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.)

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

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

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

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

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

Solution:

Every segment of the zigzag spans the full width $30,$ so it has length $\dfrac{30}{\sin\theta}$ and advances $\dfrac{30}{\tan\theta}$ horizontally.

Summing over all segments, the total length is $120$ and the total horizontal advance is $100.$ Their ratio is $\dfrac{\text{length}}{\text{horizontal}} =\dfrac{1/\sin\theta}{1/\tan\theta}=\dfrac{1}{\cos\theta}=\dfrac{120}{100}.$

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

Thus, the correct answer is A.

2023 AMC 12A Problem 15

Solution:

Problem 15 in Other Years