11. Find the standard form of the equation of the ellipse with: Vertices: (0, 15); Passes through the point (4,2). Have fun; this one ain't easy!
11. Find the standard form of the equation of the ellipse with: Vertices: (0, 15); Passes through the point (4,2). Have fun; this one ain't easy!
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
Answer question 11. Write out each step and explain what you are doing.
![### Exploring Ellipses: Interesting Problems and Solutions
**Problem 11:**
**Objective:**
Find the standard form of the equation of the ellipse with the following properties:
- Vertices: \((0, \pm 5)\)
- Passes through the point \((4,2)\)
**Hint:** This is a challenging problem designed to test your understanding of the properties and equations of ellipses. Give it your best shot!
---
**Problem 12:**
**Scenario:**
An elliptical billiard table is 8 feet long by 5 feet wide. You need to determine the location of the foci. On such a table, if a ball is placed at each focus and one ball is struck with enough force, it will always hit the other ball no matter where it banks on the table.
**Task:**
Calculate the coordinates of the foci for the given elliptical billiard table.
---
**Problem 13:**
**Scenario:**
A hall 100 feet in length is being designed as a whispering gallery, a place where whispers can be heard clearly from the other side due to acoustic properties. The foci of the hall are located 25 feet from the center.
**Questions:**
1. How high will the ceiling be at the center of the hall?
2. How high will the ceiling be at a point on the major axis 10 feet from the center?
**Hint:** Utilize the properties of ellipses and the distances from the foci to solve this problem.
---
**Explanation of Relevant Concepts:**
- **Ellipse Properties:**
An ellipse has two axes:
- Major Axis: The longer axis that passes through both foci.
- Minor Axis: The shorter axis perpendicular to the major axis at the center.
- **Standard Form Equation of an Ellipse:**
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Here, \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.
- **Foci Calculation:**
Distance from the center to each focus (c) is determined using the equation:
\[
c = \sqrt{a^2 - b^2}
\]
- **Application to Real-World Structures:**
- Whispering galleries use the reflective properties of ellipses](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fbad1c991-07cd-414d-a37c-f7238c094fb9%2Ff2262344-feef-4fac-9cad-9290ed313422%2Fyxs35sd_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Exploring Ellipses: Interesting Problems and Solutions
**Problem 11:**
**Objective:**
Find the standard form of the equation of the ellipse with the following properties:
- Vertices: \((0, \pm 5)\)
- Passes through the point \((4,2)\)
**Hint:** This is a challenging problem designed to test your understanding of the properties and equations of ellipses. Give it your best shot!
---
**Problem 12:**
**Scenario:**
An elliptical billiard table is 8 feet long by 5 feet wide. You need to determine the location of the foci. On such a table, if a ball is placed at each focus and one ball is struck with enough force, it will always hit the other ball no matter where it banks on the table.
**Task:**
Calculate the coordinates of the foci for the given elliptical billiard table.
---
**Problem 13:**
**Scenario:**
A hall 100 feet in length is being designed as a whispering gallery, a place where whispers can be heard clearly from the other side due to acoustic properties. The foci of the hall are located 25 feet from the center.
**Questions:**
1. How high will the ceiling be at the center of the hall?
2. How high will the ceiling be at a point on the major axis 10 feet from the center?
**Hint:** Utilize the properties of ellipses and the distances from the foci to solve this problem.
---
**Explanation of Relevant Concepts:**
- **Ellipse Properties:**
An ellipse has two axes:
- Major Axis: The longer axis that passes through both foci.
- Minor Axis: The shorter axis perpendicular to the major axis at the center.
- **Standard Form Equation of an Ellipse:**
\[
\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1
\]
Here, \(a\) is the semi-major axis length and \(b\) is the semi-minor axis length.
- **Foci Calculation:**
Distance from the center to each focus (c) is determined using the equation:
\[
c = \sqrt{a^2 - b^2}
\]
- **Application to Real-World Structures:**
- Whispering galleries use the reflective properties of ellipses
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