Chapter 6.5, Problem 7E

Solution

To identify: The graph of given equations.

The equation $r_3$ corresponds to graph (b).

Given information:

The given equations are $r_1=3\cos 6\theta$ , $r_2=3\cos 8\theta$ , and $r_3=3\left|\cos 3\theta \right|$ .

The given graphs are as follows.

  

Formula used:

Polar equation of a rose curve has the equation $r=a\cos n\theta$ or $r=a\sin n\theta$ has $n$ petals if $n$ is odd and $2n$ petals if $n$ is even.

Calculation:

Consider the equation $r_1=3\cos 6\theta$ .

Here, $n=6$ .so, there is $2n=12$ petals.

Consider the equation $r_2=3\cos 8\theta$ .

Here, $n=8$ . so, there is $2n=16$ petals.

Consider the equation $r_3=3\left|\cos 3\theta \right|$ .

This equation has two rose curves $r_3=3\cos 3\theta$ and $r_3=-3\cos 3\theta$ . Both the curves have different orientation.

For $r_3=3\cos 3\theta$ , $n=3$ . There is $n=3$ petals.

For $r_3=-3\cos 3\theta$ , $n=3$ . There is $n=3$ petals.

Thus, there are a total of $6$ petals in total.

There are $6$ petals in the graph (b). Therefore, $r_3$ corresponds to graph (b).