Question

Let $\mathcal{R}$ be the region in the complex plane consisting of all complex numbers $z$ that can be written as the sum of complex numbers $z_1$ and $z_2,$ where $z_1$ lies on the segment with endpoints $3$ and $4i,$ and $z_2$ has magnitude at most $1.$ What integer is closest to the area of $\mathcal{R}?$

$13$

$14$

$15$

$16$

$17$

Solution:

Adding a disk of radius $1$ to every point of the segment sweeps out all points within distance $1$ of it. The segment from $3$ to $4i$ has length $\sqrt{3^2+4^2}=5.$

This "stadium" is a $5\times2$ rectangle plus two half-disks of radius $1,$ with area $5\cdot2+\pi(1)^2=10+\pi\approx13.14.$

The closest integer is $13.$

Thus, the correct answer is A.

2022 AMC 12A Problem 13

Solution:

Problem 13 in Other Years