Difference between revisions of "2021 Fall AMC 12B Problems/Problem 15"
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− | We can see that this shape is made out of <math>12</math> of these shapes. <math>\angle{A} = 30^{\circ}</math> and <math>\angle{C} = 90^{\circ}</math>. We also know that there is a square <math>AECF</math> and two triangles <math>AEB</math> and <math>ADF</math>. With these triangles and squares we find the area of one of these shapes to be <math>9-3\sqrt{3}</math>. The area of the big shape is then <math>12(9-3\sqrt{3})=108-36\sqrt{3}</math>. | + | We can see that this shape is made out of <math>12</math> of these shapes. <math>\angle{A} = 30^{\circ}</math> and <math>\angle{C} = 90^{\circ}</math>. We also know that there is a square <math>AECF</math> and two triangles <math>AEB</math> and <math>ADF</math>. With these triangles and squares we find the area of one of these shapes to be <math>9-3\sqrt{3}</math>. The area of the big shape is then <math>12(9-3\sqrt{3})=108-36\sqrt{3}</math>. <math>108+36+3=\boxed{\qquad(\textbf{E})\ 147}</math> ~lopkiloinm |
Revision as of 17:07, 24 November 2021
Problem
Three identical square sheets of paper each with side length are stacked on top of each other. The middle sheet is rotated clockwise
about its center and the top sheet is rotated clockwise
about its center, resulting in the
-sided polygon shown in the figure below. The area of this polygon can be expressed in the form
, where
,
, and
are positive integers, and
is not divisible by the square of any prime. What is
?
IMAGE
Solution
We can see that this shape is made out of
of these shapes.
and
. We also know that there is a square
and two triangles
and
. With these triangles and squares we find the area of one of these shapes to be
. The area of the big shape is then
.
~lopkiloinm