Chapter 6.5, Problem 53E

Solution

(a)

To describe: The graph of the given polar equation.

The graph of the polar equation $r=2{\sin }^{2}2\theta +\sin 2\theta$ shows graph of rose cure have two short petals of length $1\text{unit}$ and two long petals of length $3\text{units}$ .

Given information:

The given polar equation is $r=2{\sin }^{2}2\theta +\sin 2\theta$ .

Calculation:

Graph for the polar equation $r=2{\sin }^{2}2\theta +\sin 2\theta$ .

  

Figure (1)

From the figure (1), it is observed that the polar equation represents a rose curve.

The range of $\sin 2\theta$ , $-1\le \sin 2\theta \le 1$ .

Here, the range of the polar equation $r=2{\sin }^{2}2\theta +\sin 2\theta$ is $\left[1,3\right]$ by taking maximum value of $\sin 2\theta$ as $1$ and minimum value of $\sin 2\theta$ as $-1$ .

Therefore, the rose cure have two short petals each of length $1\text{unit}$ and two long petals each of length $3\text{units}$ .

(b)

To find: If there exist any symmetry in the graph.

The graph is symmetric about the origin.

Given information:

The given polar equation is $r=2{\sin }^{2}2\theta +\sin 2\theta$ .

Calculation:

From part (a), the figure (1) represents a rose curve that is symmetric about the origin.

(c)

To find: The maximum value of $\left|r\right|$ if it exists.

The maximum value of $r$ is $3$ .

Given information:

The given polar equation is $r=2{\sin }^{2}2\theta +\sin 2\theta$ .

Calculation:

From part (a), the maximum value of $\left|r\right|$ exists.

The maximum value of $r$ is $\left|r\right|=3$ .