Question

An object moves $8$ cm in a straight line from $A$ to $B,$ turns at an angle $\alpha,$ measured in radians and chosen at random from the interval $(0, \pi),$ and moves $5$ cm in a straight line to $C.$ What is the probability that $AC \lt 7?$

$\dfrac{1}{6}$

$\dfrac{1}{5}$

$\dfrac{1}{4}$

$\dfrac{1}{3}$

$\dfrac{1}{2}$

Answer: D

Concepts:law of cosines geometric probability trigonometry

Difficulty rating: 1910

Solution:

Let $\beta = \pi - \alpha$ be the interior angle of $\triangle ABC$ at $B.$ By the Law of Cosines, $\begin{aligned} &AC^2 = 8^2 + 5^2 \\ &\quad {}- 2(8)(5)\cos\beta \\ &= 89 - 80\cos\beta. \end{aligned}$

Then $AC \lt 7$ means $89 - 80\cos\beta \lt 49,$ i.e. $\cos\beta \gt \dfrac{1}{2},$ i.e. $\beta \lt \dfrac{\pi}{3}.$

As $\alpha$ is uniform on $(0, \pi),$ so is $\beta.$ The probability is $\frac{\pi/3}{\pi} = \frac{1}{3}.$

Thus, the correct answer is D.

2003 AMC 12B Problem 21

Answer: D

Difficulty rating: 1910

Solution:

Problem 21 in Other Years