
Calculus: Early Transcendentals
8th Edition
ISBN: 9781285741550
Author: James Stewart
Publisher: Cengage Learning
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![### 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](https://content.bartleby.com/qna-images/question/6d96024e-4284-45bb-822f-9657558d02e4/96681fb8-8e97-46cf-b0db-8662f34f4526/kbmqgih_thumbnail.jpeg)
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
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- Activity 4 A.) An ellipse is the set of points such that the sum of distance from two fixed point constant. The two fixed points are called foci. • Use graph paper like that shown. It contains two small circles and a series of concentric circles from each. The concentric circles are tangent to each other as shown. • Choose the constant 13. Mark the points at the intersections of circle 9 and circle 4, because 9 + 4 = 13. Continue this process until you have marked the intersection of all circles whose sum is 13. || • Connect the points to form a smooth curve. The curve is an ellipse whose foci are the centers of the two small circles on the graph paper. Circles in the figure are named by letters, circle 1 is circle c, circle 2 is circle d, and so on and so fortharrow_forwardThe vertices of an ellipse of eccentricity 0.5 lie at the points (0, +-2). Where do the foci lie?arrow_forwardUse this graph to find each part of the ellipse. A. Orientation. I Select ] B. Center. I Select ] C. Co-Vertices. ( Select ) D. Length of Major Axis. I Select1 E. Length of Minor Axis Seleci 1 F. Foci. I Select] G. Valuc of a. Iselect1 1. Valuc of b. Select1arrow_forward
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