high is its elliptical ceiling at the center?

Short answer

The height of the ceiling at the center is$25.21$feet.

Step-by-step solution

Step 1. Given information.

The distance between the foci is $100$feet.

$$\begin{gathered} c=\frac{100}{2} \\ c=50 \end{gathered}$$

Thus, the foci are $(\pm 50,0)$.

The distance of one of the focus from the nearest wall is $6$feet.

This means that one of the vertices are at $(56,0)$

Step 2. Height of the ceiling.

The length of the whispering gallery is

$$\begin{gathered} 2a=112 \\ a=56 \end{gathered}$$

Substitute the value of $c=50$and $a=56$we get.

$$\begin{gathered} b^2=a^2-c^2 \\ {b}^2={56}^2-{50}^2 \\ {b}^2=636 \\ b=\sqrt{636} \\ b=25.21 \end{gathered}$$

The height of the ceiling at the center is$25.21feet$.