AMC12 2001 A

AMC12 2001 A · Q10

AMC12 2001 A · Q10. It mainly tests Counting & probability misc, Area & perimeter.

The plane is tiled by congruent squares and congruent pentagons as indicated. The percent of the plane that is enclosed by the pentagons is closest to

平面由全等的正方形和全等的五边形铺满，如图所示。五边形所覆盖的平面百分比最接近

(A) 50 50

(B) 52 52

(D) 56 56

(E) 58 58

Answer

Correct choice: (D)

正确答案：（D）

Solution

Consider any single tile: If the side of the small square is $a$, then the area of the tile is $\left(3a\right)^2 = 9a^2$, with $4a^2$ covered by squares, and therefore $9a^2-4a^2 = 5a^2$ by pentagons. Hence exactly $\frac{5}{9}$ of any tile is covered by pentagons, and therefore pentagons cover $\frac{5}{9}$ of the plane. When expressed as a percentage, this is $55.\overline{5}\%$, and the closest integer to this value is $\boxed{\textbf{(D) }56}$.

考虑任意一个单独的拼块：若小正方形的边长为 $a$，则该拼块的面积为 $\left(3a\right)^2=9a^2$，其中有 $4a^2$ 被正方形覆盖，因此被五边形覆盖的面积为 $9a^2-4a^2=5a^2$。因此任意拼块中恰有 $\frac{5}{9}$ 的面积被五边形覆盖，从而五边形覆盖了平面的 $\frac{5}{9}$。写成百分数为 $55.\overline{5}\%$，最接近的整数为 $\boxed{\textbf{(D) }56}$。

Topics