Question

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.)

$\arccos\tfrac{5}{6}$

$\arccos\tfrac{4}{5}$

$\arccos\tfrac{3}{10}$

$\arcsin\tfrac{4}{5}$

$\arcsin\tfrac{5}{6}$

Answer: A

Concepts:trigonometry right triangle

Difficulty rating: 1800

Solution:

Every segment of the zigzag makes angle $\theta$ with a horizontal side of the field. Therefore a segment of length $s$ advances $s\cos\theta$ meters horizontally. This remains true for the last segment even if it ends before crossing the full width of the field.

Adding the horizontal projections over the entire $120$ -meter path gives $120\cos\theta=100.$

Therefore $\cos\theta=\dfrac56,$ so $\theta=\arccos\dfrac56.$

Thus, the correct answer is A.

2023 AMC 12A Problem 15

Answer: A

Difficulty rating: 1800

Solution:

Problem 15 in Other Years