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Chapter 6: Problem 60

Use a graphing utility to graph the polar equation. Find an interval for \(\boldsymbol{\theta}\) for which the graph is traced only once. $$r=5+4 \cos \theta$$

Short Answer

Expert verified
The graph of the equation \(r = 5 + 4 \cos(\theta)\) is a lima莽on graph. The interval for 胃 in which the graph is traced only once is 0 <= 胃 < 2蟺.

Step by step solution

01

Plot the polar equation

To graph this equation using a graphing utility, input the equation \(r = 5 + 4 \cos(\theta)\) into the graphing tool. Remember to set the graph to polar coordinates first, which will normally be an option in the settings or options of the tool.
02

Interpret the graph

Observe the graph that is produced. This equation should yield a lima莽on graph because it is of the form \(r = a + b \cos(\theta)\), where a and b are both positive real numbers.
03

Determine the interval

Since we know that the cos function has a period of 2蟺 and repeats after that, we can then deduce that the graph will be traced only once over an interval of 0 <= 胃 < 2蟺. This is because the cos function hits all of its possible values within a span of 2蟺, so any point on the graph that we trace by plugging in the values of 胃 from 0 to 2蟺 will not be retraced for this specific interval.

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