Chapter 6.5, Problem 69E

Solution

a)

To find: The smallest $\theta$ interval that will produce a complete graph of $r_1=3\sin \left(\frac{7\theta }{2}\right)$ and $r_2=3\cos \left(\frac{7\theta }{2}\right)$ .

The smallest $\theta$ interval that will produce a complete graph of $r_1=3\sin \left(\frac{7\theta }{2}\right)$ and $r_2=3\cos \left(\frac{7\theta }{2}\right)$ is $4\pi$ .

Given information:

The given equations are $r_1=3\sin \left(\frac{7\theta }{2}\right)$ and $r_2=3\cos \left(\frac{7\theta }{2}\right)$ .

Calculation:

Plot the graph for $r_1=3\sin \left(\frac{7\theta }{2}\right)$ in the interval $0\le \theta \le 2\pi$ .

  

The rose curve has $7$ petals and appears incomplete.

Plot the graph for $r_1=3\sin \left(\frac{7\theta }{2}\right)$ in the interval $0\le \theta \le 4\pi$ .

  

The rose curve has $14$ petals and appears complete.

Therefore, the smallest $\theta$ interval that will produce a complete graph of $r_1=3\sin \left(\frac{7\theta }{2}\right)$ is $4\pi$ .

Plot the graph for $r_2=3\cos \left(\frac{7\theta }{2}\right)$ in the interval $0\le \theta \le 2\pi$ .

  

The rose curve has $7$ petals and appears incomplete.

Plot the graph for $r_2=3\cos \left(\frac{7\theta }{2}\right)$ in the interval $0\le \theta \le 4\pi$ .

  

The rose curve has $14$ petals and appears complete.

Therefore, the smallest $\theta$ interval that will produce a complete graph of $r_2=3\cos \left(\frac{7\theta }{2}\right)$ is $4\pi$ .

b)

To find:The number of petals in $r_1=3\sin \left(\frac{7\theta }{2}\right)$ and $r_2=3\cos \left(\frac{7\theta }{2}\right)$ .

The number of petals in both the graphs is $14$ .

Given information:

The given equations are $r_1=3\sin \left(\frac{7\theta }{2}\right)$ and $r_2=3\cos \left(\frac{7\theta }{2}\right)$ .

Calculation:

Plot the graph for $r_1=3\sin \left(\frac{7\theta }{2}\right)$ in the interval $0\le \theta \le 4\pi$ .

  

Plot the graph for $r_2=3\cos \left(\frac{7\theta }{2}\right)$ in the interval $0\le \theta \le 4\pi$ .

  

Therefore, from the graphs of the polar equations it can be deduced that the number of petals are $14$ .