Difference between revisions of "2020 AMC 12A Problems/Problem 15"
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== Problem == | == Problem == | ||
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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> | ||
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== Solution == | == Solution == | ||
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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>. | 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>. | ||
Latest revision as of 13:26, 19 January 2021
Problem
In the complex plane, let be the set of solutions to and let be the set of solutions to What is the greatest distance between a point of and a point of
Solution
Realize that will create an equilateral triangle on the complex plane with the first point at and two other points with equal magnitude at .
Also, realize that can be factored through grouping: will create points at and
Plotting the points and looking at the graph will make you realize that and are the farthest apart and through Pythagorean Theorem, the answer is revealed to be ~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 |
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