3. Identify and sketch the surface given by z = 16-y²; 0 ≤ y ≤ 2; 0≤x≤ 4. in 3-space, include ALL significant traces and significant points on the graph accurately.
3. Identify and sketch the surface given by z = 16-y²; 0 ≤ y ≤ 2; 0≤x≤ 4. in 3-space, include ALL significant traces and significant points on the graph accurately.
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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![**Problem 3: Identifying and Sketching a Surface in 3-Space**
Given the surface equation:
\[ z = 16 - y^2 \]
with the constraints:
\[ 0 \leq y \leq 2 \]
\[ 0 \leq x \leq 4 \]
Follow these steps to sketch and understand the surface:
1. **Identify the Surface:**
- The equation \( z = 16 - y^2 \) represents a parabolic cylinder because it is a parabola that extends infinitely along the x-axis.
- The parabola opens downward along the z-axis since the coefficient of \( y^2 \) is negative.
2. **Constraints:**
- The variable y ranges from 0 to 2, which limits the extent of the parabola in the y direction.
- The variable x ranges from 0 to 4, which creates a finite section of the parabolic cylinder.
3. **Significant Points:**
- At \( y = 0 \), \( z = 16 \).
- At \( y = 2 \), \( z = 16 - 2^2 = 16 - 4 = 12 \).
- Significant points along the parabolic edges within the specified range need to be identified.
4. **Traces:**
- **x-trace (when y and z vary, and x is constant):** These are parabolas of the form \( z = 16 - y^2 \).
- **y-trace (when x and z vary, and y is constant):** These are horizontal lines in the plane \( z = constant \) showing the height of the parabola at a fixed y.
5. **Sketching the Surface:**
- Draw the z-axis vertical, the y-axis horizontal in the plane, and the x-axis perpendicular to both in a 3D plot.
- Sketch the parabolic curves for different x-values (say x = 0 and x = 4) between y = 0 and y = 2.
- Ensure the parabola starts at z = 16 when y = 0 and smoothly curves downward to z = 12 when y = 2.
- Connect the parabolic curves along the x-axis to form the parabolic cylinder limited within the specified ranges.
**Conclusion:**
When you have plotted the curve as](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F6147cd51-fae9-4b86-867f-e7a7fe5750a1%2F0eef28c1-e4db-4cd5-a53e-ba3b33e1c2ff%2F0bzw2a_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem 3: Identifying and Sketching a Surface in 3-Space**
Given the surface equation:
\[ z = 16 - y^2 \]
with the constraints:
\[ 0 \leq y \leq 2 \]
\[ 0 \leq x \leq 4 \]
Follow these steps to sketch and understand the surface:
1. **Identify the Surface:**
- The equation \( z = 16 - y^2 \) represents a parabolic cylinder because it is a parabola that extends infinitely along the x-axis.
- The parabola opens downward along the z-axis since the coefficient of \( y^2 \) is negative.
2. **Constraints:**
- The variable y ranges from 0 to 2, which limits the extent of the parabola in the y direction.
- The variable x ranges from 0 to 4, which creates a finite section of the parabolic cylinder.
3. **Significant Points:**
- At \( y = 0 \), \( z = 16 \).
- At \( y = 2 \), \( z = 16 - 2^2 = 16 - 4 = 12 \).
- Significant points along the parabolic edges within the specified range need to be identified.
4. **Traces:**
- **x-trace (when y and z vary, and x is constant):** These are parabolas of the form \( z = 16 - y^2 \).
- **y-trace (when x and z vary, and y is constant):** These are horizontal lines in the plane \( z = constant \) showing the height of the parabola at a fixed y.
5. **Sketching the Surface:**
- Draw the z-axis vertical, the y-axis horizontal in the plane, and the x-axis perpendicular to both in a 3D plot.
- Sketch the parabolic curves for different x-values (say x = 0 and x = 4) between y = 0 and y = 2.
- Ensure the parabola starts at z = 16 when y = 0 and smoothly curves downward to z = 12 when y = 2.
- Connect the parabolic curves along the x-axis to form the parabolic cylinder limited within the specified ranges.
**Conclusion:**
When you have plotted the curve as
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