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。求 $n$ ？

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 $n?$

$3$

$4$

$5$

$6$

$7$

答案：B

知识点：正方体表面积比与比例

难度评级：1430

解答：

$n^3$ 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：所以 $n = 4$ 。 $\dfrac{6n^2}{6n^3} = \dfrac{1}{n} = \dfrac{1}{4},$

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.

2005 AMC 12A 第 10 题

答案：B

难度评级：1430

解答：

其他年份的第 10 题