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| − | <!-- #REDIRECT [[2020 AMC 10B Problems/Problem 14]] -->
| + | #REDIRECT [[2020 AMC 10B Problems/Problem 14]] |
| − | ==Problem==
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| − | As shown in the figure below, six semicircles lie in the interior of a regular hexagon with side length <math>2</math> so that the diameters of the semicircles coincide with the sides of the hexagon. What is the area of the shaded region––inside the hexagon but outside all of the semicircles?
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| − | <asy> size(140); fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4)); fill(arc((2,0),1,180,0)--(2,0)--cycle,white); fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white); fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white); fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white); fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white); fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white); draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0)); draw(arc((2,0),1,180,0)--(2,0)--cycle); draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle); draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle); draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle); draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle); draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle); label("$2$",(3.5,3sqrt(3)/2),NE); </asy>
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| − | <math>\textbf{(A)}\ 6\sqrt3-3\pi \qquad\textbf{(B)}\ \frac{9\sqrt3}{2}-2\pi \qquad\textbf{(C)}\ \frac{3\sqrt3}{2}-\frac{\pi}{3} \qquad\textbf{(D)}\ 3\sqrt3-\pi \\ \qquad\textbf{(E)}\ \frac{9\sqrt3}{2}-\pi</math>
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| − | | |
| − | ==Solution==
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| − | Work in progress
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| − | | |
| − | Subdivide the hexagon into 24 equilateral triangles with length 1:
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| − | <asy> size(140); fill((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--cycle,gray(0.4)); fill(arc((2,0),1,180,0)--(2,0)--cycle,white); fill(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle,white); fill(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle,white); fill(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle,white); fill(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle,white); fill(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle,white); draw((1,0)--(3,0)--(4,sqrt(3))--(3,2sqrt(3))--(1,2sqrt(3))--(0,sqrt(3))--(1,0)); draw(arc((2,0),1,180,0)--(2,0)--cycle); draw(arc((3.5,sqrt(3)/2),1,60,240)--(3.5,sqrt(3)/2)--cycle); draw(arc((3.5,3sqrt(3)/2),1,120,300)--(3.5,3sqrt(3)/2)--cycle); draw(arc((2,2sqrt(3)),1,180,360)--(2,2sqrt(3))--cycle); draw(arc((0.5,3sqrt(3)/2),1,240,420)--(0.5,3sqrt(3)/2)--cycle); draw(arc((0.5,sqrt(3)/2),1,300,480)--(0.5,sqrt(3)/2)--cycle); label("$2$",(3.5,3sqrt(3)/2),NE);
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| − | draw((1,0)--(3,2sqrt(3)));
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| − | draw((3,0)--(1,2sqrt(3)));
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| − | draw((4,sqrt(3))--(0,sqrt(3)));
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| − | draw((2,0)--(3.5,3sqrt(3)/2));
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| − | draw((3.5,sqrt(3)/2)--(2,2sqrt(3)));
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| − | draw((3.5,3sqrt(3)/2)--(0.5,3sqrt(3)/2));
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| − | draw((2,2sqrt(3))--(0.5,sqrt(3)/2));
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| − | draw((2,0)--(0.5,3sqrt(3)/2));
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| − | draw((3.5,sqrt(3)/2)--(0.5,sqrt(3)/2));
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| − | </asy>
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| − | Now note that the entire shaded region is just 6 times this part:
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| − | <asy> size(200); fill((2,sqrt(3))--(2.5,3sqrt(3)/2)--(2,2sqrt(3))--(1.5,3sqrt(3)/2)--cycle,gray(0.4));
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| − | | |
| − | fill(arc((2,0),1,180,0)--(2,0)--cycle,white);
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| − | fill(arc((2,2sqrt(3)),1,240,300)--(2,2sqrt(3))--cycle,white);
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| − | draw(arc((2,2sqrt(3)),1,240,300)--(2,2sqrt(3))--cycle);
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| − | label("$1$",(2.25,7sqrt(3)/4),NE);
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| − | draw((2,sqrt(3))--(2.5,3sqrt(3)/2)--(2,2sqrt(3))--(1.5,3sqrt(3)/2)--cycle);
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| − | draw((2.5,3sqrt(3)/2)--(1.5,3sqrt(3)/2));
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| − | | |
| − | </asy>
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| − | | |
| − | ==See Also==
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| − | | |
| − | {{AMC12 box|year=2020|ab=B|num-b=10|num-a=12}}
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| − | {{MAA Notice}}
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