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
, let the object turn clockwise.
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 $$ (Error compiling LaTeX. Unknown error_msg)\frac{\angle ABO}{180 ^\circ}$.
Let$ (Error compiling LaTeX. Unknown error_msg)B = (0, 0)\odot B = x^2 + y^2 = 25
\odot A = x^2 + (y+8)^2 = 49
O
x^2 + (y+8)^2 - x^2 - y^2 = 49 - 25
64 + 16y =24
y = - \frac52
x = \frac{5 \sqrt{3}}{2}
O = (\frac{5 \sqrt{3}}{2}, - \frac52)
\triangle BDO
30-60-90
BO = 5
BD = \frac{5 \sqrt{3}}{2}
DO = \frac52
\angle CBO = 30 ^\circ
\angle ABO = 60 ^\circ$.
Therefore the probability that$ (Error compiling LaTeX. Unknown error_msg)AC < 7\frac{\angle ABO}{180 ^\circ} = \frac{60 ^\circ}{180 ^\circ} = \boxed{\textbf{(D) } \frac13 }$
See also
2003 AMC 12B (Problems • Answer Key • Resources) | |
Preceded by Problem 20 |
Followed by Problem 22 |
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