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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![**Problem: Finding the Equation of an Ellipse**
To find the equation of the ellipse depicted on the graph, we need to analyze the graph in the image.
**Graph Description:**
- The graph features a coordinate plane with an ellipse centered at the origin (0, 0).
- The x-axis and y-axis both range from -10 to 10.
- The ellipse is vertically oriented, meaning it is taller than it is wide.
**Ellipse Characteristics:**
- The semi-major axis (along the y-direction) extends from -5 to 5 on the y-axis, giving it a length of 10. Therefore, the length of the semi-major axis is 5.
- The semi-minor axis (along the x-direction) extends from -2 to 2 on the x-axis, giving it a length of 4. Therefore, the length of the semi-minor axis is 2.
**Equation Form:**
The equation of an ellipse centered at the origin, with a semi-major axis along the y-axis and a semi-minor axis along the x-axis, is given by:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
Where:
- \(a\) is the semi-major axis length (5).
- \(b\) is the semi-minor axis length (2).
**Plugging in the Values:**
\[
\frac{x^2}{2^2} + \frac{y^2}{5^2} = 1
\]
This simplifies to:
\[
\frac{x^2}{4} + \frac{y^2}{25} = 1
\]
Thus, the equation of the ellipse is:
\[
\frac{x^2}{4} + \frac{y^2}{25} = 1
\]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14ca10a7-9a5a-4024-afbf-f0f142f43830%2F1ee1eec9-14e2-49c6-bfd2-fd3e1240bd1b%2Fy9ftbcx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem: Finding the Equation of an Ellipse**
To find the equation of the ellipse depicted on the graph, we need to analyze the graph in the image.
**Graph Description:**
- The graph features a coordinate plane with an ellipse centered at the origin (0, 0).
- The x-axis and y-axis both range from -10 to 10.
- The ellipse is vertically oriented, meaning it is taller than it is wide.
**Ellipse Characteristics:**
- The semi-major axis (along the y-direction) extends from -5 to 5 on the y-axis, giving it a length of 10. Therefore, the length of the semi-major axis is 5.
- The semi-minor axis (along the x-direction) extends from -2 to 2 on the x-axis, giving it a length of 4. Therefore, the length of the semi-minor axis is 2.
**Equation Form:**
The equation of an ellipse centered at the origin, with a semi-major axis along the y-axis and a semi-minor axis along the x-axis, is given by:
\[
\frac{x^2}{b^2} + \frac{y^2}{a^2} = 1
\]
Where:
- \(a\) is the semi-major axis length (5).
- \(b\) is the semi-minor axis length (2).
**Plugging in the Values:**
\[
\frac{x^2}{2^2} + \frac{y^2}{5^2} = 1
\]
This simplifies to:
\[
\frac{x^2}{4} + \frac{y^2}{25} = 1
\]
Thus, the equation of the ellipse is:
\[
\frac{x^2}{4} + \frac{y^2}{25} = 1
\]
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