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2003 AMC 12B Problems/Problem 21: Difference between revisions

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


<math>\mathrm{(A)}\ 448
<math>\mathrm{(A)}\ \frac{1}{6}
\qquad\mathrm{(B)}\ 486
\qquad\mathrm{(B)}\ \frac{1}{5}
\qquad\mathrm{(C)}\ 1560
\qquad\mathrm{(C)}\ \frac{1}{4}
\qquad\mathrm{(D)}\ 2001
\qquad\mathrm{(D)}\ \frac{1}{3}
\qquad\mathrm{(E)}\ 2003</math>
\qquad\mathrm{(E)}\ \frac{1}{2}</math>


== Solution ==
== Solution ==

Revision as of 12:15, 29 November 2008

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 + AC^2 - 2 AB \cdot AC \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 $\frac{\pi/3}{\pi} = \frac 13 \Rightarrow \mathrm{(D)}$.

See also

2003 AMC 12B (Problems • Answer Key • Resources)
Preceded by
Problem 20 		Followed by
Problem 22
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions
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