6:Shelch the arface in R deconbe by the equatich z=4-y² with -25y€2 and Osx=3 Giue ashort wniten descripton.
6:Shelch the arface in R deconbe by the equatich z=4-y² with -25y€2 and Osx=3 Giue ashort wniten descripton.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Title: Exploring a 3D Surface in Mathematical Modeling**
**Objective:**
To understand and sketch the surface described by the given equation within specified boundaries, and provide a concise description of the surface.
**Equation and Conditions:**
The surface in \(\mathbb{R}^3\) is described by the equation:
\[ z = 4 - y^2 \]
With the conditions:
\[ -2 \leq y \leq 2 \]
\[ 0 \leq x \leq 3 \]
**Task:**
1. **Sketch the Surface**: Visualize the surface in three-dimensional space.
2. **Provide a Description**: Offer a brief explanation of the surface characteristics.
**Description and Analysis:**
The equation \( z = 4 - y^2 \) describes a parabola in the yz-plane that opens downward, with the vertex at \( z = 4 \) when \( y = 0 \). The condition \( -2 \leq y \leq 2 \) indicates the domain for \( y \), limiting the width of this parabola.
For the \( x \)-interval \( 0 \leq x \leq 3 \), the surface extends along the x-axis, creating a parabolic cylinder bounded by these planes.
The surface can be imagined as a "sheet" stretched in the x-direction with the shape of a downward-opening parabola in the y-direction. This results in a 3D shape resembling a trough open along the x-axis. The surface is flat along the x-direction because \( x \) doesn't appear in the equation, meaning the cross-section remains constant.
**Conclusion:**
By understanding the equation and constraints, the surface is essentially a segment of a parabolic cylinder, defined by specific numeric boundaries in three-dimensional space. This exercise helps solidify the comprehension of geometric surfaces in mathematical contexts.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc740e8b2-d410-40ba-b674-1beb1e9bb454%2F7826a034-865b-44b8-944a-9765a937f1cd%2Fdthafbs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Title: Exploring a 3D Surface in Mathematical Modeling**
**Objective:**
To understand and sketch the surface described by the given equation within specified boundaries, and provide a concise description of the surface.
**Equation and Conditions:**
The surface in \(\mathbb{R}^3\) is described by the equation:
\[ z = 4 - y^2 \]
With the conditions:
\[ -2 \leq y \leq 2 \]
\[ 0 \leq x \leq 3 \]
**Task:**
1. **Sketch the Surface**: Visualize the surface in three-dimensional space.
2. **Provide a Description**: Offer a brief explanation of the surface characteristics.
**Description and Analysis:**
The equation \( z = 4 - y^2 \) describes a parabola in the yz-plane that opens downward, with the vertex at \( z = 4 \) when \( y = 0 \). The condition \( -2 \leq y \leq 2 \) indicates the domain for \( y \), limiting the width of this parabola.
For the \( x \)-interval \( 0 \leq x \leq 3 \), the surface extends along the x-axis, creating a parabolic cylinder bounded by these planes.
The surface can be imagined as a "sheet" stretched in the x-direction with the shape of a downward-opening parabola in the y-direction. This results in a 3D shape resembling a trough open along the x-axis. The surface is flat along the x-direction because \( x \) doesn't appear in the equation, meaning the cross-section remains constant.
**Conclusion:**
By understanding the equation and constraints, the surface is essentially a segment of a parabolic cylinder, defined by specific numeric boundaries in three-dimensional space. This exercise helps solidify the comprehension of geometric surfaces in mathematical contexts.
Expert Solution
Step 1
Z=4-y2 looks like a tent whose hieght is along z axis, length along 'x' axis and according to problem length is 3 unit and height is 4 unit, and section by yz plane looks like parabola.
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