2023 AMC 12A Problems/Problem 15: Difference between revisions

Revision as of 17:44, 10 November 2023

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 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(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: $\begin{align*} \cos\theta &= \frac{100 - 3\sqrt{x^2 - 900}}{120 - 3x}\\ &= \frac{100 - 3\sqrt{\frac{900}{\sin^2\theta} - 900}}{120 - \frac{90}{\sin\theta}}\\ &= \frac{100 - 90\sqrt{\csc^2\theta - 1}}{120 - \frac{90}{\sin\theta}}\\ &= \frac{100 - \frac{90}{\tan\theta}}{120 - 90\sin\theta}\\ &= \frac{100\sin\theta - 90\cos\theta}{120\sin\theta - 90}\\ &= \frac{10\sin\theta - 9\cos\theta}{12\sin\theta - 9}\\ \end{align*}$

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 3 (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 $\frac{100}{x} \times \sqrt{900+x^2} = 120$ $\sqrt{900+x^2} = \frac{6}{5}x$ $900 + x^2 = \frac{36}{25}x^2$ $x = \frac{150}{\sqrt{11}} = DP$ $AP = \sqrt{900 + x^2} = \frac{180}{\sqrt{11}}$ $cos\theta = \frac{DP}{AP} = \frac{5}{6}$ $\theta = arccos\frac{5}{6}$

(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 proportionate to the length of the incomplete base)

~dwarf_marshmallow

Video Solution 1 by OmegaLearn

https://youtu.be/NhUI-BNCIUE

@@ Line 104: / Line 104: @@
 ~ap246
+==Solution 3 (No Trig)==
+Let <math>x</math> be the length of <math>DP</math>. Apply the Pythagoras theorem on <math>\triangle{ADP}</math> to get <math>AP = \sqrt{900 + x^2}</math>, which is also the length of every zigzag segment.
+There are <math>\frac{100}{x}</math> such segments. Thus the total length formed by the zigzags is
+<cmath>\frac{100}{x} \times \sqrt{900+x^2} = 120</cmath>
+<cmath>\sqrt{900+x^2} = \frac{6}{5}x</cmath>
+<cmath>900 + x^2 = \frac{36}{25}x^2</cmath>
+<cmath>x = \frac{150}{\sqrt{11}} = DP</cmath>
+<cmath>AP = \sqrt{900 + x^2} = \frac{180}{\sqrt{11}}</cmath>
+<cmath>cos\theta = \frac{DP}{AP} = \frac{5}{6}</cmath>
+<cmath>\theta = arccos\frac{5}{6}</cmath>
+(note that <math>\frac{100}{x}</math> is not an integer, but it doesn't matter because of similar triangles. The length of the incomplete segment is always proportionate proportionate to the length of the incomplete base)
+~dwarf_marshmallow
 ==Video Solution 1 by OmegaLearn==

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