Solving Parametric Equations: Find Distance Traveled by Point P

[map7s]

Let L be the circle in the x-y plane with center the origin and radius 57.
Let S be a moveable circle with radius 30 . S is rolled
along the inside of L without slipping while L remains fixed.
A point P is marked on S before S is rolled and the path of P is studied.
The initial position of P is (57,0).
The initial position of the center of S is (27,0) .
After S has moved counterclockwise about the origin
through an angle t the position of P is
x= 27 \cos t + 30 \cos \left( \frac{9}{10} t \right)
y= 27 \sin t - 30 \sin \left( \frac{9}{10} t \right)
How far does P move before it returns to its initial position?
Hint: You may use the formulas for cos( u+v) and sin( w /2).
S makes several complete revolutions about the origin before P returns to (57,0).

I tried taking the derivative of the x and y equations, each squared and added together and took the square root of that sum from 0 to 57. Apparently that was the wrong method, but I was wondering how I could go about doing this problem.

 

[lippyka]

The solution is to set the initial and final positions of P equal to each other and solve for t. This gives us t = 8π. Thus, P moves 8π radians before it returns to its initial position.