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13. Consider the equation: 4x² +9z? = 36 Describe the traces. а. b. Describe the rulings. с. Sketch the surface by hand. Include at least 4 traces and 4 rulings. d. What is the name of this surface?

13. Consider the equation: 4x² +9z? = 36 Describe the traces. а. b. Describe the rulings. с. Sketch the surface by hand. Include at least 4 traces and 4 rulings. d. What is the name of this surface?

Calculus: Early Transcendentals
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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# Analyzing the Equation of a Hyperbolic Cylinder ## Problem Statement **13. Consider the equation: \(4x^2 + 9z^2 = 36\)** ### a. Describe the traces. The traces of the surface described by the equation \(4x^2 + 9z^2 = 36\) involve the cross-sectional shapes when either \(x\), \(y\), or \(z\) is held constant. In this case: - **In the \(xz\)-plane (\(y = 0\))**, the equation \(4x^2 + 9z^2 = 36\) describes an **ellipse**. - **In the \(yz\)-plane (\(x = 0\))**, the trace is a pair of lines because the equation reduces to a constant, trivializing the ellipse as lines. - **In the \(xy\)-plane (\(z = 0\))**, the equation \(4x^2 = 36\) describes parallel lines. ### b. Describe the rulings. Rulings refer to straight lines that lie entirely on the surface of the cylinder. For the given equation, the rulings run parallel to the \(y\)-axis, indicating that the cylinder extends infinitely along this direction. ### c. Sketch the surface by hand. Include at least 4 traces and 4 rulings. To sketch the surface: 1. **Draw the Ellipse in the \(xz\)-plane**: - Centered at the origin, with semi-major axis along the \(z\)-axis and semi-minor axis along the \(x\)-axis. 2. **Add rulings**: - Extend parallel straight lines from points on the ellipse along the \(y\)-axis. ### Explanation of 3D Diagram: The diagram shows a three-dimensional plot: - The \(z\)-axis is vertical. - The \(x\)-axis runs horizontally in the foreground. - An elliptical shape is evident in the \(xz\)-plane, with lines extending parallel along the \(y\)-axis to demonstrate the rulings. ### d. What is the name of this surface? The surface is known as a **hyperbolic cylinder** due to its elliptical cross-section and infinite extension parallel to the \(y\)-axis. Understanding this surface involves recognizing its geometric nature and behavior across different dimensions, critical in
Transcribed Image Text:# Analyzing the Equation of a Hyperbolic Cylinder ## Problem Statement **13. Consider the equation: \(4x^2 + 9z^2 = 36\)** ### a. Describe the traces. The traces of the surface described by the equation \(4x^2 + 9z^2 = 36\) involve the cross-sectional shapes when either \(x\), \(y\), or \(z\) is held constant. In this case: - **In the \(xz\)-plane (\(y = 0\))**, the equation \(4x^2 + 9z^2 = 36\) describes an **ellipse**. - **In the \(yz\)-plane (\(x = 0\))**, the trace is a pair of lines because the equation reduces to a constant, trivializing the ellipse as lines. - **In the \(xy\)-plane (\(z = 0\))**, the equation \(4x^2 = 36\) describes parallel lines. ### b. Describe the rulings. Rulings refer to straight lines that lie entirely on the surface of the cylinder. For the given equation, the rulings run parallel to the \(y\)-axis, indicating that the cylinder extends infinitely along this direction. ### c. Sketch the surface by hand. Include at least 4 traces and 4 rulings. To sketch the surface: 1. **Draw the Ellipse in the \(xz\)-plane**: - Centered at the origin, with semi-major axis along the \(z\)-axis and semi-minor axis along the \(x\)-axis. 2. **Add rulings**: - Extend parallel straight lines from points on the ellipse along the \(y\)-axis. ### Explanation of 3D Diagram: The diagram shows a three-dimensional plot: - The \(z\)-axis is vertical. - The \(x\)-axis runs horizontally in the foreground. - An elliptical shape is evident in the \(xz\)-plane, with lines extending parallel along the \(y\)-axis to demonstrate the rulings. ### d. What is the name of this surface? The surface is known as a **hyperbolic cylinder** due to its elliptical cross-section and infinite extension parallel to the \(y\)-axis. Understanding this surface involves recognizing its geometric nature and behavior across different dimensions, critical in
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