Difference between revisions of "2024 AMC 10B Problems/Problem 14"
Abhisood1234 (talk | contribs) (→Simple Coordinate Geometry) |
|||
| Line 3: | Line 3: | ||
==Problem== | ==Problem== | ||
| − | == | + | ==Simple Coordinate Geometry== |
| + | <cmath>|x|+|y| \le 8</cmath> | ||
| + | Inequalities of this form are well-known and correspond to a square in space with centre at origin and vertices at <math>(8, 0)</math>, <math>(-8, 0)</math>, <math>(0, 8)</math>, <math>(0, -8)</math>. | ||
| + | The diagonal length of this square is clearly <math>16</math>, so it has an area of | ||
| + | <cmath>\frac{1}{2} \cdot 16 \cdot 16 = 128</cmath> | ||
| + | Now, | ||
| + | <cmath>(x^2 + y^2 - 25)^2 \le 49</cmath> | ||
| + | Converting to polar form, | ||
| + | <cmath>r^2 - 25 \le 7</cmath> | ||
| + | <cmath>r \le \sqrt32</cmath> | ||
| + | And | ||
| + | <cmath>r^2 - 25 \ge -7</cmath> | ||
| + | <cmath>r \ge \sqrt18</cmath> | ||
| + | |||
| + | This corresponds to a ring in space with outer radius <math>\sqrt32</math> and inner radius <math>\sqrt18</math>. | ||
| + | Note that the outer circle is inscribed within the square, meaning it completely lies within the square. | ||
| + | |||
| + | Our probability, then, is <cmath>\frac {\pi(32 - 18)}{128}</cmath> | ||
| + | <cmath>= \frac{7\pi}{64}</cmath> | ||
| + | <math>m = 7</math> and <math>n = 64</math> | ||
| + | <cmath>m + n = 71</cmath> | ||
| + | So <math>\boxed{\textbf{(B) }71}</math> | ||
==See also== | ==See also== | ||
Revision as of 01:50, 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.
Problem
Simple Coordinate Geometry
![\[|x|+|y| \le 8\]](http://latex.artofproblemsolving.com/b/7/4/b741da773c0a2ef60cdab17e06fe48bf5bfcb015.png)
Inequalities of this 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
![\[\frac{1}{2} \cdot 16 \cdot 16 = 128\]](http://latex.artofproblemsolving.com/5/a/6/5a66fdc982311cb313225cf592a68b0cc98e45f3.png)
Now,
![\[(x^2 + y^2 - 25)^2 \le 49\]](http://latex.artofproblemsolving.com/0/a/5/0a51bc9364a32d91a5f2fc59c5d48fd738cfd599.png)
Converting to polar form,
![\[r^2 - 25 \le 7\]](http://latex.artofproblemsolving.com/3/b/9/3b91cca2301fb0aedef897e40e9629e61ce56793.png)
![\[r \le \sqrt32\]](http://latex.artofproblemsolving.com/6/0/4/604bd659dd4deb8abbde8ab226116397f6f7c2ab.png)
And
![\[r^2 - 25 \ge -7\]](http://latex.artofproblemsolving.com/9/3/e/93ef67e4111cc02f9f3359ac9c8b7a30dc0b6f32.png)
![\[r \ge \sqrt18\]](http://latex.artofproblemsolving.com/2/e/e/2ee76229284f6265f790d8025eb98aae2aa8d5d8.png)
This corresponds to a ring in space with outer radius 
and inner radius 
.
Note that the outer circle is inscribed within the square, meaning it completely lies within the square.
Our probability, then, is ![\[\frac {\pi(32 - 18)}{128}\]](http://latex.artofproblemsolving.com/4/d/d/4dd17c0c0443f3e9bfffb61b71024776ad4b9956.png)
![\[= \frac{7\pi}{64}\]](http://latex.artofproblemsolving.com/8/c/c/8cc1ae0e0723cadb3aae8e9a0ff8cebf1fd3d822.png)
and 
![\[m + n = 71\]](http://latex.artofproblemsolving.com/d/f/7/df72dee09c2097d0e0eb6d98a8740c067caedeae.png)
So 
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. 
