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Chapter 6: Problem 80

Use a graphing utility to graph the curve represented by the parametric equations. Prolate cycloid: \(x=2 \theta-4 \sin \theta, y=2-4 \cos \theta\)

Short Answer

Expert verified
The prolate cycloid has a looping cycloidal shape, which can be plotted by using the given parametric equations and generating x, y coordinates for a range of \(\theta\) values.

Step by step solution

01

Understand The Parametric Equations

The given parametric equations are \(x=2\theta -4\sin\theta\) and \(y=2-4\cos\theta\). Here, x and y are functions of the variable \(\theta\), and \(\theta\) represents the parameter, usually the angle in cycloidal motion.
02

Identify the Range of \(\theta\)

To understand how these equations will express on a graph, we must identify the range of \(\theta\). Since we are dealing with sine and cosine functions, and these functions are periodic with a period of 2蟺, we can start by identifying the range of \(\theta\) as 0 < \(\theta\) 鈮� 2蟺.
03

Generate coordinates

Next, values of x and y should be generated for various values of \(\theta\) within the defined range. The coordinates obtained can then be plotted on a graph. For instance, when \(\theta = 0\), \(x = 0\) and \(y = -2\). Other values can be obtained similarly.
04

Plot the graph

Plot the points obtained in the preceding step on a graph. Since these are parametric equations, for each value of \(\theta\), there will be a corresponding x and y point. The graph of this prolate cycloid would be a looping cycloidal path.

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