Question

一个实心木质单位立方体被一个平面切去一个实心四面体。这个平面经过同一面上的两个不相邻顶点，以及相对面上一个与前两个顶点都不相邻的顶点。切下的四面体被丢弃，剩下的部分把切面朝下放在桌上。这个物体的高度是多少？

A solid tetrahedron is sliced off a solid wooden unit cube by a plane passing through two nonadjacent vertices on one face and one vertex on the opposite face not adjacent to either of the first two vertices. The tetrahedron is discarded and the remaining portion of the cube is placed on a table with the cut surface face down. What is the height of this object?

$\dfrac{\sqrt{3}}{3}$

$\dfrac{2 \sqrt{2}}{3}$

$1$

$\dfrac{2 \sqrt{3}}{3}$

$\sqrt{2}$

答案：D

知识点：正方体体积等边三角形

难度评级：2060

解答：

被切下的四面体可以看作底面为直角等腰三角形、两直角边为 $1$ ，高为 $1$ ，所以体积为 $\dfrac13\cdot\dfrac12\cdot1=\dfrac16$ 。

切面是边长为 $\sqrt2$ 的等边三角形，面积为 $\dfrac{\sqrt3}{4}(\sqrt2)^2=\dfrac{\sqrt3}{2}$ 。设从相对顶点到切面的高为 $h$ ，由体积公式 $\dfrac13\cdot\dfrac{\sqrt3}{2}\cdot h=\dfrac16$ ，得 $h=\dfrac{\sqrt3}{3}$ 。

单位立方体的体对角线长为 $\sqrt3$ 。切面朝下放置后，高为 $\sqrt3-\dfrac{\sqrt3}{3}=\dfrac{2\sqrt3}{3}$ 。

所以正确答案是 D。

The discarded tetrahedron has a right isosceles triangle of leg $1$ as one base and height $1$ , so its volume is $\dfrac13\cdot\dfrac12\cdot1=\dfrac16$ .

The cut face is an equilateral triangle of side $\sqrt2$ , so its area is $\dfrac{\sqrt3}{4}(\sqrt2)^2=\dfrac{\sqrt3}{2}$ . If $h$ is the height from the opposite vertex to this cut face, then $\dfrac13\cdot\dfrac{\sqrt3}{2}\cdot h=\dfrac16$ , so $h=\dfrac{\sqrt3}{3}$ .

The full cube diagonal has length $\sqrt3$ . After the tetrahedron is removed and the cut face is placed down, the height is $\sqrt3-\dfrac{\sqrt3}{3}=\dfrac{2\sqrt3}{3}$ .

Thus, D is the correct answer.

2012 AMC 10B 第 23 题

答案：D

难度评级：2060

解答：

其他年份的第 23 题