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

Revision as of 15:44, 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

$[asy] size(250); import TrigMacros; rr_cartesian_axes(-2,6,-2,6,complexplane=true, usegrid = true); Label f; f.p=fontsize(6); xaxis(-1,5,Ticks(f, 1.0)); yaxis(-1,5,Ticks(f, 1.0)); dot((3,0)); dot((0,4)); draw((0,4)--(3,0), blue); draw((0.8, 4.6)..(-.6,4.8)..(-.8, 3.4),red); draw((-.8, 3.4)--(2.2, -0.6), red); draw((2.2, -0.6)..(3.6,-0.8)..(3.8,0.6), red); draw((0.8, 4.6)--(3.8,0.6),red); draw((0.8, 4.6)--(-.8, 3.4),red+dashed); draw((2.2, -0.6)--(3.8,0.6),red+ dashed); draw((3,0)--(3,-1),Arrow); label("1",(3,0)--(3,-1),E); draw((0,4)--(-.6,4.8),Arrow); label("1",(0,4)--(-.6,4.8),SW); draw((1.5,2)--(2.3,2.6),Arrow); label("1",(1.5,2)--(2.3,2.6),SE); [/asy]$

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 $\pi \approx 3$ . Therefore, the total area is $5(2) + \pi \approx 10 + 3 = 13$ (A)

~juicefruit

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

Revision as of 15:44, 12 November 2022

Problem

Solution

@@ 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>
+==Solution==
+<asy>
+size(250);
+import TrigMacros;
+rr_cartesian_axes(-2,6,-2,6,complexplane=true, usegrid = true);
+Label f;
+f.p=fontsize(6);
+xaxis(-1,5,Ticks(f, 1.0));
+yaxis(-1,5,Ticks(f, 1.0));
+dot((3,0));
+dot((0,4));
+draw((0,4)--(3,0), blue);
+draw((0.8, 4.6)..(-.6,4.8)..(-.8, 3.4),red);
+draw((-.8, 3.4)--(2.2, -0.6), red);
+draw((2.2, -0.6)..(3.6,-0.8)..(3.8,0.6), red);
+draw((0.8, 4.6)--(3.8,0.6),red);
+draw((0.8, 4.6)--(-.8, 3.4),red+dashed);
+draw((2.2, -0.6)--(3.8,0.6),red+ dashed);
+draw((3,0)--(3,-1),Arrow);
+label("1",(3,0)--(3,-1),E);
+draw((0,4)--(-.6,4.8),Arrow);
+label("1",(0,4)--(-.6,4.8),SW);
+draw((1.5,2)--(2.3,2.6),Arrow);
+label("1",(1.5,2)--(2.3,2.6),SE);
+</asy>
+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>.
+Therefore, the total area is <math>5(2) + \pi \approx 10 + 3 = 13</math> (A)
+~juicefruit