Question Statement

Man $\mathbf{A}$ is standing at one focus of a whispering gallery which is 16 feet from the nearest wall. Man $\mathbf{B}$ is standing at the other focus 100 ft away from Man $\mathbf{A}$.

1. What is the length of the whispering gallery?
2. How high is the elliptical ceiling at the centre?

---

Background and Explanation

This problem applies the geometric properties of an ellipse, where a whispering gallery's cross-section forms an ellipse with two foci. Sound waves emanating from one focus reflect off the ceiling and converge at the other focus. The key parameters are the semi-major axis $a$, semi-minor axis $b$, and focal distance $c$, related by $b^2=a^2-c^2$.

---

Solution

Finding the Length of the Gallery

First, we determine the focal distance $c$ from the distance between the two men standing at the foci:

$\begin{array}{rl}|AB|& =100\\ 2c& =100\\ c& =50\end{array}$

Next, we use the given distance from Man $\mathbf{A}$ to the nearest wall. In an ellipse, the distance from the center to a vertex is $a$, and the distance from the center to a focus is $c$. Therefore, the distance from a focus to the nearest vertex (wall) is $a-c$:

$\begin{array}{rl}a-c& =16\\ a-50& =16\\ a& =66\text{ ft}\end{array}$

The length of the whispering gallery is the major axis length $2a$:

$\text{Length}=2a=2(66)=\boxed{132\text{ ft}}$

Finding the Height at the Centre

To find the height of the elliptical ceiling at the center (which equals the semi-minor axis $b$), we use the fundamental relationship for ellipses:

$\begin{array}{rl}{b}^2& =a^2-c^2\\ & =(66{)}^2-(50{)}^2\\ & =4356-2500\\ & =1856\end{array}$

Taking the square root and simplifying:

$\begin{array}{rl}b& =\sqrt{1856}\\ & =\sqrt{64\times 29}\\ & =8\sqrt{29}\end{array}$

Therefore, the height of the elliptical ceiling at the centre is:

$\boxed{8\sqrt{29}\text{ ft}}(\approx 43.08\text{ ft})$

---

Key Formulas or Methods Used

• Distance between foci: $2c$ (where $c$ is the distance from center to each focus)
• Length of major axis: $2a$ (total length of the elliptical gallery)
• Ellipse relationship: $b^2=a^2-c^2$ or equivalently $c^2=a^2-b^2$
• Distance from focus to vertex: $a-c$ (for the nearer vertex) and $a+c$ (for the farther vertex)

---

Summary of Steps

1. Find $c$: Use the distance between the two foci (men $\mathbf{A}$ and $\mathbf{B}$): $2c=100\Rightarrow c=50$ ft.
2. Find $a$: Use the distance from focus to nearest wall: $a-c=16\Rightarrow a=66$ ft.
3. Calculate length: The gallery length equals the major axis: $2a=132$ ft.
4. Calculate height: Use $b^2=a^2-c^2$ to find the semi-minor axis $b=8\sqrt{29}$ ft, which represents the ceiling height at the center.