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Difference between revisions of "2021 Fall AMC 10B Problems/Problem 18"

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<math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math>
 
<math>(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147</math>
  
−
==Solution==
+
==Solution 1==
 
First note the useful fact that if <math>R</math> is the circumradius of a dodecagon (<math>12</math>-gon) the area of the figure is <math>3R^2.</math> If we connect the vertices of the <math>3</math> squares we get a dodecagon. The radius of circumcircle of the dodecagon is simply half the diagonal of the square, which is <math>3\sqrt{2}.</math> Thus the area of the dodecagon is <math>3 \cdot (3\sqrt{2})^2 = 3 \cdot 18 = 54.</math> But, the problem asks for the area of figure of rotated squares. This area is the area of the dodecagon, which was found, subtracting the <math>12</math> isosceles triangles, which are formed when connecting the vertices of the squares to created the dodecagon. To find this area, we need to know the base of the isosceles triangle, call this <math>x.</math> Then, we can use Law of Cosines, on the triangle that is formed from the two vertices of the square and the center of the square. After computing, we get that <math>x = 3\sqrt{3} -3.</math> Realize that the <math>12</math> isosceles are congruent with an angle measure of <math>120^{\circ},</math> this means that we can create <math>4</math> congruent equilateral triangles with side length of <math>3\sqrt3 - 3.</math> The area of the equilateral triangle is <math>\frac{\sqrt{3}}{4} \cdot (3\sqrt{3} -3)^2 =  \frac{\sqrt{3}}{4} \cdot (36 - 18\sqrt{3}) = \frac{36\sqrt{3} - 54}{4}.</math> Thus, the area of all the twelve small equilateral traingles are <math>4 \cdot \frac{36\sqrt{3} - 54}{4} = 36\sqrt{3} - 54</math>. Thus, the requested area is <math>54 - (36\sqrt{3} - 54) = 108 - 36\sqrt{3}.</math> Thus, <math>a+b+c = 108 + 36 + 3 = 147.</math> Thus, the answer is <math>\boxed{(\textbf{E}.)}.</math>
 
First note the useful fact that if <math>R</math> is the circumradius of a dodecagon (<math>12</math>-gon) the area of the figure is <math>3R^2.</math> If we connect the vertices of the <math>3</math> squares we get a dodecagon. The radius of circumcircle of the dodecagon is simply half the diagonal of the square, which is <math>3\sqrt{2}.</math> Thus the area of the dodecagon is <math>3 \cdot (3\sqrt{2})^2 = 3 \cdot 18 = 54.</math> But, the problem asks for the area of figure of rotated squares. This area is the area of the dodecagon, which was found, subtracting the <math>12</math> isosceles triangles, which are formed when connecting the vertices of the squares to created the dodecagon. To find this area, we need to know the base of the isosceles triangle, call this <math>x.</math> Then, we can use Law of Cosines, on the triangle that is formed from the two vertices of the square and the center of the square. After computing, we get that <math>x = 3\sqrt{3} -3.</math> Realize that the <math>12</math> isosceles are congruent with an angle measure of <math>120^{\circ},</math> this means that we can create <math>4</math> congruent equilateral triangles with side length of <math>3\sqrt3 - 3.</math> The area of the equilateral triangle is <math>\frac{\sqrt{3}}{4} \cdot (3\sqrt{3} -3)^2 =  \frac{\sqrt{3}}{4} \cdot (36 - 18\sqrt{3}) = \frac{36\sqrt{3} - 54}{4}.</math> Thus, the area of all the twelve small equilateral traingles are <math>4 \cdot \frac{36\sqrt{3} - 54}{4} = 36\sqrt{3} - 54</math>. Thus, the requested area is <math>54 - (36\sqrt{3} - 54) = 108 - 36\sqrt{3}.</math> Thus, <math>a+b+c = 108 + 36 + 3 = 147.</math> Thus, the answer is <math>\boxed{(\textbf{E}.)}.</math>
  
 
~NH14
 
~NH14
 +
 +
== Solution 2 ==
 +
As shown in [[:Image:2021_AMC_12B_(Nov)_Problem_15,_sol.png]], all 12 vertices of three squares form a regular dodecagon (12-gon).
 +
Denote by <math>O</math> the center of this dodecagon.
 +
 +
Hence, <math>\angle AOB = \frac{360^\circ}{12} = 30^\circ</math>.
 +
 +
Because the length of a side of a square is 6, <math>AO = 3 \sqrt{2}</math>.
 +
 +
Hence, <math>AB = 2 AO \sin \frac{\angle AOB}{2} = 3 \left( \sqrt{3} - 1 \right)</math>.
 +
 +
We notice that <math>\angle MAB = \angle MBA = 30^\circ</math>.
 +
Hence, <math>AM = \frac{AB}{2\cos \angle MAB} = 3 - \sqrt{3}</math>.
 +
 +
Therefore, the area of the region that three squares cover is
 +
<cmath>
 +
\begin{align*}
 +
& {\rm Area} \ ABCDEFGHIJKL - 12 {\rm Area} \ \triangle MAB \\
 +
& = 12 {\rm Area} \ \triangle OAB - 12 {\rm Area} \ \triangle MAB \\
 +
& = 12 \cdot \frac{1}{2} OA \cdot OB \sin \angle AOB
 +
- 12 \cdot \frac{1}{2} MA \cdot MB \sin \angle AMB \\
 +
& = 6 OA^2 \sin \angle AOB - 6 MA^2 \sin \angle AMB \\
 +
& = 108 - 36 \sqrt{3} .
 +
\end{align*}
 +
</cmath>
 +
 +
Therefore, the answer is <math>\boxed{\textbf{(E) }147}</math>.
 +
 +
~Steven Chen (www.professorchenedu.com)
  
 
==See Also==
 
==See Also==
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=19|num-b=17}}
 
{{AMC10 box|year=2021 Fall|ab=B|num-a=19|num-b=17}}
 
{{MAA Notice}}
 
{{MAA Notice}}

Revision as of 22:53, 25 November 2021

Problem

Three identical square sheets of paper each with side length $6{ }$ are stacked on top of each other. The middle sheet is rotated clockwise $30^\circ$ about its center and the top sheet is rotated clockwise $60^\circ$ about its center, resulting in the $24$-sided polygon shown in the figure below. The area of this polygon can be expressed in the form $a-b\sqrt{c}$, where $a$, $b$, and $c$ are positive integers, and $c$ is not divisible by the square of any prime. What is $a+b+c?$

$(\textbf{A})\: 75\qquad(\textbf{B}) \: 93\qquad(\textbf{C}) \: 96\qquad(\textbf{D}) \: 129\qquad(\textbf{E}) \: 147$

Solution 1

First note the useful fact that if $R$ is the circumradius of a dodecagon ($12$-gon) the area of the figure is $3R^2.$ If we connect the vertices of the $3$ squares we get a dodecagon. The radius of circumcircle of the dodecagon is simply half the diagonal of the square, which is $3\sqrt{2}.$ Thus the area of the dodecagon is $3 \cdot (3\sqrt{2})^2 = 3 \cdot 18 = 54.$ But, the problem asks for the area of figure of rotated squares. This area is the area of the dodecagon, which was found, subtracting the $12$ isosceles triangles, which are formed when connecting the vertices of the squares to created the dodecagon. To find this area, we need to know the base of the isosceles triangle, call this $x.$ Then, we can use Law of Cosines, on the triangle that is formed from the two vertices of the square and the center of the square. After computing, we get that $x = 3\sqrt{3} -3.$ Realize that the $12$ isosceles are congruent with an angle measure of $120^{\circ},$ this means that we can create $4$ congruent equilateral triangles with side length of $3\sqrt3 - 3.$ The area of the equilateral triangle is $\frac{\sqrt{3}}{4} \cdot (3\sqrt{3} -3)^2 = \frac{\sqrt{3}}{4} \cdot (36 - 18\sqrt{3}) = \frac{36\sqrt{3} - 54}{4}.$ Thus, the area of all the twelve small equilateral traingles are $4 \cdot \frac{36\sqrt{3} - 54}{4} = 36\sqrt{3} - 54$. Thus, the requested area is $54 - (36\sqrt{3} - 54) = 108 - 36\sqrt{3}.$ Thus, $a+b+c = 108 + 36 + 3 = 147.$ Thus, the answer is $\boxed{(\textbf{E}.)}.$

~NH14

Solution 2

As shown in Image:2021_AMC_12B_(Nov)_Problem_15,_sol.png, all 12 vertices of three squares form a regular dodecagon (12-gon). Denote by $O$ the center of this dodecagon.

Hence, $\angle AOB = \frac{360^\circ}{12} = 30^\circ$.

Because the length of a side of a square is 6, $AO = 3 \sqrt{2}$.

Hence, $AB = 2 AO \sin \frac{\angle AOB}{2} = 3 \left( \sqrt{3} - 1 \right)$.

We notice that $\angle MAB = \angle MBA = 30^\circ$. Hence, $AM = \frac{AB}{2\cos \angle MAB} = 3 - \sqrt{3}$.

Therefore, the area of the region that three squares cover is \begin{align*} & {\rm Area} \ ABCDEFGHIJKL - 12 {\rm Area} \ \triangle MAB \\ & = 12 {\rm Area} \ \triangle OAB - 12 {\rm Area} \ \triangle MAB \\ & = 12 \cdot \frac{1}{2} OA \cdot OB \sin \angle AOB - 12 \cdot \frac{1}{2} MA \cdot MB \sin \angle AMB \\ & = 6 OA^2 \sin \angle AOB - 6 MA^2 \sin \angle AMB \\ & = 108 - 36 \sqrt{3} . \end{align*}

Therefore, the answer is $\boxed{\textbf{(E) }147}$.

~Steven Chen (www.professorchenedu.com)

See Also

2021 Fall AMC 10B (Problems • Answer Key • Resources)
Preceded by
Problem 17 		Followed by
Problem 19
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 10 Problems and Solutions

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