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Chapter 11: Problem 38

Tabulate and plot enough points to sketch a graph of the following equations. $$r=4+4 \cos \theta$$

Short Answer

Expert verified
Question: Tabulate enough points for different values of \(\theta\) to sketch a graph of the polar equation \(r = 4 + 4\cos\theta\). Answer: The following table lists the values of \(\theta\) and their respective values of \(r\) for the given equation: | \(\theta\) | \(r\) | |-------|-----| | 0 | 8 | | \(\pi/4\) | 5.65685 | | \(\pi/2\) | 4 | | \(3\pi/4\)| 5.65685 | | \(\pi\) | 8 | | \(5\pi/4\)| 12.343145 | | \(3\pi/2\) | 16 | | \(7\pi/4\)| 12.343145 | | \(2\pi\) | 8 | These points can be plotted to form a graph resembling a "lemniscate" or an infinity symbol.

Step by step solution

01

Choose the Points for \(\theta\)

To tabulate enough points, we will choose values of \(\theta\) that range from 0 to \(2\pi\) with intervals of \(\pi/4\). This will give us a total of 9 points which should be sufficient to sketch the graph.
02

Compute the Corresponding Values of \(r\)

Now we'll compute the values of \(r\) for each selected value of \(\theta\). Use the given equation \(r = 4 + 4\cos\theta\) to find the corresponding \(r\) value: | \(\theta\) | \(r\) | |-------|-----| | 0 | 8 | | \(\pi/4\) | 5.65685 | | \(\pi/2\) | 4 | | \(3\pi/4\)| 5.65685 | | \(\pi\) | 8 | | \(5\pi/4\)| 12.343145 | | \(3\pi/2\) | 16 | | \(7\pi/4\)| 12.343145 | | \(2\pi\) | 8 |
03

Plot the Points in Polar Coordinate System

Now, we will plot the points obtained in the polar coordinate system. Notice how the points align to form the shape - this will help us sketch the graph.
04

Sketch the Graph

Finally, connect the plotted points using smooth curves to form the sketch of the graph. In this case, you should see a figure that resembles a "lemniscate" or an infinity symbol. In conclusion, by choosing 9 points ranging from 0 to \(2\pi\) for \(\theta\) and calculating their respective values of \(r\) using the given polar equation, we were able to sketch a graph representing the equation \(r = 4 + 4\cos\theta\).

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Most popular questions from this chapter

A simplified model assumes that the orbits of Earth and Mars are circular with radii of 2 and \(3,\) respectively, and that Earth completes one orbit in one year while Mars takes two years. The position of Mars as seen from Earth is given by the parametric equations $$x=(3-4 \cos \pi t) \cos \pi t+2, \quad y=(3-4 \cos \pi t) \sin \pi t$$ a. Graph the parametric equations, for \(0 \leq t \leq 2\) b. Letting \(r=(3-4 \cos \pi t),\) explain why the path of Mars as seen from Earth is a lima莽on.

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Give the property that defines all ellipses.

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