# 2023 AMC 12A Problems/Problem 15

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## 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 $\cos\theta = \frac{x}{(\frac65x)} = \frac56 \Rightarrow \theta = \boxed{\textbf{(A) }\arccos{\frac56}}$.

## 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 (⚡Solved in 56 seconds⚡)

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

## Video Solution

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