Difference between revisions of "2021 AMC 10B Problems/Problem 23"
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==Problem== | ==Problem== | ||
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> | 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> | ||
| − | + | <asy> | |
/* Made by samrocksnature */ | /* Made by samrocksnature */ | ||
draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); | draw((0,0)--(8,0)--(8,8)--(0,8)--(0,0)); | ||
| Line 10: | Line 10: | ||
fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); | fill((4,6)--(2,4)--(4,2)--(6,4)--cycle, black); | ||
filldraw(circle((2.6,3.31),0.47),gray); | filldraw(circle((2.6,3.31),0.47),gray); | ||
| − | + | </asy> | |
<math>\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72</math> | <math>\textbf{(A)} ~64 \qquad\textbf{(B)} ~66 \qquad\textbf{(C)} ~68 \qquad\textbf{(D)} ~70 \qquad\textbf{(E)} ~72</math> | ||
==Solution== | ==Solution== | ||
Revision as of 18:43, 11 February 2021
Problem
A square with side length 
is colored white except for 
black isosceles right triangular regions with legs of length 
in each corner of the square and a black diamond with side length 
in the center of the square, as shown in the diagram. A circular coin with diameter 
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 
![[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]](http://latex.artofproblemsolving.com/7/8/2/782e019eee5da15727a6caf03020a1dfd545b11d.png)
