AMC10 2004 A

AMC10 2004 A · Q21

AMC10 2004 A · Q21. It mainly tests Fractions, Circle theorems.

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 $8/13$ 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.)

两条不同的直线穿过三个半径分别为3、2和1的同心圆的圆心。图中阴影区域的面积是不阴影区域面积的$8/13$。这两条直线形成的锐角的弧度量是多少？（注：$\pi$弧度等于180度。）

(A) $\pi/8$ $\pi/8$

(B) $\pi/7$ $\pi/7$

(D) $\pi/5$ $\pi/5$

(E) $\pi/4$ $\pi/4$

Answer

Correct choice: (B)

正确答案：（B）

Solution

(B) Let $\theta$ be the acute angle between the two lines. The area of shaded Region 1 in the diagram is $$ 2\left(\frac{1}{2}\theta(1)^2\right)=\theta. $$ The area of shaded Region 2 is $$ 2\left(\frac{1}{2}(\pi-\theta)(2^2-1^2)\right)=3\pi-3\theta. $$ The area of shaded Region 3 is $$ 2\left(\frac{1}{2}\theta(3^2-2^2)\right)=5\theta. $$ Hence the total area of the shaded regions is $3\pi+3\theta$. The area bounded by the largest circle is $9\pi$, so $$ \frac{3\pi+3\theta}{9\pi}=\frac{8}{8+13}. $$ Solving for $\theta$ gives $\theta=\pi/7$.

（B）设 $\theta$ 为两条直线之间的锐角。图中阴影区域1的面积为 $$ 2\left(\frac{1}{2}\theta(1)^2\right)=\theta. $$ 阴影区域2的面积为 $$ 2\left(\frac{1}{2}(\pi-\theta)(2^2-1^2)\right)=3\pi-3\theta. $$ 阴影区域3的面积为 $$ 2\left(\frac{1}{2}\theta(3^2-2^2)\right)=5\theta. $$ 因此阴影部分的总面积为 $3\pi+3\theta$。最大圆所围成的面积为 $9\pi$，所以 $$ \frac{3\pi+3\theta}{9\pi}=\frac{8}{8+13}. $$ 解得 $\theta=\pi/7$。

Topics