Problem:
An object moves 8 cm in a straight line from A to B, turns at an angle α, measured in radians and chosen at random from the interval (0,π), and moves 5 cm in a straight line to C. What is the probability that AC<7 ?
Answer Choices:
A. 61
B. 51
C. 41
D. 31
E. 21
Solution:
Let β=π−α. Apply the Law of Cosines to △ABC to obtain
(AC)2=82+52−2(8)(5)cosβ=89−80cosβ
Thus AC<7 if and only if
89−80cosβ<49, that is, if and only if cosβ>21
Therefore we must have 0<β<3π, and the requested probability is ππ/3=31.
The problems on this page are the property of the MAA's American Mathematics Competitions