AMC 8 2006 Problem 18

AMC8 2006 年第 18 题

立体几何★★★☆☆

A cube with 3-inch edges is made using 27 cubes with 1-inch edges. Nineteen of the smaller cubes are white and eight are black. If the eight black cubes are placed at the corners of the larger cube, what fraction of the surface area of the larger cube is white?

一个边长为 3 英寸的正方体是由 27 个边长为 1 英寸的正方体组成的。其中 19 个小正方体为白色，8 个是黑色。若这 8 个黑色正方体放在大的正方体的 8 个角落，那么大正方体的表面积中，白色部分的比例是多少？

A. $\displaystyle \frac{1}{9}$
B. $\displaystyle \frac{1}{4}$
C. $\displaystyle \frac{4}{9}$
D. $\displaystyle \frac{5}{9}$
E. $\displaystyle \frac{19}{27}$

正确答案 · Correct Answer

解析 · Solution

Solution 1

The surface area of the cube is 6(3)(3)=54. Each of the eight black cubes has 3 faces on the outside, making 3(8)=24 black faces. Therefore there are 54-24=30 white faces. To find the ratio, we evaluate

\displaystyle \frac{30}{54} = \frac{5}{9}

继续练习

免费刷 2006 整卷 →