AMC 8 2014 Problem 19

AMC8 2014 年第 19 题

立体几何★★★☆☆

A cube with 3-inch edges is to be constructed from 27 smaller cubes with 1-inch edges. Twenty-one of the cubes are colored red and 6 are colored white. If the 3-inch cube is constructed to have the smallest possible white surface area showing, what fraction of the surface area is white?

一个边长为 3 英寸的大正方体是由 27 个边长为 1 英寸的小正方体组成的。其中 21 个小正方体是红色的，6 个是白色的。若要求大的正方体的表面积中白色部分越少越好，那么表面积的白色部分所占总表面积的比例是多少？

A. $\displaystyle \frac{5}{54}$
B. $\displaystyle \frac{1}{9}$
C. $\displaystyle \frac{5}{27}$
D. $\displaystyle \frac{2}{9}$
E. $\displaystyle \frac{1}{3}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

The amount of white surface area is smallest when you place one white cube in the interior of the larger cube. Place each of the other 5 5 white cubes at the center of a face so that 1 1 white face and 8 8 red faces are visible on that face. The total surface area of the larger cube is 6 ⋅ 3 2 = 54 6⋅3 2 =54 square inches, so the fraction of the surface area that is white is 5 54 54 5 . Answer: 𝐴 A .

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