AMC10 2005 A

AMC10 2005 A · Q11

AMC10 2005 A · Q11. It mainly tests Linear equations, 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 $n$ 单位的木制立方体所有六个面都被涂成红色，然后被切成 $n^3$ 个单位立方体。单位立方体总面数中有恰好四分之一是红色的。$n$ 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

(B) The unit cubes have a total of $6n^3$ faces, of which $6n^2$ are red. Therefore $$ \frac{1}{4}=\frac{6n^2}{6n^3}=\frac{1}{n}, $$ so $n=4$.

（B）单位小立方体共有 $6n^3$ 个面，其中 $6n^2$ 个是红色。因此 $$ \frac{1}{4}=\frac{6n^2}{6n^3}=\frac{1}{n}, $$ 所以 $n=4$。

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