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.

Short answer

The height of the ceiling at the center is approximately 51.96 feet.

Step-by-step solution

Identify Given Information

The whispering gallery is shaped like an ellipse. We are given that the length of the gallery is 120 feet, meaning the major axis is 120 feet long. We are also given that the distance from the center to each focus is 30 feet.

Understand the Ellipse's Structure

The key variables for an ellipse are its semi-major axis \( a \), semi-minor axis \( b \), and the distance from the center to a focus \( c \). Here, \( 2a = 120 \), so \( a = 60 \). The distance from the center to a focus is \( c = 30 \).

Apply the Ellipse Equation

For an ellipse, the relationship between \( a \), \( b \), and \( c \) is given by: \[ c^2 = a^2 - b^2 \]. Using the given values, \( c = 30 \) and \( a = 60 \), substitute into this equation.

Solve for \( b^2 \)

Substitute \( a = 60 \) and \( c = 30 \) into the equation: \[ 30^2 = 60^2 - b^2 \]. Simplify this to \[ 900 = 3600 - b^2 \].

Determine \( b \)

Rearrange the equation to solve for \( b^2 \): \( b^2 = 3600 - 900 = 2700 \). Thus, \( b = \sqrt{2700} \).

Calculate the Height of the Ceiling

The height of the ceiling at the center is the value of \( b \), which is the semi-minor axis length: \( b = \sqrt{2700} = 30\sqrt{3} \approx 51.96 \). Thus, the height of the ceiling at the center is approximately 51.96 feet.

Key concepts

Elliptical Geometry

Elliptical geometry is a fascinating branch of mathematics focused on the shape of ellipses. An ellipse is an elongated circle and is defined by two focal points. The total distance from one focal point to any point on the edge of the ellipse and back to the other focal point is constant. This unique property of ellipses has practical applications, such as in whispering galleries.
These structures are designed with elliptical shapes to harness sound reflections advantages. Understanding the basic components of an ellipse, such as the two axes and the foci, is critical for solving related calculus and geometry problems.
In contextual problems, like architectural designs, this geometry ensures even sound distribution, making interactions in places like whispering galleries more effective.

Reflection of Sound Waves

Reflection of sound waves is a key phenomenon that allows whispering galleries to function. When a sound hits the ceiling of an elliptical gallery, it reflects off in such a way that if it comes from one focus, it will always pass through the other. This is because of the constant sum property of ellipses.
This property ensures that sound waves travel efficiently between two points, allowing whispers to be heard clearly over distances, just as in the problem discussed. Such understanding is crucial in designing spaces where acoustics play a significant role, ensuring sound reaches the desired focal point.
Acoustics in architectural designs, like theatres or concert halls, often use similar principles to optimize sound travel.

Ellipse Equation

The ellipse equation defines the relation between the major and minor axes and the focal points, expressed as: \[ c^2 = a^2 - b^2 \] where \( a \) is the semi-major axis, \( b \) is the semi-minor axis, and \( c \) the distance from the center to a focus. This equation is a fundamental tool for solving ellipse-related problems. To understand it better, visualize swapping the focal length \( c \), or modifying \( a \) or \( b \), and observing its effect on the ellipse's shape.
Applying the equation appropriately, as in calculating the semi-minor axis from the given dimensions of the gallery, shows its practical utility in architectural and engineering applications.

Semi-Major Axis

The semi-major axis of an ellipse is half of its longest diameter. It runs from the center to the edge along the longest direction of the ellipse. In our earlier problem, the semi-major axis is 60 feet, derived from half the total length of 120 feet.
Understanding the semi-major axis is crucial because it directly influences the ellipse's overall size and shape. It determines part of the overall dimensions affecting things like room capacity or sound wave length in practical applications.

• Used to find focus positions.
• Part of the ellipse equation \( c^2 = a^2 - b^2 \).
• Crucial for applications in design and architecture.

Semi-Minor Axis

The semi-minor axis complements the semi-major axis, representing the shortest distance from the center to the edge of the ellipse. In practical geometry applications, it's pivotal in calculating height or other critical inside dimensions like in the whispering gallery ceiling. In the given exercise, solving the equation \( c^2 = a^2 - b^2 \) allowed us to find \( b \). The calculated value of \( b \) determines the ceiling height at the center. This measurement is essential for understanding how tall objects can enter that space or simulating sound movement within that area.
So, comprehending and utilizing the semi-minor axis is fundamental in mathematical problem-solving and practical design considerations.