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 "2020 AMC 12A Problems/Problem 15"

m (→‎Solution)
m (minor spacing changes)
Line 1: Line 1:
 
== Problem ==
 
== Problem ==
−
 
 
In the complex plane, let <math>A</math> be the set of solutions to <math>z^{3}-8=0</math> and let <math>B</math> be the set of solutions to <math>z^{3}-8z^{2}-8z+64=0.</math> What is the greatest distance between a point of <math>A</math> and a point of <math>B?</math>
 
In the complex plane, let <math>A</math> be the set of solutions to <math>z^{3}-8=0</math> and let <math>B</math> be the set of solutions to <math>z^{3}-8z^{2}-8z+64=0.</math> What is the greatest distance between a point of <math>A</math> and a point of <math>B?</math>
  
Line 6: Line 5:
  
 
== Solution ==
 
== Solution ==
−
 
 
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>.  
 
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>.  
  

Revision as of 13:26, 19 January 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

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions. AMC logo.png

Retrieved from "https://artofproblemsolving.com/wiki/index.php?title=2020_AMC_12A_Problems/Problem_15&oldid=142782"