2003 AMC 12B Problems/Problem 21
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 1 (Trigonometry)
By the Law of Cosines,
It follows that , and the probability is .
Solution 2 (Analytic Geometry)
, let the object turn clockwise.
Let , .
Note that the possible points of create a semi-circle of radius and center . The area where is enclosed by a circle of radius and center . The probability that is .
The function of is , the function of is .
is the point that satisfies the system of equations:
, , , ,
Note that is a triangle, as , , . As a result , .
Therefore the probability that 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.