AMC8 2014

AMC8 2014 · Q19

AMC8 2014 · Q19. It mainly tests 3D geometry (volume), Optimization (basic).

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 $3$ 英寸的立方体由 $27$ 个边长为 $1$ 英寸的小立方体组成。这些小立方体中有 $21$ 个是红色的，$6$ 个是白色的。如果要构建的 $3$ 英寸立方体使得白色表面积尽可能小，那么白色表面积占总表面积的几分之几？

(A) \frac{5}{54} \frac{5}{54}

(B) \frac{1}{9} \frac{1}{9}

(D) \frac{2}{9} \frac{2}{9}

(E) \frac{1}{3} \frac{1}{3}

Answer

Correct choice: (A)

正确答案：（A）

Solution

For the least possible surface area that is white, we should have 1 cube in the center, and the other 5 with only 1 face exposed. This gives 5 square inches of white, surface area. Since the cube has a surface area of 54 square inches, our answer is $\boxed{\textbf{(A) }\frac{5}{54}}$.

为了使白色表面积最小，我们应该有 1 个白色立方体放在中心，其他 5 个白色立方体的暴露面只有 1 个面。这样，白色表面积是 5 平方英寸。由于整个立方体的表面积是 54 平方英寸，答案是 $\boxed{\textbf{(A) }\frac{5}{54}}$。

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