Question

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 $\dfrac{8}{13}$ of the area of the unshaded region. What is the radian measure of the acute angle formed by the two lines? (Note: $\pi$ radians is $180$ degrees.)

$\dfrac{\pi}{8}$

$\dfrac{\pi}{7}$

$\dfrac{\pi}{6}$

$\dfrac{\pi}{5}$

$\dfrac{\pi}{4}$

Solution:

Let $\theta$ be the acute angle. The shaded region has three parts: two acute sectors of the unit disk with total area $\theta,$ two obtuse sectors of the ring between radii $1$ and $2$ with total area $3(\pi - \theta),$ and two acute sectors of the ring between radii $2$ and $3$ with total area $5\theta.$

Adding these gives a shaded area of $\theta + 3(\pi - \theta) + 5\theta = 3\pi + 3\theta.$

The shaded region is $\dfrac{8}{13}$ of the unshaded region, so it is $\dfrac{8}{21}$ of the total area $9\pi.$ Then $3\pi + 3\theta = \dfrac{8}{21}(9\pi) = \dfrac{24\pi}{7},$ which gives $\theta = \dfrac{\pi}{7}.$

Thus, the correct answer is B.

2004 AMC 10A Problem 21

Solution:

Problem 21 in Other Years