Difference between revisions of "2003 AMC 12B Problems/Problem 21"
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By the [[Law of Cosines]], | By the [[Law of Cosines]], | ||
<cmath>\begin{align*} | <cmath>\begin{align*} | ||
− | AB^2 + BC^2 - 2 AB \cdot BC \cos \ | + | AB^2 + BC^2 - 2 AB \cdot BC \cos \alpha = 89 - 80 \cos \alpha = AC^2 &< 49\\ |
− | \cos \ | + | \cos \alpha &< \frac 12\\ |
\end{align*}</cmath> | \end{align*}</cmath> | ||
Revision as of 22:13, 18 August 2016
Problem
An object moves cm in a straight line from to , turns at an angle , measured in radians and chosen at random from the interval , and moves cm in a straight line to . What is the probability that ?
Solution
By the Law of Cosines,
It follows that , and the probability is .
See also
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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.