The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line shown in
The curve traced out by a point P on the circumference of a circle as the circle rolls along a straight line shown in
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 text in the image reads:
"The curve traced out by a point \( P \) on the circumference of a circle as the circle rolls along a straight line shown in"
(Note: The image does not contain any graphs or diagrams, so no additional explanation is needed for visual elements.)
![The image depicts a mathematical concept known as a **cycloid** and highlights the shaded region \( A \), which lies below an arch formed by the cycloid. A cycloid is generated by tracing a fixed point, \( P \), on the circumference of a circle as it rolls along a straight line.
### Parametric Equations for a Cycloid:
Given a circle of radius \( R \), the cycloid is represented parametrically by:
\[
x(t) = R(t - \sin t) , \quad y(t) = R(1 - \cos t)
\]
### Problem:
Determine the area of the region \( A \) when the radius \( R \) is 4.
### Options:
1. \(\text{area}(A) = 48\pi\)
2. \(\text{area}(A) = 50\pi\)
3. \(\text{area}(A) = 47\pi\)
4. \(\text{area}(A) = 46\pi\)
5. \(\text{area}(A) = 49\pi\)
The diagram includes the circle that generates the cycloid, the path traced by the point \( P \), and the shaded region representing the area in question. The axes are labelled \( x \) and \( y \), with the circle shown as rolling along the \( x \)-axis.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9a433dd5-6857-42c0-ae40-09bd40f154d3%2Ff80e1626-cc48-49b8-9a17-3430c9b2e343%2F4tc7gci_processed.jpeg&w=3840&q=75)
Transcribed Image Text:The image depicts a mathematical concept known as a **cycloid** and highlights the shaded region \( A \), which lies below an arch formed by the cycloid. A cycloid is generated by tracing a fixed point, \( P \), on the circumference of a circle as it rolls along a straight line.
### Parametric Equations for a Cycloid:
Given a circle of radius \( R \), the cycloid is represented parametrically by:
\[
x(t) = R(t - \sin t) , \quad y(t) = R(1 - \cos t)
\]
### Problem:
Determine the area of the region \( A \) when the radius \( R \) is 4.
### Options:
1. \(\text{area}(A) = 48\pi\)
2. \(\text{area}(A) = 50\pi\)
3. \(\text{area}(A) = 47\pi\)
4. \(\text{area}(A) = 46\pi\)
5. \(\text{area}(A) = 49\pi\)
The diagram includes the circle that generates the cycloid, the path traced by the point \( P \), and the shaded region representing the area in question. The axes are labelled \( x \) and \( y \), with the circle shown as rolling along the \( x \)-axis.
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