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Difference between revisions of "1992 AJHSME Problems/Problem 24"

(Created page with "==Problem== Four circles of radius <math>3</math> are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to <asy> fill((3...")
 
 
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<math>\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27</math>
 
<math>\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27</math>
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==Solution==
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The side of the square of made up of two radii, and the area of the square is <math>(3+3)^2=36</math>. The unshaded region of the square is composed of four quarter circles, and has the area of one circle, which is <math>9\pi</math>. The area of the shaded region is <math>36-9\pi</math> which less than <math>36-9(3)=9</math>, closest to <math>\boxed{\text{(A)}\ 7.7}</math>.
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==See Also==
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{{AJHSME box|year=1992|num-b=23|num-a=25}}
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{{MAA Notice}}

Latest revision as of 23:10, 4 July 2013

Problem

Four circles of radius $3$ are arranged as shown. Their centers are the vertices of a square. The area of the shaded region is closest to

[asy] fill((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle,lightgray); fill(arc((3,3),(0,3),(3,0),CCW)--(3,3)--cycle,white); fill(arc((3,-3),(3,0),(0,-3),CCW)--(3,-3)--cycle,white); fill(arc((-3,-3),(0,-3),(-3,0),CCW)--(-3,-3)--cycle,white); fill(arc((-3,3),(-3,0),(0,3),CCW)--(-3,3)--cycle,white); draw(circle((3,3),3)); draw(circle((3,-3),3)); draw(circle((-3,-3),3)); draw(circle((-3,3),3)); draw((3,3)--(3,-3)--(-3,-3)--(-3,3)--cycle); [/asy]

$\text{(A)}\ 7.7 \qquad \text{(B)}\ 12.1 \qquad \text{(C)}\ 17.2 \qquad \text{(D)}\ 18 \qquad \text{(E)}\ 27$

Solution

The side of the square of made up of two radii, and the area of the square is $(3+3)^2=36$. The unshaded region of the square is composed of four quarter circles, and has the area of one circle, which is $9\pi$. The area of the shaded region is $36-9\pi$ which less than $36-9(3)=9$, closest to $\boxed{\text{(A)}\ 7.7}$.

See Also

1992 AJHSME (Problems • Answer Key • Resources)
Preceded by
Problem 23 		Followed by
Problem 25
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AJHSME/AMC 8 Problems and Solutions

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