Difference between revisions of "2024 AMC 10B Problems/Problem 14"
Numerophile (talk | contribs) (→Solution 2) |
Elephant200 (talk | contribs) m (→Diagram) |
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// By Elephant200 | // By Elephant200 | ||
// Feel free to adjust the code | // Feel free to adjust the code | ||
+ | |||
size(10cm); | size(10cm); | ||
Line 22: | Line 23: | ||
pair C = (-8, 0); | pair C = (-8, 0); | ||
pair D = (0, -8); | pair D = (0, -8); | ||
− | draw(A--B--C--D--cycle); | + | draw(A--B--C--D--cycle, linewidth(1.5)); |
label("$(8,0)$", A, NE); | label("$(8,0)$", A, NE); | ||
label("$(0,8)$", B, NE); | label("$(0,8)$", B, NE); | ||
− | label("$(-8,0)$", C, | + | label("$(-8,0)$", C, NW); |
− | label("$(0,-8)$", D, | + | label("$(0,-8)$", D, SE); |
− | filldraw(circle((0,0),4*sqrt(2)), gray); | + | filldraw(circle((0,0),4*sqrt(2)), gray, linewidth(1.5)); |
− | filldraw(circle((0,0),3*sqrt(2)), white); | + | filldraw(circle((0,0),3*sqrt(2)), white, linewidth(1.5)); |
− | draw((- | + | draw((-10, 0)--(10,0),EndArrow(5)); |
− | draw(( | + | draw((10, 0)--(-10,0),EndArrow(5)); |
− | draw((0,- | + | draw((0,-10)--(0,10), EndArrow(5)); |
− | draw((0, | + | draw((0,10)--(0,-10),EndArrow(5)); |
</asy> | </asy> | ||
~Elephant200 | ~Elephant200 |
Revision as of 15:27, 14 November 2024
- The following problem is from both the 2024 AMC 10B #14 and 2024 AMC 12B #9, so both problems redirect to this page.
Contents
[hide]Problem
A dartboard is the region B in the coordinate plane consisting of points such that . A target T is the region where . A dart is thrown at a random point in B. The probability that the dart lands in T can be expressed as , where and are relatively prime positive integers. What is ?
Diagram
~Elephant200
Solution 1
Inequalities of the form are well-known and correspond to a square in space with centre at origin and vertices at , , , . The diagonal length of this square is clearly , so it has an area of Now, Converting to polar form, and
The union of these inequalities is the circular region for which every circle in has a radius between and , inclusive. The area of such a region is thus The requested probability is therefore yielding We have
-anonymous, countmath1
Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)
https://youtu.be/YqKmvSR1Ckk?feature=shared
~ Pi Academy
Video Solution 2 by SpreadTheMathLove
https://www.youtube.com/watch?v=24EZaeAThuE
Solution 2
~Kathan
See also
2024 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 13 |
Followed by Problem 15 | |
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 |
2024 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 8 |
Followed by Problem 10 |
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 |
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.