Find an arc length parametrization r₁ (s) of the curve r(t) = (71,3/2, 103/2), with the parameter s measuring from (0, 0, 0). (Use symbolic notation and fractions where needed.)
Find an arc length parametrization r₁ (s) of the curve r(t) = (71,3/2, 103/2), with the parameter s measuring from (0, 0, 0). (Use symbolic notation and fractions where needed.)
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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Question
![**Title: Arc Length Parametrization of a Curve**
**Concept**: To find an arc length parametrization \( \mathbf{r}_1(s) \) of the given curve \( \mathbf{r}(t) \), where the parameter \( s \) measures the arc length from the point \( (0, 0, 0) \).
**Given Curve**:
\[
\mathbf{r}(t) = \left\langle 7t, \frac{10}{3}t^{3/2}, \frac{10}{\sqrt{3}}t^{3/2} \right\rangle
\]
**Instruction**: Use symbolic notation and fractions where needed.
**Solution**:
The solution presented in the image was:
\[
\mathbf{r}_1(s) = \left\langle \frac{550}{\sqrt{901}}, \frac{16500}{\sqrt{901}} \right\rangle
\]
This solution has been marked as "Incorrect".
**Approach**:
To find the correct arc length parametrization, we typically follow these steps:
1. **Compute \(\mathbf{r}'(t)\)**:
The derivative of \( \mathbf{r}(t) \) with respect to \( t \).
2. **Compute the magnitude of \( \mathbf{r}'(t) \)**:
\[ \left| \mathbf{r}'(t) \right| \]
3. **Express \( s \) as a function of \( t \)**:
Integrate the magnitude of \( \mathbf{r}'(t) \) to find \( s \).
4. **Solve for \( t \) in terms of \( s \) and substitute back to get \( \mathbf{r}_1(s) \)**.
Since the solution provided in the image was incorrect, revisiting these steps will be necessary to find the accurate parameterization.
---
**Note for educators**: This process helps students understand the conversion between parametric curves and their arc-length parametrizations, reinforcing concepts in differential calculus, vector calculus, and integral calculus.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff95e890f-a45d-49ff-801b-f76192665ba7%2F8b85869e-5535-4805-a582-c55dc80c988b%2Fzacr72_processed.png&w=3840&q=75)
Transcribed Image Text:**Title: Arc Length Parametrization of a Curve**
**Concept**: To find an arc length parametrization \( \mathbf{r}_1(s) \) of the given curve \( \mathbf{r}(t) \), where the parameter \( s \) measures the arc length from the point \( (0, 0, 0) \).
**Given Curve**:
\[
\mathbf{r}(t) = \left\langle 7t, \frac{10}{3}t^{3/2}, \frac{10}{\sqrt{3}}t^{3/2} \right\rangle
\]
**Instruction**: Use symbolic notation and fractions where needed.
**Solution**:
The solution presented in the image was:
\[
\mathbf{r}_1(s) = \left\langle \frac{550}{\sqrt{901}}, \frac{16500}{\sqrt{901}} \right\rangle
\]
This solution has been marked as "Incorrect".
**Approach**:
To find the correct arc length parametrization, we typically follow these steps:
1. **Compute \(\mathbf{r}'(t)\)**:
The derivative of \( \mathbf{r}(t) \) with respect to \( t \).
2. **Compute the magnitude of \( \mathbf{r}'(t) \)**:
\[ \left| \mathbf{r}'(t) \right| \]
3. **Express \( s \) as a function of \( t \)**:
Integrate the magnitude of \( \mathbf{r}'(t) \) to find \( s \).
4. **Solve for \( t \) in terms of \( s \) and substitute back to get \( \mathbf{r}_1(s) \)**.
Since the solution provided in the image was incorrect, revisiting these steps will be necessary to find the accurate parameterization.
---
**Note for educators**: This process helps students understand the conversion between parametric curves and their arc-length parametrizations, reinforcing concepts in differential calculus, vector calculus, and integral calculus.
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