Problem:
An object moves 8 cm in a straight line from A to B, turns at an angle Ξ±, measured in radians and chosen at random from the interval (0,Ο), and moves 5 cm in a straight line to C. What is the probability that AC<7 ?
Answer Choices:
A. 61β
B. 51β
C. 41β
D. 31β
E. 21β
Solution:
Let Ξ²=ΟβΞ±. Apply the Law of Cosines to β³ABC to obtain
(AC)2=82+52β2(8)(5)cosΞ²=89β80cosΞ²
Thus AC<7 if and only if
89β80cosΞ²<49, that is, if and only if cosΞ²>21β
Therefore we must have 0<Ξ²<3Οβ, and the requested probability is ΟΟ/3β=31ββ.
The problems on this page are the property of the MAA's American Mathematics Competitions