2005 AMC 12A 第 10 题

Question

一个边长为

n

的木立方体六个面都涂成红色，然后切成

n^3

个单位立方体。所有单位立方体的面中，恰有四分之一是红色的。

n

是多少？

A wooden cube

n

units on a side is painted red on all six faces and then cut into

n^3

unit cubes. Exactly one-fourth of the total number of faces of the unit cubes are red. What is

n?

3

4

5

6

7

分别数小立方体面的总数和红色面的数量。

Count the total number of small-cube faces and the number that are red

总共有

6n^3

个面，其中红色面有

6n^2

个。

There are

6n^3

faces in all and

6n^2

red ones

n^3

个单位立方体共有

6n^3

个面，其中红色面正是原立方体的表面，共

6n^2

个。

令红色面所占比例为四分之一：

\dfrac{6n^2}{6n^3} = \dfrac{1}{n} = \dfrac{1}{4}\text{，}

所以

n = 4

。

所以正确答案是 B。

The

n^3

unit cubes have

6n^3

faces total. The red faces are exactly the surface of the original cube,

6n^2

of them.

Setting the red fraction to one-fourth,

\dfrac{6n^2}{6n^3} = \dfrac{1}{n} = \dfrac{1}{4},

so

n = 4.

Thus, the correct answer is B.

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