Chapter 8.4, Problem 58E

(a)

To determine

To show: The parametric equations for the prolate cycloid are same as the equations for the curtate cycloid.

Explanation of Solution

Given information:

In exercise $57$ if the point $P$ lies outside the circle at a distance $b$ from the center with $b>a$ , then the curve traced by $P$ is called a prolate cycloid.

Proof:

Draw the curve traced by $P$ is given in figure below:

  

Figure(1)

As the path traced by the point $P$ is same as the path by for the curtate cycloid.

So, the equation of prolate cycloid is same as the equation for the curtate cycloid.

Therefore, the equation of the prolate cycloid is $x=a\theta -b\sin \theta$ and $y=a-b\cos \theta$ .

(b)

To determine

To graph: The prolate cycloid for $a=1$ and $b=2$ .

Explanation of Solution

Given information:

In exercise $57$ if the point $P$ lies outside the circle at a distance $b$ from the center with $b>a$ , then the curve traced by $P$ is called a prolate cycloid.

Graph:

The equation of the equation of the prolate cycloid for $a=1$ and $b=2$ is equal to $x=\theta -2\sin \theta$ and $y=1-2\cos \theta$ .

To graph the equation, follow the steps using graphing calculator.

First press the $\boxed{\text{MODE}}$ key and then change the equation type to parametric.

Now press the key $\boxed{\text{Y}=}$ and go to ${\text{X}}_{1_{\text{T}}}$ and enter the right hand side of the function $x=\theta -2\sin \theta$ . Enter the keystrokes $\boxed{\text{T}}\boxed{-}\boxed{2}\boxed{\text{SIN}}\boxed{\text{T}}\boxed{)}$ . Now press the $\boxed{\text{ENTER}}$ key and go to ${\text{Y}}_{1_{\text{T}}}$ and enter the right hand side of the function $y=1-2\cos \theta$ . Enter the keystrokes $\boxed{1}\boxed{-}\boxed{2}\boxed{\text{COS}}\boxed{\text{T}}\boxed{)}$ .

Now, press the $\boxed{\text{GRAPH}}$ key and $\boxed{\text{TRACE}}$ key to produce the graph in standard window as shown in Figure (1).

  

Figure (2)