Difference between revisions of "2007 AMC 10B Problems/Problem 13"
Scrabbler94 (talk | contribs) (undo previous edits) |
|||
(5 intermediate revisions by 2 users not shown) | |||
Line 7: | Line 7: | ||
==Solution== | ==Solution== | ||
− | You can find the area of half the intersection by subtracting the isosceles triangle in the sector from the whole sector. This sector is | + | You can find the area of half the intersection by subtracting the isosceles triangle in the sector from the whole sector. This sector is one-fourth of the area of the circle with radius <math>2,</math> and the isosceles triangle is a right triangle. Therefore, the area of half the intersection is |
<cmath>\frac{1}{4} 4\pi - \frac{1}{2}(2)(2) = \pi - 2</cmath> | <cmath>\frac{1}{4} 4\pi - \frac{1}{2}(2)(2) = \pi - 2</cmath> | ||
That means the area of the whole intersection is <math>\boxed{\mathrm{(D) \ } 2(\pi-2)}</math> | That means the area of the whole intersection is <math>\boxed{\mathrm{(D) \ } 2(\pi-2)}</math> |
Latest revision as of 20:42, 15 February 2024
Problem
Two circles of radius are centered at
and at
What is the area of the intersection of the interiors of the two circles?
Solution
You can find the area of half the intersection by subtracting the isosceles triangle in the sector from the whole sector. This sector is one-fourth of the area of the circle with radius and the isosceles triangle is a right triangle. Therefore, the area of half the intersection is
That means the area of the whole intersection is
See Also
2007 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 12 |
Followed by Problem 14 | |
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.