AMC12 2005 A

AMC12 2005 A · Q10

AMC12 2005 A · Q10. It mainly tests Fractions, 3D geometry (volume).

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 $n$?

一个边长为 $n$ 单位的木立方体，六个面都涂成红色，然后切成 $n^3$ 个单位立方体。单位立方体所有面的总数中，恰好有四分之一是红色的。$n$ 是多少？

(A) 3 3

(B) 4 4

(D) 6 6

(E) 7 7

Answer

Correct choice: (B)

正确答案：（B）

Solution

There are $6n^3$ sides total on the unit cubes, and $6n^2$ are painted red. $\dfrac{6n^2}{6n^3}=\dfrac{1}{4} \Rightarrow n=4 \rightarrow \mathrm {B}$

单位立方体共有 $6n^3$ 个面，其中被涂成红色的有 $6n^2$ 个。 $\dfrac{6n^2}{6n^3}=\dfrac{1}{4} \Rightarrow n=4 \rightarrow \mathrm {B}$

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