Difference between revisions of "2011 AMC 12B Problems/Problem 24"
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== Problem == | == Problem == | ||
| − | Let <math>P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)</math>. What is the minimum perimeter | + | Let <math>P(z) = z^8 + \left(4\sqrt{3} + 6\right)z^4 - \left(4\sqrt{3} + 7\right)</math>. What is the minimum perimeter among all the <math>8</math>-sided polygons in the complex plane whose vertices are precisely the zeros of <math>P(z)</math>? |
<math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6}</math> | <math>\textbf{(A)}\ 4\sqrt{3} + 4 \qquad \textbf{(B)}\ 8\sqrt{2} \qquad \textbf{(C)}\ 3\sqrt{2} + 3\sqrt{6}</math> | ||
Revision as of 17:07, 25 October 2011
Problem
Let 
. What is the minimum perimeter among all the 
-sided polygons in the complex plane whose vertices are precisely the zeros of 
?
Solution
Answer: (B)
First of all, we need to find all 
such that 
So 
or 
or 
Now we have a solution at 
if we look at them in polar coordinate, further more, the 8-gon is symmetric (it is a regular octagon) . So we only need to find the side length of one and multiply by .
So answer 
distance from 
to 
Side length 
Hence, answer is 
.
See also
| 2011 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 23 |
Followed by Problem 25 |
| 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 | |
