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==Problem==
Two distinct lines pass through the center of three concentric circles of radii 3, 2, and 1. The area of the shaded region in the diagram is <math>\frac{8}{13}</math> of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: <math>\pi</math> radians is <math>180</math> degrees.)

<center>[[Image:AMC10_2004A_21.png]]</center>

<math> \mathrm{(A) \ } \frac{\pi}{8} \qquad \mathrm{(B) \ } \frac{\pi}{7} \qquad \mathrm{(C) \ } \frac{\pi}{6} \qquad \mathrm{(D) \ } \frac{\pi}{5} \qquad \mathrm{(E) \ } \frac{\pi}{4} </math>

==Solution==
Let the area of the shaded region be <math>S</math>, the area of the unshaded region be <math>U</math>, and the acute angle that is formed by the two lines be <math>\theta</math>. We can set up two equations between <math>S</math> and <math>U</math>:

<math>S+U=9\pi</math>

<math>S=\dfrac{8}{13}U</math>

Thus <math>\dfrac{21}{13}U=9\pi</math>, and <math>U=\dfrac{39\pi}{7}</math>, and thus <math>S=\dfrac{8}{13}\cdot \dfrac{39\pi}{7}=\dfrac{24\pi}{7}</math>.

Now we can make a formula for the area of the shaded region in terms of <math>\theta</math>:

<math>\dfrac{2\theta}{2\pi}*\pi +\dfrac{2(\pi-\theta)}{2\pi}*(4\pi-\pi)+\dfrac{2\theta}{2\pi}(9\pi-4\pi)=\theta +3\pi-3\theta+5\theta=3\theta+3\pi=\dfrac{24\pi}{7}</math>

Thus <math>3\theta=\dfrac{3\pi}{7}\Rightarrow \theta=\dfrac{\pi}{7}\Rightarrow \mathrm{(B) \ }</math>

== See also ==
{{AMC10 box|year=2004|ab=A|num-b=20|num-a=22}}

[[Category:Introductory Geometry Problems]]