Difference between revisions of "2004 AMC 12B Problems/Problem 10"
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From the [[Pythagorean Theorem]] for the right triangle <math>OXZ</math> we have <math>a^2 + c^2 = b^2</math>, hence <math>b^2-c^2=a^2</math> and thus the shaded area is <math>\boxed{\mathrm{(A)\ }\pi a^2}</math>. | From the [[Pythagorean Theorem]] for the right triangle <math>OXZ</math> we have <math>a^2 + c^2 = b^2</math>, hence <math>b^2-c^2=a^2</math> and thus the shaded area is <math>\boxed{\mathrm{(A)\ }\pi a^2}</math>. | ||
+ | |||
+ | ==Solution 2== | ||
+ | Set <math>c=0,</math> then the area is just <math>\boxed{\pi a^2}.</math> | ||
== See also == | == See also == |
Revision as of 13:34, 23 April 2023
- The following problem is from both the 2004 AMC 12B #10 and 2004 AMC 10B #12, so both problems redirect to this page.
Contents
Problem
An annulus is the region between two concentric circles. The concentric circles in the figure have radii and , with . Let be a radius of the larger circle, let be tangent to the smaller circle at , and let be the radius of the larger circle that contains . Let , , and . What is the area of the annulus?
Solution
The area of the large circle is , the area of the small one is , hence the shaded area is .
From the Pythagorean Theorem for the right triangle we have , hence and thus the shaded area is .
Solution 2
Set then the area is just
See also
2004 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 9 |
Followed by Problem 11 |
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 12 Problems and Solutions |
2004 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.