Let $Lbe$ the circle $in$ the $x-y$ plane with center the origin and radius $19.$

Let $Sbe$ a moveable circle with radius $3.Sis$ rolled

along the inside $ofL$ without slipping while $L$ remains fixed.

A point $Pis$ marked $onS$ before $Sis$ rolled and the path $ofPis$ studied.

The initial position $ofPis(19,0).$

The initial position $of$ the center $ofSis(16,0).$

After $S$ has moved counterclockwise about the origin

through $an$ angle $t$ the position $ofPis$

$x=16cost+3cos(\frac{16}{3}t)$

$y=16sint-3sin(\frac{16}{3}t)$

How far does $P$ move before $it$ returns $to$ its initial position?

Hint: You may use the formulas for $cos(u+v)$ and $sin(\frac{w}{2}).$

$S$ makes several complete revolutions about the origin before $P$ returns $to(19,0).$