Problem 58 - Algebra and Trigonometry Real Mathematics, Real People

Question

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

Step-by-Step Solution

Answer

The graph of the given parametric equations should be a curve that reflects the prolate cycloid shape. This is achieved by using a graphing utility correctly set to polynomial mode, entering the given equations, and adjusting settings as necessary.

Step 1: Set Up the Graphing Utility

First, open the graphing utility and ensure it's in polynomial mode if available. Choose the parametric option and put the equations for \( x \) and \( y \) into the graphing utility directly as: \( x=2 \theta-4 \sin \theta \) and \( y=2-4 \cos \theta \) respectively.

Step 2: Adjust the Interval and the Increment

The parameter \(\theta \) typically ranges from 0 to 2\( \pi \) for one complete cycle. However, if you want to observe multiple cycles, you can extend this range. Similarly, you can adjust the increment of \(\theta \) according to the precision you want. Normally you can start with an increment of 0.1 or 0.05.

Step 3: Graph and Interpret the Curve

After setting up the utility and adjusting the interval and increment, graph the curve. The graph should be a curve that reflects the given equations. The utility plots the points, defined by the pair of equations as it increases \(\theta \) from the lower bound to the upper bound step by step. By this way, it generates the curve of the prolate cycloid.