Problem 316 - Calculus Volume 3

Question

Whispering galleries are rooms designed with elliptical ceilings. A person standing at one focus can whisper and be heard by a person standing at the other focus because all the sound waves that reach the ceiling are reflected to the other person. If a whispering gallery has a length of 120 feet and the foci are located 30 feet from the center, find the height of the ceiling at the center.

Step-by-Step Solution

Answer

The height of the ceiling at the center is \(30\sqrt{3}\) feet.

Step 1: Understanding the Problem

The problem describes an elliptical ceiling where sound reflects from one focus to the other due to the properties of ellipses. We are given the length of the room (major axis, 120 feet) and the distance from the center to each focus (30 feet) and need to find the height of the ceiling at the center.

Step 2: Identify Major and Minor Axis

In an ellipse, the length of the major axis is the total width of the ellipse at its widest point, which is given as 120 feet. The distance from the center to a focus is 30 feet. We denote the semi-major axis as \(a\) and the semi-minor axis as \(b\). Since the total major axis is 120 feet, \(a = \frac{120}{2} = 60\) feet.

Step 3: Using Ellipse Properties

In an ellipse, the relationship between the semi-major axis \(a\), semi-minor axis \(b\), and the distance from the center to a focus \(c\) is given by \(c^2 = a^2 - b^2\). Here, \(c = 30\) feet.

Step 4: Solving for the Semi-minor Axis

Using the equation \(c^2 = a^2 - b^2\), substitute the known values: \(30^2 = 60^2 - b^2\). \(900 = 3600 - b^2\). Thus, \(b^2 = 3600 - 900 = 2700\). Then, \(b = \sqrt{2700} = 30\sqrt{3}\).

Step 5: Calculate the Height at the Center

The height of the ceiling at the center of the ellipse is the length of the semi-minor axis \(b\). Therefore, the height is \(30\sqrt{3}\) feet.

Key Concepts

Whispering Gallery

A whispering gallery is a fascinating architectural feature often found in elliptical-shaped rooms. These galleries have unique acoustics, allowing sound to bounce efficiently from one focal point to another. Imagine standing at one focus and whispering; those whispers can be clearly heard at the other focus. This phenomenon occurs because of the unique properties of an ellipse, where the paths of sound waves are perfectly directed to ensure that one focus receives sound from the source at the opposite focus. Whispering galleries can be found in famous locations like St. Paul's Cathedral in London. They highlight how mathematics and architectural design harmonize to create spaces with powerful acoustic abilities.

Ellipse Properties

Ellipses are captivating geometric shapes characterized by their oval form. Unlike a circle, which has a single center point, ellipses have two foci. The property of ellipses that makes them special is that the sum of the distances from any point on the ellipse to the two foci is constant. This property is what enables the sound in a whispering gallery to travel in such an efficient manner. Since sound traveling reflects off the curved surface and converges back onto the other focus, the ellipse allows even a whisper to be delivered clearly. This property is not just useful for architectural wonders but is also important in many scientific applications, such as the orbits of planets.

Sound Reflection

Sound reflection in a whispering gallery exemplifies the practical use of geometric principles. When a sound wave is emitted from a source located at one focus of an elliptical gallery, the wave reflects off the surface and passes through the other focus. This is due to the unique path that follows the ellipse's property, efficiently channeling sound. The angle at which the sound wave hits the surface equals the angle at which it reflects off. This principle is akin to how light reflects in optics, following the law that the angle of incidence is equal to the angle of reflection. Hence, the elliptical ceiling functions as a perfect conduit for sound, emphasizing the unison of physics and geometry.

Semi-Major and Semi-Minor Axis

Understanding the terms semi-major and semi-minor axis is crucial when dealing with ellipses. The semi-major axis (\(a\)) is half the length of the longest diameter of the ellipse, while the semi-minor axis (\(b\)) is half the length of the shortest diameter. In the context of our problem, the room spans 120 feet across the major axis, thus \(a = 60\) feet. The foci are set 30 feet from the center, where \(c = 30\) feet. Using the relationship \(c^2 = a^2 - b^2\), we solve for \(b\) to find it is \(30\sqrt{3}\) feet. This reveals the height at the center. These axes help define the shape and size of the ellipse, playing a critical role in its applications, from architecture to astronomy.