Difference between revisions of "2008 AMC 10A Problems/Problem 17"
(→Problem) |
|||
Line 2: | Line 2: | ||
An [[equilateral triangle]] has side length <math>6</math>. What is the [[area]] of the region containing all points that are outside the triangle but not more than <math>3</math> units from a point of the triangle? | An [[equilateral triangle]] has side length <math>6</math>. What is the [[area]] of the region containing all points that are outside the triangle but not more than <math>3</math> units from a point of the triangle? | ||
− | <math>\mathrm{(A)}\ 36+24\sqrt{3}\qquad\mathrm{(B)}\ 54+9\pi\qquad\mathrm{(C)}\ 54+18\sqrt{3}+6\pi\qquad\mathrm{(D)}\ \left(2\sqrt{3}+3\right)^2\pi\\mathrm{(E)}\ 9\left(\sqrt{3}{+1\right)^2\pi</math> | + | <math>\mathrm{(A)}\ 36+24\sqrt{3}\qquad\mathrm{(B)}\ 54+9\pi\qquad\mathrm{(C)}\ 54+18\sqrt{3}+6\pi\qquad\mathrm{(D)}\ \left(2\sqrt{3}+3\right)^2\pi\\mathrm{(E)}\ 9\left(\sqrt{3}{+1\right})^2\pi</math> |
==Solution== | ==Solution== |
Revision as of 11:03, 21 January 2016
Problem
An equilateral triangle has side length . What is the area of the region containing all points that are outside the triangle but not more than units from a point of the triangle?
$\mathrm{(A)}\ 36+24\sqrt{3}\qquad\mathrm{(B)}\ 54+9\pi\qquad\mathrm{(C)}\ 54+18\sqrt{3}+6\pi\qquad\mathrm{(D)}\ \left(2\sqrt{3}+3\right)^2\pi\\mathrm{(E)}\ 9\left(\sqrt{3}{+1\right})^2\pi$ (Error compiling LaTeX. Unknown error_msg)
Solution
The region described contains three rectangles of dimensions , and three degree arcs of circles of radius . Thus the answer is
See also
2008 AMC 10A (Problems • Answer Key • Resources) | ||
Preceded by Problem 16 |
Followed by Problem 18 | |
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.