Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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![### Understanding the Length of a Helix
The helix of radius \( R \), height \( h \), and \( N \) complete turns has the parametrization
\[
\mathbf{r}(t) = \left\langle R \cos \left( \frac{2\pi Nt}{h} \right) , R \sin \left( \frac{2\pi Nt}{h} \right) , t \right\rangle , \quad 0 \leq t \leq h
\]
To calculate the length \( s \) of a helix if \( R = 3 \), \( h = 7 \), and \( N = 8 \), we follow the steps below. (Use symbolic notation and fractions where needed.)
The first calculation provided in the image is:
\[
s = 7 \sqrt{ 1 + \frac{1152\pi}{49} }
\]
However, the response indicates that the answer is incorrect.
To compute the correct length of the helix, let's break down the necessary steps:
1. **Parametrize the Helix Equation**
Given the parameterization of the helix:
\[
\mathbf{r}(t) = \left\langle R \cos \left( \frac{2\pi Nt}{h} \right), R \sin \left( \frac{2\pi Nt}{h} \right), t \right\rangle
\]
where \( R = 3 \), \( h = 7 \), and \( N = 8 \).
2. **Calculate the Derivative**
Compute the derivative \( \mathbf{r}'(t) \):
\[
\mathbf{r}'(t) = \left\langle -R \frac{2\pi N}{h} \sin \left( \frac{2\pi Nt}{h} \right), R \frac{2\pi N}{h} \cos \left( \frac{2\pi Nt}{h} \right), 1 \right\rangle
= \left\langle -3 \frac{16\pi}{7} \sin \left( \frac{16\pi t}{7} \right), 3 \frac{16\pi}{7} \cos \left( \frac{16\pi t}{](https://content.bartleby.com/qna-images/question/460d6b43-992c-4555-b09d-3f6791ba763b/fbf643e4-805e-45a4-8926-f6e8750cd715/fax93dd_processed.jpeg)
Transcribed Image Text:### Understanding the Length of a Helix
The helix of radius \( R \), height \( h \), and \( N \) complete turns has the parametrization
\[
\mathbf{r}(t) = \left\langle R \cos \left( \frac{2\pi Nt}{h} \right) , R \sin \left( \frac{2\pi Nt}{h} \right) , t \right\rangle , \quad 0 \leq t \leq h
\]
To calculate the length \( s \) of a helix if \( R = 3 \), \( h = 7 \), and \( N = 8 \), we follow the steps below. (Use symbolic notation and fractions where needed.)
The first calculation provided in the image is:
\[
s = 7 \sqrt{ 1 + \frac{1152\pi}{49} }
\]
However, the response indicates that the answer is incorrect.
To compute the correct length of the helix, let's break down the necessary steps:
1. **Parametrize the Helix Equation**
Given the parameterization of the helix:
\[
\mathbf{r}(t) = \left\langle R \cos \left( \frac{2\pi Nt}{h} \right), R \sin \left( \frac{2\pi Nt}{h} \right), t \right\rangle
\]
where \( R = 3 \), \( h = 7 \), and \( N = 8 \).
2. **Calculate the Derivative**
Compute the derivative \( \mathbf{r}'(t) \):
\[
\mathbf{r}'(t) = \left\langle -R \frac{2\pi N}{h} \sin \left( \frac{2\pi Nt}{h} \right), R \frac{2\pi N}{h} \cos \left( \frac{2\pi Nt}{h} \right), 1 \right\rangle
= \left\langle -3 \frac{16\pi}{7} \sin \left( \frac{16\pi t}{7} \right), 3 \frac{16\pi}{7} \cos \left( \frac{16\pi t}{
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