Chapter 9, Problem 107CR

a.

To determine

To find:

How would the path of the parametric equation change for, $x=4\cos 2t,y=3\sin 2t$ .

Answer to Problem 107CR

The changes in the path of the parametric equation $x=4\cos 2t,y=3\sin 2t$ is represented in table form.

Explanation of Solution

Given information:

The path of the moving object is modelled by the parametric equations, $x=4\cos t\text{ and }y=3\sin t$

Based on the values of $x\text{ and }y$ , the graph will have the same orientation but will be traced quickly twice as fast as the original speed

For $x=4\cos t\text{ and }y=3\sin t$

$\theta$	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{3\pi }{4}$	$\pi$	$\frac{5\pi }{4}$	$\frac{3\pi }{2}$	$\frac{7\pi }{4}$	$2\pi$
$x$	4	2.828	0	$-2.828$	$-4$	$-2.828$	0	2.828	4
$y$	0	2.121	3	2.121	0	$-2.121$	$-3$	$-2.121$	0

For $x=4\cos 2t,y=3\sin 2t$

$\theta$	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{3\pi }{4}$	$\pi$	$\frac{5\pi }{4}$	$\frac{3\pi }{2}$	$\frac{7\pi }{4}$	$2\pi$
$x$	4	0	$-4$	0	4	0	$-4$	0	4
$y$	0	3	0	$-3$	0	3	0	$-3$	0

Therefore, this is how the path change for $x=4\cos 2t,y=3\sin 2t$ .

b.

To determine

To find:

How would the path of the parametric equation change for, $x=5\cos t,y=3\sin t$ .

Answer to Problem 107CR

The changes in the path of the parametric equation $x=5\cos t,y=3\sin t$ is represented in table form.

Explanation of Solution

Given information:

The path of the moving object is modelled by the parametric equations, $x=4\cos t\text{ and }y=3\sin t$

Based on the values of $x\text{ and }y$ , the graph path will have the same orientation but the path is now flatter because the length of the major axis increased from 8 to 10 units

For $x=4\cos t\text{ and }y=3\sin t$

$\theta$	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{3\pi }{4}$	$\pi$	$\frac{5\pi }{4}$	$\frac{3\pi }{2}$	$\frac{7\pi }{4}$	$2\pi$
$x$	4	2.828	0	$-2.828$	$-4$	$-2.828$	0	2.828	4
$y$	0	2.121	3	2.121	0	$-2.121$	$-3$	$-2.121$	0

For $x=5\cos t,y=3\sin t$

$\theta$	0	$\frac{\pi }{4}$	$\frac{\pi }{2}$	$\frac{3\pi }{4}$	$\pi$	$\frac{5\pi }{4}$	$\frac{3\pi }{2}$	$\frac{7\pi }{4}$	$2\pi$
$x$	5	3.536	0	$-3.536$	$-5$	$-3.536$	0	3.536	5
$y$	0	2.121	3	$-2.121$	0	$-2.121$	$-3$	$-2.121$	0

Therefore, this is how the path change for $x=5\cos t,y=3\sin t$ .