Question

一个实心立方体边长为 $3$ 英寸。在每个面的中心切出一个 $2$ 英寸乘 $2$ 英寸的正方形孔。每个切口的边都与立方体的边平行，并且每个孔都贯穿整个立方体。剩余立体的体积是多少立方英寸？

A solid cube has side length $3$ inches. A $2$ -inch by $2$ -inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

$7$

$8$

$10$

$12$

$15$

答案：A

知识点：体积容斥原理立体几何

难度评级：1790

解答：

三个被切出的长方体都在立方体中心相交。

交集是边长为 $2$ 的立方体。因此被切除区域的体积为 $\begin{align*}3 \cdot 2 \cdot 2 \cdot 3 - 2 \cdot 2^3 &= 36 - 16 \\&= 20.\end{align*}$

中心区域被所有 $3$ 个孔都包含，所以要减去重复计入的两次。

剩余体积为 $3^3 - 20 = 27 - 20 = 7.$

所以正确答案是 A。

Note that all the cut out solids intersect in the middle of the cube.

This region of intersection is a cube with side length $2.$ Then the volume of the cutout region is $\begin{align*}3 \cdot 2 \cdot 2 \cdot 3 - 2 \cdot 2^3 &= 36 - 16 \\&= 20.\end{align*}$

We have to subtract out the center region twice since it is included in all $3$ regions.

The remaining volume is then $3^3 - 20 = 27 - 20 = 7.$

Thus, A is the correct answer.

2010 AMC 10A 第 17 题

答案：A

难度评级：1790

解答：

其他年份的第 17 题