Difference between revisions of "2021 AMC 10A Problems/Problem 19"
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+ | == Video Solution by The Power Of Logic (Graphing) == | ||
+ | https://youtu.be/-pa72wBA85Y | ||
==Video Solution by TheBeautyofMath== | ==Video Solution by TheBeautyofMath== |
Revision as of 00:37, 21 February 2021
Contents
Problem
The area of the region bounded by the graph ofis
, where
and
are integers. What is
?
Solution 1
In order to attack this problem, we need to consider casework:
Case 1:
Substituting and simplifying, we have , i.e.
, which gives us a circle of radius
centered at
.
Case 2:
Substituting and simplifying again, we have , i.e.
. This gives us a circle of radius
centered at
.
Case 3:
Doing the same process as before, we have , i.e.
. This gives us a circle of radius
centered at
.
Case 4:
One last time: we have , i.e.
. This gives us a circle of radius
centered at
.
After combining all the cases and drawing them on the Cartesian Plane, this is what the diagram looks like:
Now, the area of the shaded region is just a square with side length
with four semicircles of radius
.
The area is
. The answer is
which is
Solution by Bryguy
Diagram anonymous (?): https://artofproblemsolving.com/wiki/index.php/File:Image_2021-02-11_111327.png (someone please help link file thanks)
Video Solution (Using absolute value properties to graph)
~ pi_is_3.14
Video Solution by The Power Of Logic (Graphing)
Video Solution by TheBeautyofMath
~IceMatrix
See Also
2021 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 18 |
Followed by Problem 20 | |
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.