AMC10 2000 A

AMC10 2000 A · Q18

AMC10 2000 A · Q18. It mainly tests Area & perimeter, Geometric probability (basic).

Charlyn walks completely around the boundary of a square whose sides are each 5 km long. From any point on her path she can see exactly 1 km horizontally in all directions. What is the area of the region consisting of all points Charlyn can see during her walk, expressed in square kilometers and rounded to the nearest whole number?

查琳完全绕着一个边长各为5千米的正方形的边界走一圈。从她路径上的任意一点，她都能在所有方向水平看清1千米。求查琳在行走过程中能看到的区域的面积（以平方千米为单位，四舍五入到最接近的整数）。

(A) 24 24

(B) 27 27

(D) 40 40

(E) 42 42

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): At any point on Charlyn’s walk, she can see all the points inside a circle of radius $1$ km. The portion of the viewable region inside the square consists of the interior of the square except for a smaller square with side length $3$ km. This portion of the viewable region has area $(25-9)\ \text{km}^2$. The portion of the viewable region outside the square consists of four rectangles, each $5$ km by $1$ km, and four quarter-circles, each with a radius of $1$ km. This portion of the viewable region has area $4\left(5+\frac{\pi}{4}\right)=(20+\pi)\ \text{km}^2$. The area of the entire viewable region is $36+\pi\approx 30\ \text{km}^2$.

答案（C）：在 Charlyn 行走路径上的任意一点，她都能看到以该点为圆心、半径为 $1$ km 的圆内所有点。正方形内部的可视区域等于正方形内部去掉一个边长为 $3$ km 的小正方形后剩余的部分。该部分面积为 $(25-9)\ \text{km}^2$。正方形外部的可视区域由四个长方形（每个为 $5$ km 乘 $1$ km）以及四个四分之一圆（每个半径为 $1$ km）组成。该部分面积为 $4\left(5+\frac{\pi}{4}\right)=(20+\pi)\ \text{km}^2$。整个可视区域的面积为 $36+\pi\approx 30\ \text{km}^2$。

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