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Difference between revisions of "2021 Fall AMC 12B Problems/Problem 15"

(Created page with "==Problem== Three identical square sheets of paper each with side length <math>6</math> are stacked on top of each other. The middle sheet is rotated clockwise <math>30^\circ<...")
 
(→‎Solution)
Line 12: Line 12:
 
pair C=(3,3);
 
pair C=(3,3);
 
pair D=(3,2.59807621);
 
pair D=(3,2.59807621);
−
draw(A--B--C--D);
+
draw(A--B--C--D--A);
 
label("A",A,W);
 
label("A",A,W);
 
label("B",B,N);
 
label("B",B,N);
−
label("C",B,N);
+
label("C",C,N);
−
label("C",B,E);
+
label("D",D,E);
 
</asy>
 
</asy>

Revision as of 15:50, 24 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 $a+b+c$?

IMAGE

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

Solution

[asy] pair A=(0,0); pair B=(2.59807621,3); pair C=(3,3); pair D=(3,2.59807621); draw(A--B--C--D--A); label("A",A,W); label("B",B,N); label("C",C,N); label("D",D,E); [/asy]

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