Difference between revisions of "2024 AMC 10B Problems/Problem 14"
Countmath1 (talk | contribs) m (→Solution 1) |
Megaboy6679 (talk | contribs) m |
||
Line 113: | Line 113: | ||
~[https://artofproblemsolving.com/wiki/index.php/User:Athmyx Athmyx] | ~[https://artofproblemsolving.com/wiki/index.php/User:Athmyx Athmyx] | ||
+ | |||
+ | ==Solution 3== | ||
+ | [[Image: 2024_AMC_12B_P09.jpeg|thumb|center|600px|]] | ||
+ | ~Kathan | ||
==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)== | ==Video Solution 1 by Pi Academy (Fast and Easy ⚡🚀)== | ||
Line 123: | Line 127: | ||
https://www.youtube.com/watch?v=24EZaeAThuE | https://www.youtube.com/watch?v=24EZaeAThuE | ||
− | |||
− | |||
− | |||
==See also== | ==See also== |
Revision as of 18:49, 15 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
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 and lands 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 intersection 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
Solution 2 (Calculus)
Expressing the Area of Region \( B \)
Region \( B \) consists of points where \( |x| + |y| \le 8 \)
In each quadrant, this can be expressed by the following functions:
First quadrant: \( y = 8 - x \) Second quadrant: \( y = 8 + x \) Third quadrant: \( y = -8 - x \) Fourth quadrant: \( y = -8 + x \)
In the first quadrant, \( x \) ranges from 0 to 8, and \( y \) ranges from 0 to \( 8 - x \). Thus, the area in the first quadrant is: The total area of region \( B \) is:
Expressing the Area of Region \( T \) Region \( T \) is defined by the inequality \( (x^2 + y^2 - 25)^2 \le 49 \), which can be rewritten as:
To find the area, we switch to polar coordinates with \( x = r \cos \theta \) and \( y = r \sin \theta \), where \( x^2 + y^2 = r^2 \). Here, \( r \) ranges from \( \sqrt{18} \) to \( \sqrt{32} \), and \( \theta \) ranges from 0 to \( 2\pi \).
The area of \( T \) can then be found by:
The probability \( P \) that a dart lands in region \( T \) is the area of \( T \) divided by the area of \( B \):
So the probability is of the form \( \frac{m}{n} \pi \), where \( m = 7 \) and \( n = 64 \), so \( m + n = 7 + 64 = 71 \).
Solution 3
~Kathan
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
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.