Question 33
Alice is sitting in a teacup ride with infinitely many layers of spinning disks. The largest disk has radius $5$. Each succeeding disk has its center attached to a point on the circumference of the previous disk and has a radius equal to $2/3$ of the previous disk. Each disk spins around its center (relative to the disk it is attached to) at a rate of $\pi/6$ radians per second. Initially, at $t=0$, the centers of the disks are aligned on a single line, going outward. Alice is sitting at the limit point of all these disks. After $12$ seconds, what is the length of the trajectory that Alice has traced out?