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

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In the graph below, the solutions to <math>z^{3}-8=0</math> are shown in red, and the solutions to <math>z^{3}-8z^{2}-8z+64=0</math> are shown in blue. The greatest distance between one red point and one blue point is shown in a black dashed line segment.
 
In the graph below, the solutions to <math>z^{3}-8=0</math> are shown in red, and the solutions to <math>z^{3}-8z^{2}-8z+64=0</math> are shown in blue. The greatest distance between one red point and one blue point is shown in a black dashed line segment.
  
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<b>File in progress</b>
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[[File:2020 AMC 12A Problem 15.png|center]]
  
 
~MRENTHUSIASM
 
~MRENTHUSIASM

Revision as of 12:10, 17 April 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

Realize that $z^{3}-8=0$ will create an equilateral triangle on the complex plane with the first point at $2+0i$ and two other points with equal magnitude at $-1{\pm}i\sqrt{3}$.

Also, realize that $z^{3}-8z^{2}-8z+64$ can be factored through grouping: $z^{3}-8z^{2}-8z+64=(z-8)(z^{2}-8).$ $(z-8)(z^{2}-8)$ will create points at $8+0i$ and $\pm2\sqrt{2}+0i.$

Plotting the points and looking at the graph will make you realize that $-1{\pm}i\sqrt{3}$ and $8+0i$ are the farthest apart and through Pythagorean Theorem, the answer is revealed to be $\sqrt{\sqrt{3}^{2}+(8-(-1))^{2}}=\sqrt{84}=\boxed{\textbf{(D) } 2\sqrt{21}.}$ ~lopkiloinm

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

~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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