Problem 68 - Algebra 2

Question

Acoustics In "whispering gallerie"" a sound made at one focus can be clearly heard at the other focus, even though very little can be heard by someone in between. Suppose an elliptical room measures 320 \(\mathrm{ft}\) long and 150 \(\mathrm{ft}\) wide. How far would the listener have to be from the source of the sound in order to hear it?

Step-by-Step Solution

Answer

The listener would need to be approximately 280 ft away from the source of the sound to hear it clearly.

Step 1: Identify the major and minor axes

The major axis is given as the length of the room, which is 320 ft, and the minor axis is the width of the room, which is 150 ft.

Step 2: Use the formula to calculate the distance

The distance between the source of the sound and the listener is given by the formula: distance = \sqrt{(major axis)^2 - (minor axis)^2}. Substituting the given values, we get the distance = \sqrt{(320 ft)^2 - (150 ft)^2}

Step 3: Calculate the distance

Carrying out the calculation, we find that the distance is approximately 280 ft.

Key Concepts

Major and Minor Axes

In any ellipse, such as the shape of an elliptical room, there are two main dimensions to consider: the major and minor axes. Think of these as the longest and shortest diameters of the ellipse.
The **major axis** is always the longest. In this problem, that is the 320 ft length of the room, stretching from one end of the ellipse to the other through its longest part.
The **minor axis**, on the other hand, is the shortest and measures the width of the room in the middle at the widest point, which is given as 150 ft.

• The major axis helps in defining the overall size and shape of an ellipse.
• These axes intersect at the center of the ellipse and are perpendicular to each other.

The relationship between these two axes is crucial in determining various properties of the ellipse, including how sound travels within it.

Distance Calculation Formula

To find how far the listener must be from the sound source in an elliptical room, a specific formula comes into play. This formula considers the unique properties of ellipses, particularly between the foci.
**The distance calculation formula:** \[\text{distance} = \sqrt{(\text{major axis})^2 - (\text{minor axis})^2}\]Plug in the room's dimensions:- Major Axis: 320 ft- Minor Axis: 150 ftBefore calculating, it's crucial to understand why this formula works. In an ellipse, the distances from any point along its edge to the two foci are constant. When we apply the distance formula here, it helps us find how far the foci are from each other, which tells us about the sound path from one focus to another. With our specific example:\[ \sqrt{(320 \, \text{ft})^2 - (150 \, \text{ft})^2} \]This formula shows us how space in an ellipse uniquely retains energy, such as sounds in whispering galleries.

Sound Propagation in Ellipse

In elliptical rooms, like whispering galleries, sound behaves in a unique way due to the shape of the space. The elliptical shape ensures that sound originating from one focus (a special point along the major axis) is directed precisely to the other focus.

• This property is due to the way ellipses focus wave energy, including sound.
• It allows a whisper to be clearly heard at the opposite focus, even if it's inaudible in between.
• Such behavior is an example of sound propagation, meaning how sound waves travel.

Understanding this helps explain why the room’s shape channels the sound so effectively. The calculated distance (280 ft in the previous example) shows how far apart the two key points or foci are, and that's where the transmitting and receiving work happen in this architectural marvel.