Difference between revisions of "2015 AMC 10B Problems/Problem 12"
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<math>\textbf{(A) }11\qquad \textbf{(B) }12\qquad \textbf{(C) }13\qquad \textbf{(D) }14\qquad \textbf{(E) }15</math> | <math>\textbf{(A) }11\qquad \textbf{(B) }12\qquad \textbf{(C) }13\qquad \textbf{(D) }14\qquad \textbf{(E) }15</math> | ||
− | ==Solution== | + | ==Video Solution 1== |
+ | https://youtu.be/RZDFs3qrw7Y | ||
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+ | ~Education, the Study of Everything= | ||
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+ | =Solution== | ||
The equation of the circle is <math>(x-5)^2+(y-5)^2=100</math>. Plugging in the given conditions we have <math>(x-5)^2+(-x-5)^2 \leq 100</math>. Expanding gives: <math>x^2-10x+25+x^2+10x+25\leq 100</math>, which simplifies to | The equation of the circle is <math>(x-5)^2+(y-5)^2=100</math>. Plugging in the given conditions we have <math>(x-5)^2+(-x-5)^2 \leq 100</math>. Expanding gives: <math>x^2-10x+25+x^2+10x+25\leq 100</math>, which simplifies to | ||
<math>x^2\leq 25</math> and therefore | <math>x^2\leq 25</math> and therefore |
Revision as of 16:13, 2 August 2022
Problem
For how many integers is the point inside or on the circle of radius centered at ?
Video Solution 1
~Education, the Study of Everything=
Solution=
The equation of the circle is . Plugging in the given conditions we have . Expanding gives: , which simplifies to and therefore and . So ranges from to , for a total of integer values.
Note by Williamgolly: Alternatively, draw out the circle and see that these points must be on the line .
See Also
2015 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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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