2016 AMC 12B Problems/Problem 18
Problem
What is the area of the region enclosed by the graph of the equation
Solution
Consider the case when , . Notice the circle intersect the axes at points and . Find the area of this circle in the first quadrant. The area is made of a semi-circle with radius of and a triangle:
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);draw((arc(1/2,1/2),sqrt(2)/2,-45,135). dotted); for(int i=0;i<4;++i){draw(rotate(i*90,(0,0))*arc((1/2,1/2),sqrt(2)/2,-45,135));dot(rotate(i*90,(0,0))*(1/2,1/2));} (Error making remote request. Unknown error_msg)
Because of symmetry, the area is the same in all four quadrants. The answer is
See Also
2016 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 17 |
Followed by Problem 19 |
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All AMC 12 Problems and Solutions |
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