Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) 16x² +9y² + 92² = 144 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at x = 0 using y and z.) | X X X
Use traces to sketch the surface. (If an answer does not exist, enter DNE. Select Update Graph to see your response plotted on the screen. Select the Submit button to grade your response.) 16x² +9y² + 92² = 144 (Write an equation for the cross section at z = 0 using x and y.) (Write an equation for the cross section at y = 0 using x and z.) (Write an equation for the cross section at x = 0 using y and z.) | X X X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Graphing Quadratic Surfaces: Analyzing Cross Sections**
### Instructions for Understanding and Sketching a Quadratic Surface
The given equation for the surface is:
\[ 16x^2 + 9y^2 + 9z^2 = 144 \]
This represents an ellipsoid. To sketch and understand this quadratic surface, we will analyze its cross sections by setting \( z = 0 \), \( y = 0 \), and \( x = 0 \) respectively.
#### Equations for Cross Sections:
1. **Cross section at \( z = 0 \):**
\[ 16x^2 + 9y^2 = 144 \]
Divide by 144:
\[ \frac{16x^2}{144} + \frac{9y^2}{144} = 1 \]
Simplify:
\[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]
2. **Cross section at \( y = 0 \):**
\[ 16x^2 + 9z^2 = 144 \]
Divide by 144:
\[ \frac{16x^2}{144} + \frac{9z^2}{144} = 1 \]
Simplify:
\[ \frac{x^2}{9} + \frac{z^2}{16} = 1 \]
3. **Cross section at \( x = 0 \):**
\[ 9y^2 + 9z^2 = 144 \]
Divide by 144:
\[ \frac{9y^2}{144} + \frac{9z^2}{144} = 1 \]
Simplify:
\[ \frac{y^2}{16} + \frac{z^2}{16} = 1 \]
\[ \frac{y^2}{4} + \frac{z^2}{4} = 1 \]
### Interactive Graphing Tool
To visualize the ellipsoid and its cross sections, enter the respective equations into the provided fields and click "Update Graph."
#### Detailed Descriptions of Graphing Interface:
- **Graph Controls:**
- **Update Graph**: Plots the entered equations on the 3D graph.
- **Student](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb2f5b0ad-df1d-48c6-a9a3-5338b8167143%2F5bf0e676-622e-4e2d-8b86-5d2726debce4%2Flyew5va_processed.png&w=3840&q=75)
Transcribed Image Text:**Graphing Quadratic Surfaces: Analyzing Cross Sections**
### Instructions for Understanding and Sketching a Quadratic Surface
The given equation for the surface is:
\[ 16x^2 + 9y^2 + 9z^2 = 144 \]
This represents an ellipsoid. To sketch and understand this quadratic surface, we will analyze its cross sections by setting \( z = 0 \), \( y = 0 \), and \( x = 0 \) respectively.
#### Equations for Cross Sections:
1. **Cross section at \( z = 0 \):**
\[ 16x^2 + 9y^2 = 144 \]
Divide by 144:
\[ \frac{16x^2}{144} + \frac{9y^2}{144} = 1 \]
Simplify:
\[ \frac{x^2}{9} + \frac{y^2}{16} = 1 \]
2. **Cross section at \( y = 0 \):**
\[ 16x^2 + 9z^2 = 144 \]
Divide by 144:
\[ \frac{16x^2}{144} + \frac{9z^2}{144} = 1 \]
Simplify:
\[ \frac{x^2}{9} + \frac{z^2}{16} = 1 \]
3. **Cross section at \( x = 0 \):**
\[ 9y^2 + 9z^2 = 144 \]
Divide by 144:
\[ \frac{9y^2}{144} + \frac{9z^2}{144} = 1 \]
Simplify:
\[ \frac{y^2}{16} + \frac{z^2}{16} = 1 \]
\[ \frac{y^2}{4} + \frac{z^2}{4} = 1 \]
### Interactive Graphing Tool
To visualize the ellipsoid and its cross sections, enter the respective equations into the provided fields and click "Update Graph."
#### Detailed Descriptions of Graphing Interface:
- **Graph Controls:**
- **Update Graph**: Plots the entered equations on the 3D graph.
- **Student
Transcribed Image Text:**Identify the Surface**
This question asks the learner to identify a specific type of surface from a list of options. It appears as a multiple-choice question commonly used in educational assessments, particularly in the field of mathematics or geometry. Here are the details provided:
**Options to identify the surface:**
1. Parabolic cylinder
2. Ellipsoid
3. Hyperboloid of one sheet
4. Elliptic cylinder (Selected option)
5. Elliptic paraboloid
6. Elliptic cone
7. Hyperbolic paraboloid
8. Hyperboloid of two sheets
**Explanation:**
- The fourth option (Elliptic cylinder) is selected.
- There is a red cross (X) mark at the bottom of the list, indicating that the selected option is incorrect.
Educational websites would use such questions to test understanding of three-dimensional geometric surfaces. The correct choice would be essential in reinforcing the distinction between various types of surfaces like cylinders, ellipsoids, hyperboloids, and paraboloids.
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