2003 AMC 12B Problems/Problem 21

Problem

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 $AC < 7$?

$\mathrm{(A)}\ \frac{1}{6} \qquad\mathrm{(B)}\ \frac{1}{5} \qquad\mathrm{(C)}\ \frac{1}{4} \qquad\mathrm{(D)}\ \frac{1}{3} \qquad\mathrm{(E)}\ \frac{1}{2}$

Solution

By the Law of Cosines, \begin{align*} AB^2 + BC^2 - 2 AB \cdot BC \cos \alpha = 89 - 80 \cos \alpha = AC^2 &< 49\\ \cos \alpha &> \frac 12\\ \end{align*}

It follows that $0 < \alpha < \frac {\pi}3$, and the probability is $\frac{\pi/3}{\pi} = \frac 13 \Rightarrow \mathrm{(D)}$.

Fake-solve Solution

Notice that if the angle chosen is $0$ radians, then the distance $AC$ is $8+5=13$. If the angle chosen is $2\pi$ radians, then the distance $AC$ is $8-5=3$. Now, note that $|7-13|=6$ and $|7-3|=4$. Finally, since $\frac 46 = \frac 23$, our answer is $1-\frac 23 = \frac 13\Rightarrow \mathrm{(D)}$.

Solution by franzliszt

See also

2003 AMC 12B (ProblemsAnswer KeyResources)
Preceded by
Problem 20
Followed by
Problem 22
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All AMC 12 Problems and Solutions

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