Difference between revisions of "2015 AMC 10B Problems/Problem 12"
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==Problem== | ==Problem== | ||
− | For how many integers <math>x</math> is the point <math>(x, -x)</math> inside or on the circle of radius 10 centered at <math>(5, 5)</math>? | + | For how many integers <math>x</math> is the point <math>(x, -x)</math> inside or on the circle of radius <math>10</math> centered at <math>(5, 5)</math>? |
− | <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> |
+ | |||
+ | ==Video Solution== | ||
+ | https://youtu.be/RZDFs3qrw7Y | ||
+ | |||
+ | ~Education, the Study of Everything= | ||
==Solution== | ==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 | ||
− | <math>x\leq 5</math> | + | <math>x\leq 5</math> and <math>x\geq -5</math>. So <math>x</math> ranges from <math>-5</math> to <math>5</math>, for a total of <math>\boxed{\mathbf{(A)}\ 11}</math> integer values. |
+ | |||
+ | Note by Williamgolly: | ||
+ | Alternatively, draw out the circle and see that these points must be on the line <math>y=-x</math>. | ||
==See Also== | ==See Also== | ||
{{AMC10 box|year=2015|ab=B|num-b=11|num-a=13}} | {{AMC10 box|year=2015|ab=B|num-b=11|num-a=13}} | ||
{{MAA Notice}} | {{MAA Notice}} |
Latest revision as of 13:20, 16 July 2024
Contents
Problem
For how many integers is the point
inside or on the circle of radius
centered at
?
Video Solution
~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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.