4) Draw any four level curves (contours) of the surface given by the formula: -3z + 4x - 2y² = 0. Describe the shape of the surface in words.
4) Draw any four level curves (contours) of the surface given by the formula: -3z + 4x - 2y² = 0. Describe the shape of the surface in words.
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**: Draw any four level curves (contours) of the surface given by the formula:
\[ -3z + 4x - 2y^2 = 0 \]
**Description**: Describe the shape of the surface in words.
---
**Solution**:
1. **Formula Rearrangement**:
We start by rewriting the given formula to better understand it. The formula is:
\[ -3z + 4x - 2y^2 = 0 \]
Rearanging for \( z \), we have:
\[ z = \frac{4x}{3} - \frac{2y^2}{3} \]
Now we can see that \( z \) is dependent on \( x \) and \( y \).
2. **Level Curves (Contours)**:
To draw level curves, we set \( z \) to different constant values:
- For \( z = 0 \): \[ \frac{4x}{3} - \frac{2y^2}{3} = 0 \quad \Rightarrow \quad 4x = 2y^2 \quad \Rightarrow \quad x = \frac{y^2}{2} \]
This is a parabola opening along the x-axis.
- For \( z = 1 \): \[ 1 = \frac{4x}{3} - \frac{2y^2}{3} \quad \Rightarrow \quad 1 = \frac{4x - 2y^2}{3} \quad \Rightarrow \quad 4x - 2y^2 = 3 \quad \Rightarrow \quad 2x = y^2 + \frac{3}{2} \]
This is a parabola opening along the x-axis, but shifted.
- For \( z = -1 \): \[ -1 = \frac{4x}{3} - \frac{2y^2}{3} \quad \Rightarrow \quad -3 = 4x - 2y^2 \quad \Rightarrow \quad 4x = 2y^2 - 3 \quad \Rightarrow \quad x = \frac{y^2}{2} - \frac{3}{4} \]
This is a downwards parab](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1597ed32-e45d-4583-a76f-92703bb626d9%2F76f52290-0475-4ac7-8c89-4329b6a3c8c3%2Fmgbobx_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem**: Draw any four level curves (contours) of the surface given by the formula:
\[ -3z + 4x - 2y^2 = 0 \]
**Description**: Describe the shape of the surface in words.
---
**Solution**:
1. **Formula Rearrangement**:
We start by rewriting the given formula to better understand it. The formula is:
\[ -3z + 4x - 2y^2 = 0 \]
Rearanging for \( z \), we have:
\[ z = \frac{4x}{3} - \frac{2y^2}{3} \]
Now we can see that \( z \) is dependent on \( x \) and \( y \).
2. **Level Curves (Contours)**:
To draw level curves, we set \( z \) to different constant values:
- For \( z = 0 \): \[ \frac{4x}{3} - \frac{2y^2}{3} = 0 \quad \Rightarrow \quad 4x = 2y^2 \quad \Rightarrow \quad x = \frac{y^2}{2} \]
This is a parabola opening along the x-axis.
- For \( z = 1 \): \[ 1 = \frac{4x}{3} - \frac{2y^2}{3} \quad \Rightarrow \quad 1 = \frac{4x - 2y^2}{3} \quad \Rightarrow \quad 4x - 2y^2 = 3 \quad \Rightarrow \quad 2x = y^2 + \frac{3}{2} \]
This is a parabola opening along the x-axis, but shifted.
- For \( z = -1 \): \[ -1 = \frac{4x}{3} - \frac{2y^2}{3} \quad \Rightarrow \quad -3 = 4x - 2y^2 \quad \Rightarrow \quad 4x = 2y^2 - 3 \quad \Rightarrow \quad x = \frac{y^2}{2} - \frac{3}{4} \]
This is a downwards parab
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