Problem 370

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Problem 370: (2023 AMC12A, 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 in a length that is $120$ meters? (This figure is not drawn to scale. Do not assume that the zigzag path has exactly four segments as shown; there could be more or fewer.)

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

Solution

Exercise 370: Solution

Usain is walking for exercise by zigzagging across a $90$-meter by $50$-meter rectangular field, beginning at point $A$ and ending on the segment $\overline{BC}$, as shown in the figure $(APQS)$. Every segment of his path makes the same angle $\theta$ with the long sides of the field: $\angle PAB=\angle QPC=\angle SQB=\cdots=\theta$. What angle $\theta$ will produce a path of length $150$ meters? (The figure is not drawn to scale. Do not assume that the zigzag path has exactly three segments as shown; there could be more or fewer.)

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