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Difference between revisions of "2020 AMC 12A Problems/Problem 15"

m (→‎Solution)
(Described the detailed steps to find the roots of both equations. lopkiloinm should be the primary author and I am secondary. I will use Asymptote for diagram later.)
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<math>\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9+\sqrt{3}</math>
 
<math>\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9+\sqrt{3}</math>
  
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== Solution ==
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== Solution 1 ==
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Realize that <math>z^{3}-8=0</math> will create an equilateral triangle on the complex plane with the first point at <math>2+0i</math> and two other points with equal magnitude at <math>-1{\pm}i\sqrt{3}</math>.  
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We solve each equation separately:
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<ol style="margin-left: 1.5em;">
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  <li>We solve <math>z^{3}-8=0</math> by De Moivre's Theorem.<p>
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Let <math>z=r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta,</math> where <math>r</math> is the magnitude of <math>z</math> such that <math>r\geq0,</math> and <math>\theta</math> is the argument of <math>z</math> such that <math>0\leq\theta<2\pi.</math> <p>
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We have <cmath>z^3=r^3\operatorname{cis}(3\theta)=8(1+0i),</cmath> from which
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<ul style="list-style-type:square;">
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  <li><math>r^3=8,</math> so <math>r=2.</math></li><p>
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  <li><math>\begin{cases}
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\begin{aligned}
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\cos(3\theta) &= 1 \\
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\sin(3\theta) &= 0
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\end{aligned},
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\end{cases}</math> so <math>3\theta=0,2\pi,4\pi,</math> or <math>\theta=0,\frac{2\pi}{3},\frac{4\pi}{3}.</math> </li><p>
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</ul>
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The set of solutions to <math>z^{3}-8=0</math> is <math>\boldsymbol{A=\left\{2,-1+\sqrt{3}i,-1-\sqrt{3}i\right\}}.</math> In the complex plane, they are the vertices of an equilateral triangle whose circumcircle has center <math>0</math> and radius <math>2.</math></li><p>
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  <li>We solve <math>z^{3}-8z^{2}-8z+64=0</math> by factoring by grouping.</li><p>
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We have
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<cmath>\begin{align*}
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z^2(z-8)-8(z-8)&=0 \\
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\bigl(z^2-8\bigr)\bigl(z-8\bigr)&=0.
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\end{align*}</cmath>
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The set of solutions to <math>z^{3}-8z^{2}-8z+64=0</math> is <math>\boldsymbol{B=\left\{2\sqrt{2}i,-2\sqrt{2}i,8\right\}}.</math>
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</ol>
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In the graph below, the greatest distance between a point of <math>A</math> and a point of <math>B</math> is the distance between <math>-1\pm\sqrt{3}i</math> to <math>8,</math> as shown in the dashed line segments. By the Distance Formula, the answer is <cmath>\sqrt{(8-(-1))^2+\left(\pm\sqrt{3}-0\right)^2}=\sqrt{84}=\boxed{\textbf{(D) } 2\sqrt{21}}.</cmath>
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<b>DIAGRAM NEEDED.</b>
  
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Also, realize that <math>z^{3}-8z^{2}-8z+64</math> can be factored through grouping: <math>z^{3}-8z^{2}-8z+64=(z-8)(z^{2}-8).</math> <math>(z-8)(z^{2}-8)</math> will create points at <math>8+0i</math> and <math>\pm2\sqrt{2}+0i.</math>
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~lopkiloinm ~MRENTHUSIASM
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Plotting the points and looking at the graph will make you realize that <math>-1{\pm}i\sqrt{3}</math> and <math>8+0i</math> are the farthest apart and through Pythagorean Theorem, the answer is revealed to be <math>\sqrt{\sqrt{3}^{2}+(8-(-1))^{2}}=\sqrt{84}=\boxed{\textbf{(D) } 2\sqrt{21}.}</math>
 
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~lopkiloinm
 
  
 
==Remark==
 
==Remark==

Revision as of 04:21, 11 September 2021

Problem

In the complex plane, let $A$ be the set of solutions to $z^{3}-8=0$ and let $B$ be the set of solutions to $z^{3}-8z^{2}-8z+64=0.$ What is the greatest distance between a point of $A$ and a point of $B?$

$\textbf{(A) } 2\sqrt{3} \qquad \textbf{(B) } 6 \qquad \textbf{(C) } 9 \qquad \textbf{(D) } 2\sqrt{21} \qquad \textbf{(E) } 9+\sqrt{3}$

Solution 1

We solve each equation separately:

  1. We solve $z^{3}-8=0$ by De Moivre's Theorem.

    Let $z=r(\cos\theta+i\sin\theta)=r\operatorname{cis}\theta,$ where $r$ is the magnitude of $z$ such that $r\geq0,$ and $\theta$ is the argument of $z$ such that $0\leq\theta<2\pi.$

    We have \[z^3=r^3\operatorname{cis}(3\theta)=8(1+0i),\] from which

    • $r^3=8,$ so $r=2.$

    • $\begin{cases} \begin{aligned} \cos(3\theta) &= 1 \\ \sin(3\theta) &= 0 \end{aligned}, \end{cases}$ so $3\theta=0,2\pi,4\pi,$ or $\theta=0,\frac{2\pi}{3},\frac{4\pi}{3}.$

    The set of solutions to $z^{3}-8=0$ is $\boldsymbol{A=\left\{2,-1+\sqrt{3}i,-1-\sqrt{3}i\right\}}.$ In the complex plane, they are the vertices of an equilateral triangle whose circumcircle has center $0$ and radius $2.$

  2. We solve $z^{3}-8z^{2}-8z+64=0$ by factoring by grouping.

    3. We have \begin{align*} z^2(z-8)-8(z-8)&=0 \\ \bigl(z^2-8\bigr)\bigl(z-8\bigr)&=0. \end{align*} The set of solutions to $z^{3}-8z^{2}-8z+64=0$ is $\boldsymbol{B=\left\{2\sqrt{2}i,-2\sqrt{2}i,8\right\}}.$

In the graph below, the greatest distance between a point of $A$ and a point of $B$ is the distance between $-1\pm\sqrt{3}i$ to $8,$ as shown in the dashed line segments. By the Distance Formula, the answer is \[\sqrt{(8-(-1))^2+\left(\pm\sqrt{3}-0\right)^2}=\sqrt{84}=\boxed{\textbf{(D) } 2\sqrt{21}}.\] DIAGRAM NEEDED.

~lopkiloinm ~MRENTHUSIASM

Remark

In the graph below, the solutions to $z^{3}-8=0$ are shown in red, and the solutions to $z^{3}-8z^{2}-8z+64=0$ are shown in blue. The greatest distance between one red point and one blue point is shown in a black dashed line segment.

File:2020 AMC 12A Problem 15.png

Graph in Desmos: https://www.desmos.com/calculator/uylcxkffak

~MRENTHUSIASM

See Also

2020 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 14 		Followed by
Problem 16
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

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