Problem:
Usain is walking for exercise by zigzagging across a -meter by -meter rectangular field, beginning at point and ending on the segment . He wants to increase the distance walked by zigzagging as shown in the figure below . What angle will produce a length that is meters? (The figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; it could be more or fewer.)
Answer Choices:
A.
B.
C.
D.
E.
Solution:
With the point labels shown in the figure, let be the foot of the perpendicular from to . Then
Similar equations hold for other segments of the zigzag walk compared to the straight walk. Usain will achieve his objective if the reciprocal of this ratio is . Therefore and , which is about .
Note: The number of zigzags depends on the dimensions of the rectangle. When and , the path consists of three segments, with lengths (in meters) of approximately 54.27 , 54.27 , and 11.46 , summing to 120 .
The problems on this page are the property of the MAA's American Mathematics Competitions