Problem:
Rectangle PQRS lies in a plane with PQ=RS=2 and QR=SP=6. The rectangle is rotated 90∘ clockwise about R, then rotated 90∘ clockwise about the point that S moved to after the first rotation. What is the length of the path traveled by point P?
Answer Choices:
A. (23+5)π
B. 6π
C. (3+10)π
D. (3+25)π
E. 210π
Solution:
Let P′ and S′ denote the positions of P and S, respectively, after the rotation about R, and let P′′ denote the final position of P. In the rotation that moves P to position P′, the point P rotates 90∘ on a circle with center R and radius PR=22+62=210. The length of the arc traced by P is (1/4)(2π⋅210)=π10. Next, P′ rotates to P′′ through a 90∘ arc on a circle with center S′ and radius S′P′=6. The length of this arc is 41(2π⋅6)=3π. The total distance traveled by P is