Difference between revisions of "2023 AMC 12A Problems/Problem 15"

Revision as of 19:58, 19 May 2024

Problem

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 he zigzag path has exactly four segments as shown; there could be more or fewer.)

$[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label("$\theta$", (-41.5,-10.5)); label("$\theta$", (-13,10.5)); label("$\theta$", (15.5,-10.5)); label("$\theta$", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label("$D$",(-58,15)); label("$A$",(-58,-15)); label("$C$",(58,15)); label("$B$",(58,-15)); label("$S$",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); label("$P$",(-22.5,23)); label("$Q$",(5,-23)); label("$R$",(32.5,23)); [/asy]$

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

Solution 1

By "unfolding" $APQRS$ into a straight line, we get a right angled triangle $ABS'$ .

$[asy] import olympiad; draw((-50,15)--(50,15)); draw((50,15)--(50,-15)); draw((50,-15)--(-50,-15)); draw((-50,-15)--(-50,15)); draw((-50,-15)--(-22.5,15)); draw((-22.5,15)--(5,-15)); draw((5,-15)--(32.5,15)); draw((32.5,15)--(50,-4.090909090909)); label("$\theta$", (-41.5,-10.5)); label("$\theta$", (-13,10.5)); label("$\theta$", (15.5,-10.5)); label("$\theta$", (43,10.5)); dot((-50,15)); dot((-50,-15)); dot((50,15)); dot((50,-15)); dot((50,-4.09090909090909)); label("$D$",(-58,15)); label("$A$",(-58,-15)); label("$C$",(58,15)); label("$B$",(58,-15)); label("$S$",(58,-4.0909090909)); dot((-22.5,15)); dot((5,-15)); dot((32.5,15)); dot((5,45)); dot((32.5,75)); dot((50,94.09090909090909)); draw((-22.5,15)--(50,94.09090909090909)); draw((50,-4.09090909090909)--(50,94.09090909090909)); label("$P$",(-22.5,23)); label("$Q$",(5,-23)); label("$R$",(32.5,23)); label("$Q'$",(5,35)); label("$R'$",(32.5,85)); label("$S'$",(58,94.09090909090909)); [/asy]$

$cos(\theta)=\frac{100}{120}$

$\theta=\boxed{\textbf{(A) } \arccos\left(\frac{5}{6}\right)}$

~lptoggled

Solution 2 (also simple)

Drop an altitude from $P$ to $AB$ and let its base be $x$ . Note that if we repeat this for $Q$ and $R$ , all four right triangles (including $\triangle{RSC}$ ) will have the same trig ratios. By proportion, the hypotenuse $AP$ is $\frac{x}{100}(120) = \frac65 x$ , so $\cos\theta = \frac{x}{(\frac65x)} = \frac56 \Rightarrow \theta = \boxed{\textbf{(A) }\arccos{\frac56}}$ .

~IbrahimNadeem

Solution 3(Trig Bash)

We can let $x$ be the length of one of the full segments of the zigzag. We can then notice that $\sin\theta = \frac{30}{x}$ . By Pythagorean Theorem, we see that $DP = \sqrt{x^2 - 900}$ . This implies that: $RC = 100 - 3\sqrt{x^2 - 900}.$ We also realize that $RS = 120 - 3x$ , so this means that: $\cos\theta = \frac{100 - 3\sqrt{x^2 - 900}}{120 - 3x}.$ We can then substitute $x = \frac{30}{\sin\theta}$ , so this gives: $\begin{align*} \cos\theta &= \frac{100 - 3\sqrt{x^2 - 900}}{120 - 3x}\\ &= \frac{100 - 3\sqrt{\frac{900}{\sin^2\theta} - 900}}{120 - \frac{90}{\sin\theta}}\\ &= \frac{100 - 90\sqrt{\csc^2\theta - 1}}{120 - \frac{90}{\sin\theta}}\\ &= \frac{100 - \frac{90}{\tan\theta}}{120 - \frac{90}{\sin\theta}}\\ &= \frac{100\sin\theta - 90\cos\theta}{120\sin\theta - 90}\\ &= \frac{10\sin\theta - 9\cos\theta}{12\sin\theta - 9}\\ \end{align*}$

Now we have: $\cos\theta = \frac{10\sin\theta - 9\cos\theta}{12\sin\theta - 9},$ meaning that: $12\sin\theta\cos\theta - 9\cos\theta = 10\sin\theta - 9\cos\theta \implies \cos\theta = \frac{10}{12} = \frac56.$ This means that $\theta = \arccos\left(\frac56\right)$ , giving us $\boxed{\textbf{A}}$

~ap246

Solution 4 (No Trig)

Let $x$ be the length of $DP$ . Apply the Pythagoras theorem on $\triangle{ADP}$ to get $AP = \sqrt{900 + x^2}$ , which is also the length of every zigzag segment.

There are $\frac{100}{x}$ such segments. Thus the total length formed by the zigzags is $\frac{100}{x} \times \sqrt{900+x^2} = 120$ $\sqrt{900+x^2} = \frac{6}{5}x$ $900 + x^2 = \frac{36}{25}x^2$ $x = \frac{150}{\sqrt{11}} = DP$ $AP = \sqrt{900 + x^2} = \frac{180}{\sqrt{11}}$ $\cos\theta = \frac{DP}{AP} = \frac{5}{6}$ $\theta=\boxed{\textbf{(A) } \arccos\left(\frac{5}{6}\right)}$

(note that $\frac{100}{x}$ is not an integer, but it doesn't matter because of similar triangles. The length of the incomplete segment is always proportionate to the length of the incomplete base)

~dwarf_marshmallow

Solution 5 (Intuitive and Quick)

Imagine that Usain walks at a constant speed. The horizontal component of Usain's velocity does not change. (Imagine a beam of light reflecting off of mirrors. A mirror only changes the velocity of light in the direction perpendicular to the mirror.) The horizontal component of Usain's velocity divided by his total velocity must be $\frac{100}{120}$ . Therefore $\theta=\boxed{\textbf{(A) } \arccos\left(\frac{5}{6}\right)}$ . ~numerophile

Solution 6

Although the diagram is not fully accurate, we can use it to some extent.

It's given that the length of the rectangle is $100$ and the total length of the path Usain is taking is $120$ , so Usain will walk $\frac{120}{100} = \frac{6}{5}$ longer than he would if he were just to walk along the rectangle. Therefore for each point along his path, he will travel $\frac{6}{5}$ farther than he would if he were just to walk along the rectangle.

Drop a perpendicular from point $P$ to point $Q$ on $\overline{AB}$ . Let $AQ = x$ : it follows that $QP = 30$ and $PA = \frac{6}{5}x.$ By the Pythagorean Theorem,

$x^2 + 900 = \frac{36}{25}x,$

so $x = \frac{150}{\sqrt{11}}$ and $\frac{6x}{5} = \frac{180}{\sqrt{11}}$ .

Now, $\cos\theta = \frac{\frac{150}{\sqrt{11}}}{\frac{180}{\sqrt{11}}} = \frac{5}{6}$ and $\theta = \boxed{\textbf{(A) } \arccos\left(\frac{5}{6}\right)}.$

-Benedict T (countmath1)

Video Solution 1 by OmegaLearn

https://youtu.be/NhUI-BNCIUE

Video Solution (⚡Solved in 56 seconds⚡)

https://youtu.be/jkujDM5aW3w ~Education, the Study of Everything

Video Solution

https://youtu.be/S5H8JEImiA8

~Steven Chen (Professor Chen Education Palace, www.professorchenedu.com)

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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 = \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 = \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.)

2023 AMC 12A (Problems • Answer Key • Resources)
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