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 BC. He wants to increase the distance walked by zigzagging as shown in the figure below (APQRS). What angle θ=∠PAB=∠QPC=∠RQB=⋯ will produce in a length that is 120 meters? (Do not assume the zigzag path has exactly four segments as shown; there could be more or fewer.)[asy]
import olympiad;
draw((-50,15)--(50,15));
draw((50,15)--(50,-15));
draw((50,-15)--(-50,-15));
draw((-50,-15)--(-50,15));
draw((-50,-15)--(-22.5,15));
draw((-22.5,15)--(5,-15));
draw((5,-15)--(32.5,15));
draw((32.5,15)--(50,-4.090909090909));
label("θ", (-41.5,-10.5));
label("θ", (-13,10.5));
label("θ", (15.5,-10.5));
label("θ", (43,10.5));
dot((-50,15));
dot((-50,-15));
dot((50,15));
dot((50,-15));
dot((50,-4.09090909090909));
label("D",(-58,15));
label("A",(-58,-15));
label("C",(58,15));
label("B",(58,-15));
label("S",(58,-4.0909090909));
dot((-22.5,15));
dot((5,-15));
dot((32.5,15));
label("P",(-22.5,23));
label("Q",(5,-23));
label("R",(32.5,23));
[/asy]<spanclass=′latex−bold′>(A)</span> arccos65<spanclass=′latex−bold′>(B)</span> arccos54<spanclass=′latex−bold′>(C)</span> arccos103<spanclass=′latex−bold′>(D)</span> arcsin54<spanclass=′latex−bold′>(E)</span> arcsin65 AMCAMC 122023 AMC2023 AMC 12A