Sketch the solid described by the given inequalities. Oses n/2, r szs4
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:** Visualizing Solids with Inequalities
**Objective:**
Understand how to sketch and interpret the solid described by a set of inequalities in a 3D coordinate system.
**Given Inequalities:**
\[ 0 \leq \theta \leq \pi/2, \, r \leq z \leq 4 \]
**Explanation of Diagrams:**
The series of four 3D plots offer different visual perspectives of the described solid bounded by the specified inequalities.
**Top Left Graph:**
- **Axes:** x, y, z
- **Description:** This plot shows a cylindrical shape with a curved surface that forms part of a circle in the x-y plane, with height extending along the z-axis from \( r \) to 4.
- **Key Features:** The inequality \( r \leq z \) creates a slanted surface on the top, resembling a circular segment.
**Top Right Graph:**
- **Axes:** y, -x, z
- **Description:** This view emphasizes the conical nature of the surface, highlighting how the radius \( r \) influences the height \( z \), forming a cone truncated by \( z = 4 \).
**Bottom Left Graph:**
- **Axes:** x, y, z
- **Description:** Offering another angle, this plot displays how the cone intersects with the plane \( z = 4 \), under the given angular constraint \( 0 \leq \theta \leq \pi/2 \).
**Bottom Right Graph:**
- **Axes:** x, y, z
- **Description:** Displays the cylindrical portion of the solid, emphasizing the boundary created at height \( z = 4 \) and the extent of the solid within the angular range.
**Conclusion:**
These visualizations aid in understanding the intersection and boundaries of solids defined by inequalities within a cylindrical coordinate system. Analyzing the plots enhances comprehension of geometry in three-dimensional space.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1c7104a7-391d-43e8-bb65-99aa718e4fa3%2Fe0dc1a4c-a08f-4d9a-a3ac-a788b48268ce%2F8m1vw1l_processed.png&w=3840&q=75)
Transcribed Image Text:**Title:** Visualizing Solids with Inequalities
**Objective:**
Understand how to sketch and interpret the solid described by a set of inequalities in a 3D coordinate system.
**Given Inequalities:**
\[ 0 \leq \theta \leq \pi/2, \, r \leq z \leq 4 \]
**Explanation of Diagrams:**
The series of four 3D plots offer different visual perspectives of the described solid bounded by the specified inequalities.
**Top Left Graph:**
- **Axes:** x, y, z
- **Description:** This plot shows a cylindrical shape with a curved surface that forms part of a circle in the x-y plane, with height extending along the z-axis from \( r \) to 4.
- **Key Features:** The inequality \( r \leq z \) creates a slanted surface on the top, resembling a circular segment.
**Top Right Graph:**
- **Axes:** y, -x, z
- **Description:** This view emphasizes the conical nature of the surface, highlighting how the radius \( r \) influences the height \( z \), forming a cone truncated by \( z = 4 \).
**Bottom Left Graph:**
- **Axes:** x, y, z
- **Description:** Offering another angle, this plot displays how the cone intersects with the plane \( z = 4 \), under the given angular constraint \( 0 \leq \theta \leq \pi/2 \).
**Bottom Right Graph:**
- **Axes:** x, y, z
- **Description:** Displays the cylindrical portion of the solid, emphasizing the boundary created at height \( z = 4 \) and the extent of the solid within the angular range.
**Conclusion:**
These visualizations aid in understanding the intersection and boundaries of solids defined by inequalities within a cylindrical coordinate system. Analyzing the plots enhances comprehension of geometry in three-dimensional space.
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