Question

飞镖盘是坐标平面中的区域 $B$ ，由满足 $|x| + |y| \le 8$ 的点 $(x, y)$ 组成。目标区域 $T$ 满足 $(x^2 + y^2 - 25)^2 \le 49$ 。一支飞镖随机落在 $B$ 中。它落在 $T$ 中的概率可表示为 $\dfrac{m}{n} \cdot \pi$ ，其中 $m$ 和 $n$ 是互质的正整数。求 $m + n$ 。

A dartboard is the region $B$ in the coordinate plane consisting of points $(x, y)$ such that $|x| + |y| \le 8.$ A target $T$ is the region where $(x^2 + y^2 - 25)^2 \le 49.$ A dart is thrown and lands at a random point in $B.$ The probability that the dart lands in $T$ can be expressed as $\dfrac{m}{n} \cdot \pi,$ where $m$ and $n$ are relatively prime positive integers. What is $m + n?$

$39$

$71$

$73$

$75$

$135$

答案：B

知识点：几何概率圆环切线

难度评级：1660

解答：

$B$ 是正方形 $|x| + |y| \le 8$ ，面积为 $2 \cdot 8^2 = 128$ 。目标条件 $(x^2 + y^2 - 25)^2 \le 49$ 等价于 $|x^2 + y^2 - 25| \le 7$ ，也就是 $18 \le x^2 + y^2 \le 32$ ，这是面积为 $\pi(32 - 18) = 14\pi$ 的圆环。它是否在 $B$ 中？原点到边 $x + y = 8$ 的距离是 $\tfrac{8}{\sqrt2} = 4\sqrt2 = \sqrt{32}$ ，正好是外半径，所以圆环在正方形内。概率为 $\tfrac{14\pi}{128} = \tfrac{7}{64}\pi$ ，因此 $m + n = 71$ 。正确答案是 B。

$B$ is the square $|x| + |y| \le 8,$ with area $2 \cdot 8^2 = 128.$ The target condition $(x^2 + y^2 - 25)^2 \le 49$ unpacks to $|x^2 + y^2 - 25| \le 7,$ that is $18 \le x^2 + y^2 \le 32,$ an annulus of area $\pi(32 - 18) = 14\pi.$ Does it fit inside $B?$ The distance from the origin to an edge $x + y = 8$ is $\tfrac{8}{\sqrt2} = 4\sqrt2 = \sqrt{32},$ exactly the outer radius, so yes, the annulus sits inside the square. The probability is $\tfrac{14\pi}{128} = \tfrac{7}{64}\pi,$ giving $m + n = 71.$ Therefore, the answer is B.

2024 AMC 10B 第 14 题

答案：B

难度评级：1660

解答：

其他年份的第 14 题