Question

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 $m + n?$

$39$

$71$

$73$

$75$

$135$

Solution:

The dartboard $|x| + |y| \le 8$ is a square with diagonals $16,$ so its area is $\tfrac12 \cdot 16 \cdot 16 = 128.$ The target condition $(x^2 + y^2 - 25)^2 \le 49$ means $-7 \le x^2 + y^2 - 25 \le 7,$ i.e. $18 \le x^2 + y^2 \le 32,$ an annulus of area $\pi(32 - 18) = 14\pi.$

The distance from the origin to a side of the square (for instance $x + y = 8$ ) is $\dfrac{8}{\sqrt2} = \sqrt{32},$ exactly the annulus's outer radius. So the annulus is tangent to the square and lies entirely within $B.$ The probability is $\dfrac{14\pi}{128} = \dfrac{7}{64}\pi,$ giving $m + n = 7 + 64 = 71.$

Thus, the correct answer is B.

2024 AMC 12B Problem 9

Solution:

Problem 9 in Other Years