The tower's radius actually plays no part in the problem.
The elevator will obviously move exactly 200 feet from the bottom of the tower to the top.
To calculate the length of the stairs, imagine it "uncoiled" from the tower, and flattened out on a flat surface. You will be looking at a right triangle with the height of 200 feet, and the angles of 30 and 60, where the length of the hypotenuse is what we are trying to find.
We can use a simple trigonometric equasion:
sin(A)/a = sin(B)/b, or
sin(90)/H = sin(30)/200 and
1/H = (1/2)/200, giving us
H = 400.
So the length of the stairway is only twice the length of the elevator shaft.