Difference between revisions of "2023 AMC 12A Problems/Problem 15"
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Usain is walking for exercise by zigzagging across a <math>100</math>-meter by <math>30</math>-meter rectangular field, beginning at point <math>A</math> and ending on the segment <math>\overline{BC}</math>. He wants to increase the distance walked by zigzagging as shown in the figure below <math>(APQRS)</math>. What angle <math>\theta</math><math>\angle PAB=\angle QPC=\angle RQB=\cdots</math> will produce in a length that is <math>120</math> meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.) | Usain is walking for exercise by zigzagging across a <math>100</math>-meter by <math>30</math>-meter rectangular field, beginning at point <math>A</math> and ending on the segment <math>\overline{BC}</math>. He wants to increase the distance walked by zigzagging as shown in the figure below <math>(APQRS)</math>. What angle <math>\theta</math><math>\angle PAB=\angle QPC=\angle RQB=\cdots</math> will produce in a length that is <math>120</math> meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.) | ||
− | [ | + | [Pls finish] |
+ | <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("$\theta$", (-41.5,-11)); | ||
+ | label("$\theta$", (-13,11)); | ||
+ | </asy> | ||
<math>\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}</math> | <math>\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}</math> |
Revision as of 11:34, 10 November 2023
Contents
Question
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 in a length that is
meters? (This figure is not drawn to scale. Do not assume that he zigzag path has exactly four segments as shown; there could be more or fewer.)
[Pls finish]
Solution 1
By "unfolding" into a straight line, we get a right angled triangle
.
~lptoggled
Video Solution 1 by OmegaLearn
Solution 2(Trig Bash)
We can let be the length of one of the full segments of the zigzag. We can then notice that
. By Pythagorean Theorem, we see that
. This implies that:
We also realize that
, so this means that:
We can then substitute
, so this gives:
Now we have: meaning that:
This means that
, giving us
~ap246
See also
2023 AMC 12A (Problems • Answer Key • Resources) | |
Preceded by Problem 14 |
Followed by Problem 16 |
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
All AMC 12 Problems and Solutions |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.