Chapter 6.5, Problem 45E

Solution

The length of each petal of the polar curve.

The maximum length of the petal is $6\text{units}$ and minimum is $2\text{units}$ .

Given information:

The given polar equation is $r=2+4\sin 2\theta$ .

Formula used:

The range for $\sin 2\theta$ is, $0\le \sin 2\theta \le 1$ .

Calculation:

Plot the graph for the polar equation $r=2+4\sin 2\theta$ .

  

Maximum value of $\sin 2\theta =1$ in the range $\left[0,\frac{\pi }{2}\right]$ .

Thus, the maximum length of the petals occurs in the first quadrant which is $6\text{units}$ .

Minimum value of $\sin 2\theta =0$ in the range $\left[\frac{\pi }{2},\pi \right]$ .

Thus, the minimum length of the petals occurs in the second quadrant which is $2\text{units}$ .

Therefore, the maximum length of the petal is $6\text{units}$ and minimum is $2\text{units}$ .