Calculus: Early Transcendentals
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
Bartleby Related Questions Icon

Related questions

bartleby

Concept explainers

Question
100%
### Whispering Gallery: Sound Reflection in an Ellipse

A person in a whispering gallery standing at one focus of the ellipse 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 104 meters, and the foci are located 20 meters from the center, find the height of the ceiling at the center.

#### Parameters:
- Length of the whispering gallery: 104 meters
- Distance of the foci from the center: 20 meters

**Task:** Determine the height of the ceiling at the center of the ellipse.

```plaintext
Source: OpenStax Precalculus
```

#### Answer Input:
- Height: [  ] (Select an answer)

This problem involves understanding and applying the geometric properties of an ellipse to find the required height. 

### Solution Approach

To solve this problem, we need to use the standard properties of an ellipse. The total length of the ellipse will help us determine the semi-major and semi-minor axes, essential to finding the height of the ceiling:

1. **Identify the major and minor axes**:
   - The total length (major axis) = 104 meters.
   - Hence, the semi-major axis (a) = 104 / 2 = 52 meters.

2. **Determine the distance to the foci**:
   - The distance from the center to each focus (c) = 20 meters.

3. **Use the relationship in ellipses**:
   - For an ellipse, \( a^2 = b^2 + c^2 \)
   - Here, \( a \) (semi-major axis) = 52 meters and \( c \) (distance to the foci) = 20 meters.

4. **Calculate the semi-minor axis (b)**:
   \[
   52^2 = b^2 + 20^2 \\
   2704 = b^2 + 400 \\
   b^2 = 2704 - 400 \\
   b^2 = 2304 \\
   b = \sqrt{2304} = 48 \text{ meters}
   \]

5. **The height of the ceiling at the center is equal to b**:
   - Height: 48 meters

This example illustrates the application of geometric properties of ellipses
expand button
Transcribed Image Text:### Whispering Gallery: Sound Reflection in an Ellipse A person in a whispering gallery standing at one focus of the ellipse 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 104 meters, and the foci are located 20 meters from the center, find the height of the ceiling at the center. #### Parameters: - Length of the whispering gallery: 104 meters - Distance of the foci from the center: 20 meters **Task:** Determine the height of the ceiling at the center of the ellipse. ```plaintext Source: OpenStax Precalculus ``` #### Answer Input: - Height: [ ] (Select an answer) This problem involves understanding and applying the geometric properties of an ellipse to find the required height. ### Solution Approach To solve this problem, we need to use the standard properties of an ellipse. The total length of the ellipse will help us determine the semi-major and semi-minor axes, essential to finding the height of the ceiling: 1. **Identify the major and minor axes**: - The total length (major axis) = 104 meters. - Hence, the semi-major axis (a) = 104 / 2 = 52 meters. 2. **Determine the distance to the foci**: - The distance from the center to each focus (c) = 20 meters. 3. **Use the relationship in ellipses**: - For an ellipse, \( a^2 = b^2 + c^2 \) - Here, \( a \) (semi-major axis) = 52 meters and \( c \) (distance to the foci) = 20 meters. 4. **Calculate the semi-minor axis (b)**: \[ 52^2 = b^2 + 20^2 \\ 2704 = b^2 + 400 \\ b^2 = 2704 - 400 \\ b^2 = 2304 \\ b = \sqrt{2304} = 48 \text{ meters} \] 5. **The height of the ceiling at the center is equal to b**: - Height: 48 meters This example illustrates the application of geometric properties of ellipses
Expert Solution
Check Mark
Knowledge Booster
Background pattern image
Calculus
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, calculus and related others by exploring similar questions and additional content below.
Similar questions
Recommended textbooks for you
Text book image
Calculus: Early Transcendentals
Calculus
ISBN:9781285741550
Author:James Stewart
Publisher:Cengage Learning
Text book image
Thomas' Calculus (14th Edition)
Calculus
ISBN:9780134438986
Author:Joel R. Hass, Christopher E. Heil, Maurice D. Weir
Publisher:PEARSON
Text book image
Calculus: Early Transcendentals (3rd Edition)
Calculus
ISBN:9780134763644
Author:William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz
Publisher:PEARSON
Text book image
Calculus: Early Transcendentals
Calculus
ISBN:9781319050740
Author:Jon Rogawski, Colin Adams, Robert Franzosa
Publisher:W. H. Freeman
Text book image
Precalculus
Calculus
ISBN:9780135189405
Author:Michael Sullivan
Publisher:PEARSON
Text book image
Calculus: Early Transcendental Functions
Calculus
ISBN:9781337552516
Author:Ron Larson, Bruce H. Edwards
Publisher:Cengage Learning