2023 AMC 12A Problem 15

Usain is walking for exercise by zigzagging across a 100-meter by 30-meter rectangular field, beginning at point A and ending on the segment \overline{BC}. He wants to increase the distance walked by zigzagging as shown in the figure below (APQRS). What angle \theta=\angle PAB =\angle QPC =\angle RQB=\cdots will produce a length that is 120 meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)

\textbf{(A)} \arccos\tfrac{5}{6} \qquad \textbf{(B)} \arccos\tfrac{4}{5} \qquad \textbf{(C)} \arccos\tfrac{3}{10} \qquad \textbf{(D)} \arcsin\tfrac{4}{5} \qquad \textbf{(E)} \arcsin\tfrac{5}{6}

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.