(b) Find a rectangular-coordinate X where x equation for the curve by eliminating the parameter. ✔ 0
(b) Find a rectangular-coordinate X where x equation for the curve by eliminating the parameter. ✔ 0
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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Transcribed Image Text:The image contains a mathematical question and graphs related to finding a rectangular-coordinate equation by eliminating a parameter.
### Graphs:
- **Left Graph:**
- The x-axis is labeled from 1 to 6.
- The y-axis is labeled from -10 to 4.
- There is a downward curved red arrow starting at y=4 and curving downwards past y=-10.
- **Right Graph:**
- The x-axis is labeled from -6 to -3.
- The y-axis seems to have the same range but is not labeled.
- There is an upward curved red arrow starting from below y=-10 upwards past y=0.
### Question:
(b) Find a rectangular-coordinate equation for the curve by eliminating the parameter.
### Input Box:
- A text box symbolized by a blank space.
- Incorrect and correct indicators with a red 'x' indicating an incorrect entry and a green checkmark for a correct entry.
- Constraint: \( x \geq 0 \)
### Need Help Section:
- Two buttons labeled "Read It" and "Watch It" suggesting additional resources.
This content aims to teach how to eliminate parameters in parametric equations to obtain rectangular equations, presented with visual aid through the graphs.
![A pair of parametric equations is given:
\[
x = \sqrt{t}, \quad y = 4 - t
\]
(a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as \( t \) increases.
**Graphs Description:**
1. **Top Left Graph:**
- The graph shows a curve in the first quadrant, starting from the point (0, 4).
- As \( t \) increases, the curve moves upwards and to the right.
- The curve has a positive slope, and arrows indicate the direction towards increasing \( x \) and \( y \).
2. **Top Right Graph:**
- This graph displays a curve in the second quadrant, starting from roughly (-2, 10).
- As \( t \) increases, the curve moves downward and to the right.
- The curve has a negative slope, with arrows pointing downwards.
3. **Bottom Left Graph:**
- A downward opening curve in the first quadrant, beginning at (0, 4) and moving downwards as \( x \) approaches 3.
- The curve has a negative slope, with an arrow indicating the direction of decrease in \( y \).
4. **Bottom Right Graph:**
- This curve appears in the fourth quadrant, starting from approximately (-4, 0).
- As \( t \) increases, the curve moves upwards and to the right.
- The curve shows positive growth with an arrow indicating the direction towards increasing \( x \).
Each graph is plotted on a standard Cartesian plane with labeled x and y axes, and the scales set appropriately for the curves illustrated. The arrows on each curve highlight the progression of the curve as the parameter \( t \) increases.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F61d8af2b-7fdf-4a8e-a1bf-75f5ea698050%2Ffea038a6-f33c-4f7e-9270-9feab5274ce1%2Ftuxumft_processed.jpeg&w=3840&q=75)
Transcribed Image Text:A pair of parametric equations is given:
\[
x = \sqrt{t}, \quad y = 4 - t
\]
(a) Sketch the curve represented by the parametric equations. Use arrows to indicate the direction of the curve as \( t \) increases.
**Graphs Description:**
1. **Top Left Graph:**
- The graph shows a curve in the first quadrant, starting from the point (0, 4).
- As \( t \) increases, the curve moves upwards and to the right.
- The curve has a positive slope, and arrows indicate the direction towards increasing \( x \) and \( y \).
2. **Top Right Graph:**
- This graph displays a curve in the second quadrant, starting from roughly (-2, 10).
- As \( t \) increases, the curve moves downward and to the right.
- The curve has a negative slope, with arrows pointing downwards.
3. **Bottom Left Graph:**
- A downward opening curve in the first quadrant, beginning at (0, 4) and moving downwards as \( x \) approaches 3.
- The curve has a negative slope, with an arrow indicating the direction of decrease in \( y \).
4. **Bottom Right Graph:**
- This curve appears in the fourth quadrant, starting from approximately (-4, 0).
- As \( t \) increases, the curve moves upwards and to the right.
- The curve shows positive growth with an arrow indicating the direction towards increasing \( x \).
Each graph is plotted on a standard Cartesian plane with labeled x and y axes, and the scales set appropriately for the curves illustrated. The arrows on each curve highlight the progression of the curve as the parameter \( t \) increases.
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