Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
The helix of radius R, height h, and N complete turns has the parametrization Nt r(t) = (Rcos (2¹). R sin (2¹),1). ),₁). h h Calculate the length s of a helix if R = 3, h = 7, and N = 8. (Use symbolic notation and fractions where needed.) S= Incorrect 7√/1+ 0 ≤t≤h 1152 49

The helix of radius R, height h, and N complete turns has the parametrization Nt r(t) = (Rcos (2¹). R sin (2¹),1). ),₁). h h Calculate the length s of a helix if R = 3, h = 7, and N = 8. (Use symbolic notation and fractions where needed.) S= Incorrect 7√/1+ 0 ≤t≤h 1152 49

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter9: Systems Of Equations And Inequalities
Section: Chapter Questions
Problem 39RE
See similar textbooks
icon
Related questions
Question

This is my only attempt 

### 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}{
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}{
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 3 steps with 13 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Similar questions
  • SEE MORE QUESTIONS
Recommended textbooks for you
Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
Algebra
ISBN:
9781133382119
Author:
Swokowski
Publisher:
Cengage
SEE MORE TEXTBOOKS