Difference between revisions of "2010 AMC 10B Problems/Problem 16"
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+ | == Problem== | ||
+ | A square of side length <math>1</math> and a circle of radius <math>\dfrac{\sqrt{3}}{3}</math> share the same center. What is the area inside the circle, but outside the square? | ||
+ | <math>\textbf{(A)}\ \dfrac{\pi}{3}-1 \qquad \textbf{(B)}\ \dfrac{2\pi}{9}-\dfrac{\sqrt{3}}{3} \qquad \textbf{(C)}\ \dfrac{\pi}{18} \qquad \textbf{(D)}\ \dfrac{1}{4} \qquad \textbf{(E)}\ \dfrac{2\pi}{9}</math> | ||
+ | |||
+ | ==Solution 1== | ||
+ | The radius of the circle is <math>\frac{\sqrt{3}}{3} = \sqrt{\frac{1}{3}}</math>. Half the diagonal of the square is <math>\frac{\sqrt{1^2+1^2}}{2} = \frac{\sqrt{2}}{2} = \sqrt{\frac12}</math>. We can see that the circle passes outside the square, but the square is NOT completely contained in the circle. Therefore the picture will look something like this: | ||
+ | |||
+ | <center><asy> | ||
+ | unitsize(5cm); | ||
+ | defaultpen(linewidth(.8pt)+fontsize(10pt)); | ||
+ | dotfactor=3; | ||
+ | |||
+ | real r=sqrt(1/3); | ||
+ | pair O=(0,0); | ||
+ | pair W=(0.5,0.5), X=(0.5,-0.5), Y=(-0.5,-0.5), Z=(-0.5,0.5); | ||
+ | pair A=(-sqrt(1/12),0.5), B=(sqrt(1/12),0.5); | ||
+ | pair V=(0,0.5); | ||
+ | path outer=Circle(O,r); | ||
+ | draw(outer); | ||
+ | draw(W--X--Y--Z--cycle); | ||
+ | draw(O--A); | ||
+ | draw(O--B); | ||
+ | draw(V--O); | ||
+ | |||
+ | pair[] ps={A,B,V,O}; | ||
+ | dot(ps); | ||
+ | |||
+ | label("$O$",O,SW); | ||
+ | label("$\frac{\sqrt{3}}{3}$",O--B,SE); | ||
+ | label("$A$",A,NW); | ||
+ | label("$B$",B,NE); | ||
+ | label("$X$",V,NW); | ||
+ | label("$a$",B--V,S); | ||
+ | label("$\frac12$",O--V,W); | ||
+ | </asy></center> | ||
+ | |||
+ | Then we proceed to find: 4 <math>\cdot</math> (area of sector marked off by the two radii - area of the triangle with sides on the square and the two radii). | ||
+ | |||
+ | First we realize that the radius perpendicular to the side of the square between the two radii marking off the sector splits <math>AB</math> in half. Let this half-length be <math>a</math>. Also note that <math>OX=\frac12</math> because it is half the sidelength of the square. Because this is a right triangle, we can use the [[Pythagorean Theorem]] to solve for <math>a.</math> | ||
+ | |||
+ | <cmath>a^2+\left( \frac12 \right) ^2 = \left( \frac{\sqrt{3}}{3} \right) ^2</cmath> | ||
+ | |||
+ | Solving, <math>a= \frac{\sqrt{3}}{6}</math> and <math>2a=\frac{\sqrt{3}}{3}</math>. Since <math>AB=AO=BO</math>, <math>\triangle AOB</math> is an equilateral triangle and the central angle is <math>60^{\circ}</math>. Therefore the sector has an area <math>\pi \left( \frac{\sqrt{3}}{3} \right) ^2 \left( \frac{60}{360} \right) = \frac{\pi}{18}</math>. | ||
+ | |||
+ | Now we turn to the triangle. Since it is equilateral, we can use the formula for the [[area of an equilateral triangle]] which is | ||
+ | |||
+ | <cmath>\frac{s^2\sqrt{3}}{4} = \frac{\frac13 \sqrt{3}}{4} = \frac{\sqrt{3}}{12}</cmath> | ||
+ | |||
+ | Putting it together, we get the answer to be <math>4\cdot\left( \frac{\pi}{18}-\frac{\sqrt{3}}{12} \right)= \boxed{\textbf{(B)}\ \frac{2\pi}{9}-\frac{\sqrt{3}}{3}}</math> | ||
+ | |||
+ | ==Solution 2 (Answer Choices)== | ||
+ | |||
+ | At once, we can eliminate <math>(C)</math>, <math>(D)</math>, and <math>(E)</math> because the answer should be <math>a \times \pi + b</math>, where <math>a</math> and <math>b</math> are real numbers (not necessarily rational) not equal to <math>0</math>. Now, all that's left is <math>\frac{\pi}{3}-1</math> and <math>\frac{2 \pi}{9} - \frac{\sqrt{3}}{3}</math>. <math>\frac{\pi}{3}-1</math> is simply the area of the circle minus the square, which is definitely wrong, so the answer is <math>\boxed{\textbf{(B)} \frac{2 \pi}{9} - \frac{\sqrt{3}}{3}}</math> | ||
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/FQO-0E2zUVI | ||
+ | |||
+ | ~IceMatrix | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2010|ab=B|num-b=15|num-a=17}} | ||
+ | {{MAA Notice}} |
Latest revision as of 18:52, 27 January 2021
Problem
A square of side length and a circle of radius share the same center. What is the area inside the circle, but outside the square?
Solution 1
The radius of the circle is . Half the diagonal of the square is . We can see that the circle passes outside the square, but the square is NOT completely contained in the circle. Therefore the picture will look something like this:
Then we proceed to find: 4 (area of sector marked off by the two radii - area of the triangle with sides on the square and the two radii).
First we realize that the radius perpendicular to the side of the square between the two radii marking off the sector splits in half. Let this half-length be . Also note that because it is half the sidelength of the square. Because this is a right triangle, we can use the Pythagorean Theorem to solve for
Solving, and . Since , is an equilateral triangle and the central angle is . Therefore the sector has an area .
Now we turn to the triangle. Since it is equilateral, we can use the formula for the area of an equilateral triangle which is
Putting it together, we get the answer to be
Solution 2 (Answer Choices)
At once, we can eliminate , , and because the answer should be , where and are real numbers (not necessarily rational) not equal to . Now, all that's left is and . is simply the area of the circle minus the square, which is definitely wrong, so the answer is
Video Solution
~IceMatrix
See Also
2010 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 15 |
Followed by Problem 17 | |
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 AMC 10 Problems and Solutions |
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