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Difference between revisions of "2007 AIME I Problems/Problem 2"

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== Problem ==
 
== Problem ==
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The [[complex number]] <math>z</math> is equal to <math>9+bi</math>, where <math>b</math> is a [[positive]] [[real number]] and <math>i^{2}=-1</math>.  Given that the imaginary parts of <math>z^{2}</math> and <math>z^{3}</math> are the same, what is <math>b</math> equal to?
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A 100 foot long moving walkway moves at a constant rate of 6 feet per second.  Al steps onto the start of the walkway and stands.  Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second.  Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second.  At a certain time, one of these three persons is exactly halfway between the other two.  At that time, find the distance in feet between the start of the walkway and the middle person.
  
 
== Solution ==
 
== Solution ==
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Squaring, we find that <math>(9 + bi)^2 = 81 + 18bi - b^2</math>. Cubing and ignoring the real parts of the result, we find that <math>(81 + 18bi - b^2)(9 + bi) = \ldots + (9\cdot 18 + 81)bi - b^3i</math>.
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{{solution}}
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Setting these two equal, we get that <math>18bi = 243bi - b^3i</math>, so <math>b(b^2 - 225) = 0</math> and <math>b = -15, 0, 15</math>. Since <math>b > 0</math>, the solution is <math>015</math>.
 
  
 
== See also ==
 
== See also ==
 
{{AIME box|year=2007|n=I|num-b=1|num-a=3}}
 
{{AIME box|year=2007|n=I|num-b=1|num-a=3}}
  
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[[Category:Intermediate Complex Numbers Problems]]
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[[Category:Intermediate Algebra Problems]]

Revision as of 19:11, 14 March 2007

Problem

A 100 foot long moving walkway moves at a constant rate of 6 feet per second. Al steps onto the start of the walkway and stands. Bob steps onto the start of the walkway two seconds later and strolls forward along the walkway at a constant rate of 4 feet per second. Two seconds after that, Cy reaches the start of the walkway and walks briskly forward beside the walkway at a constant rate of 8 feet per second. At a certain time, one of these three persons is exactly halfway between the other two. At that time, find the distance in feet between the start of the walkway and the middle person.

Solution

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See also

2007 AIME I (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15
All AIME Problems and Solutions
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