Question

一块直径为 $12$ 英尺的圆形草地被一条宽 $3$ 英尺的笔直碎石路切过，其中一条路边经过草地中心。剩余草地的面积（平方英尺）为

A circular grass plot $12$ feet in diameter is cut by a straight gravel path $3$ feet wide, one edge of which passes through the center of the plot. The number of square feet in the remaining grass area is

$36\pi-34$

$30\pi-15$

$36\pi-33$

$35\pi-9\sqrt3$

$30\pi-9\sqrt3$

答案：E

知识点：圆面积扇形面积分割特殊直角三角形

难度评级：2230

小提示：

用与小路垂直的圆直径将小路平分

Bisect the path by the perpendicular diameter of the circular plot

大提示：

小路的每一半由一个 $30^\circ$ 扇形和一个 $30^\circ$ - $60^\circ$ - $90^\circ$ 三角形组成

Each half of the path is a $30^\circ$ sector together with a $30^\circ$ - $60^\circ$ - $90^\circ$ triangle

解答：

草地半径为 $6$ 。在小路的另一条边上，圆心到该边的垂直距离为 $3$ ，所以通向交点的半径与经过圆心的路边成 $30^\circ$ 角。小路的一半由半径为 $6$ 的 $30^\circ$ 扇形和直角边长为 $3$ 与 $3\sqrt3$ 的直角三角形组成。其面积为 $\begin{aligned} &\frac{30}{360}\pi(6^2) +\frac12(3)(3\sqrt3)\\ &\qquad=3\pi+\frac{9\sqrt3}{2} \end{aligned}\text{。}$ 因此整条小路的面积为 $6\pi+9\sqrt3$ ，剩余草地面积为 $36\pi-(6\pi+9\sqrt3) =30\pi-9\sqrt3\text{。}$

所以正确答案是 E。

The plot has radius $6.$ At the path’s other edge, the perpendicular distance from the center is $3,$ so the radius to an intersection point makes a $30^\circ$ angle with the edge through the center. Half of the path consists of a $30^\circ$ sector of radius $6$ and a right triangle with legs $3$ and $3\sqrt3.$ Its area is $\begin{aligned} &\frac{30}{360}\pi(6^2) +\frac12(3)(3\sqrt3)\\ &\qquad=3\pi+\frac{9\sqrt3}{2}. \end{aligned}$ Thus the whole path has area $6\pi+9\sqrt3,$ and the remaining grass area is $36\pi-(6\pi+9\sqrt3) =30\pi-9\sqrt3.$

Therefore, the correct answer is E.

1973 AMC 12 第 25 题

答案：E

难度评级：2230

小提示：

大提示：

解答：

其他年份的第 25 题