Chapter 6.5, Problem 47E

Solution

To find: The length of each petal of the polar curve.

The maximum length of the petal is $5\text{units}$ and minimum is $3\text{units}$ .

Given information:

The given polar equation is $r=1-4\cos 5\theta$ .

Formula used:

The range for $\cos 5\theta$ is, $-1\le \cos 5\theta \le 1$ .

Calculation:

Plot the graph for the polar equation $r=1-4\cos 5\theta$ .

  

Maximum value of $\cos 5\theta =1$ .

Calculate value of $r$ .

  $\begin{array}{c}r=1-4\cos 5\theta \\ =1-4\\ =-3\end{array}$

Thus, the minimum length of the petals is $3\text{units}$ .

Minimum value of $\cos 5\theta =-1$ .

  $\begin{array}{c}r=1-4\cos 5\theta \\ =1-4\left(-1\right)\\ =5\end{array}$

Thus, the maximum length of the petals is $5\text{units}$ .

Therefore, the maximum length of the petal is $5\text{units}$ and minimum is $3\text{units}$ .