Difference between revisions of "2016 AMC 10B Problems/Problem 21"
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<math>\textbf{(A)}\ \pi+\sqrt{2}\qquad\textbf{(B)}\ \pi+2\qquad\textbf{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}</math> | <math>\textbf{(A)}\ \pi+\sqrt{2}\qquad\textbf{(B)}\ \pi+2\qquad\textbf{(C)}\ \pi+2\sqrt{2}\qquad\textbf{(D)}\ 2\pi+\sqrt{2}\qquad\textbf{(E)}\ 2\pi+2\sqrt{2}</math> | ||
+ | |||
+ | ==Solution== | ||
+ | WLOG note that if a point in the first quadrant satisfies the equation, so do its corresponding points in the other three quadrants. Therefore we can assume that <math>x, y \ge 0</math> and multiply by <math>4</math> at the end. | ||
+ | |||
+ | We can rearrange the equation to get <math>x^2-x+y^2-y=0 \Rightarrow (x-\tfrac12)^2+(y-\tfrac12)^2=(\tfrac{\sqrt2}{2})^2</math>, which describes a circle with center <math>(\tfrac12, \tfrac12)</math> and radius <math>\tfrac{\sqrt2}{2}.</math> It's clear we now want to find the union of four circles with overlap. | ||
+ | |||
+ | <asy>draw((0,-1.5)--(0,1.5),EndArrow);draw((-1.5,0)--(1.5,0),EndArrow);draw((0,1)--(1,0)--(0,-1)--(-1,0)--cycle,dotted); | ||
+ | for(int i=0;i<4;++i){draw(rotate(i*90,(0,0))*arc((1/2,1/2),sqrt(1/2),-45,135));dot(rotate(i*90,(0,0))*(1/2,1/2));}</asy> | ||
+ | There are several ways to find the area, but note that if you connect <math>(0, 1)</math> to its other three respective points in the other three quadrants, you get a square of area <math>2</math>, along with four half-circles of diameter <math>\sqrt{2}</math>, for a total area of <math>2+2\cdot(\tfrac{\sqrt2}{2})^2\pi = \pi + 2</math> which is <math>\textbf{(B)}</math>. | ||
+ | |||
+ | ==See Also== | ||
+ | {{AMC10 box|year=2016|ab=B|num-b=20|num-a=22}} | ||
+ | {{MAA Notice}} |
Revision as of 11:53, 21 February 2016
Problem
What is the area of the region enclosed by the graph of the equation
Solution
WLOG note that if a point in the first quadrant satisfies the equation, so do its corresponding points in the other three quadrants. Therefore we can assume that and multiply by at the end.
We can rearrange the equation to get , which describes a circle with center and radius It's clear we now want to find the union of four circles with overlap.
There are several ways to find the area, but note that if you connect to its other three respective points in the other three quadrants, you get a square of area , along with four half-circles of diameter , for a total area of which is .
See Also
2016 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 20 |
Followed by Problem 22 | |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.