Difference between revisions of "2021 AMC 10B Problems/Problem 23"

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==Problem==
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A square with side length <math>8</math> is colored white except for <math>4</math> black isosceles right triangular regions with legs of length <math>2</math> in each corner of the square and a black diamond with side length <math>2\sqrt{2}</math> in the center of the square, as shown in the diagram. A circular coin with diameter <math>1</math> is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as <math>\frac{1}{196}(a+b\sqrt{2}+\pi)</math>
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[asy]
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/* Made by samrocksnature */
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draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0));
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fill((2,0)--(0,2)--(0,0)--cycle, black);
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fill((6,0)--(8,0)--(8,2)--cycle, black);
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fill((8,6)--(8,8)--(6,8)--cycle, black);
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fill((0,6)--(2,8)--(0,8)--cycle, black);
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fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black);
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filldraw(circle((2.6,3.31),0.47),gray);
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[/asy]
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<math>\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72</math>
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==Solution==

Revision as of 19:43, 11 February 2021

Problem

A square with side length $8$ is colored white except for $4$ black isosceles right triangular regions with legs of length $2$ in each corner of the square and a black diamond with side length $2\sqrt{2}$ in the center of the square, as shown in the diagram. A circular coin with diameter $1$ is dropped onto the square and lands in a random location where the coin is completely contained within the square. The probability that the coin will cover part of the black region of the square can be written as $\frac{1}{196}(a+b\sqrt{2}+\pi)$ [asy] /* Made by samrocksnature */ draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); fill((2,0)--(0,2)--(0,0)--cycle, black); fill((6,0)--(8,0)--(8,2)--cycle, black); fill((8,6)--(8,8)--(6,8)--cycle, black); fill((0,6)--(2,8)--(0,8)--cycle, black); fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); filldraw(circle((2.6,3.31),0.47),gray); [/asy]

$\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72$

Solution