Art of Problem Solving
AoPS Online
Math texts, online classes, and more
for students in grades 5-12.
Visit AoPS Online ‚
Books for Grades 5-12 Online Courses
Beast Academy
Engaging math books and online learning
for students ages 6-13.
Visit Beast Academy ‚
Books for Ages 6-13 Beast Academy Online
AoPS Academy
Small live classes for advanced math
and language arts learners in grades 2-12.
Visit AoPS Academy ‚
Find a Physical Campus Visit the Virtual Campus

Difference between revisions of "2022 AMC 12A Problems/Problem 13"

(Created page with "==Problem== Let <math>\mathcal{R}</math> be the region in the complex plane consisting of all complex numbers <math>z</math> that can be written as the sum of complex numbers...")
 
Line 4: Line 4:
  
 
<math>\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17</math>
 
<math>\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17</math>
 +
 +
==Solution==
 +
 +
If <math>z</math> is a complex number and <math>z = a + bi</math>, then the magnitude (length) of <math>z</math> is <math>\sqrt{a^2 + b^2}</math>. Therefore, <math>z_1</math> has a magnitude of 5. If <math>z_2</math> has a magnitude of at most one, that means for each point on the segment given by <math>z_1</math>, the bounds of the region <math>\mathcal{R}</math> could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of <math>\pi \approx 3</math>.
 +
<math>\\$ The total area is, therefore, </math>5(2) + \pi \approx 10 + 3 = 13$ (A)

Revision as of 15:40, 12 November 2022

Problem

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 $\mathcal{R}$?

$\textbf{(A) } 13 \qquad \textbf{(B) } 14 \qquad \textbf{(C) } 15 \qquad \textbf{(D) } 16 \qquad \textbf{(E) } 17$

Solution

If $z$ is a complex number and $z = a + bi$, then the magnitude (length) of $z$ is $\sqrt{a^2 + b^2}$. Therefore, $z_1$ has a magnitude of 5. If $z_2$ has a magnitude of at most one, that means for each point on the segment given by $z_1$, the bounds of the region $\mathcal{R}$ could be at most 1 away. Alone the line, excluding the endpoints, a rectangle with a width of 2 and a length of 5, the magnitude, would be formed. At the endpoints, two semicircles will be formed with a radius of 1 for a total area of $\pi \approx 3$. $\$ The total area is, therefore,$ (Error compiling LaTeX. Unknown error_msg)5(2) + \pi \approx 10 + 3 = 13$ (A)

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2022_AMC_12A_Problems/Problem_13&oldid=181157"