Question

一个单位立方体被切两刀，形成三个三棱柱，其中两个全等，如图 $1$ 所示。然后按图 $2$ 中虚线所示用同样方式切割该立方体，得到九块。包含顶点 $W$ 的那一块体积是多少？

A unit cube is cut twice to form three triangular prisms, two of which are congruent, as shown in Figure $1.$ The cube is then cut in the same manner along the dashed lines shown in Figure $2.$ This creates nine pieces. What is the volume of the piece that contains vertex $W?$

$\dfrac{1}{12}$

$\dfrac{1}{9}$

$\dfrac{1}{8}$

$\dfrac{1}{6}$

$\dfrac{1}{4}$

答案：A

知识点：棱锥体积立体几何

难度评级：1910

小提示：

含 $W$ 的那块是一个棱锥；确定它的底面和顶点。

The piece at $W$ is a pyramid; identify its base and apex

大提示：

它的底面积记为 $B$ ，高记为 $h$ ；使用公式 $V = \dfrac{1}{3}Bh$ 。

If its base area is $B$ and height is $h,$ use $V = \dfrac{1}{3}Bh$

解答：

两组互相垂直的切割都从上底面的棱一直切到下底面的中线。在 $W$ 附近，它们切出一个正四棱锥：它的底面是下底面中与 $W$ 相邻的那四分之一，顶点位于下底面中心正上方一个单位处。

它的底面是边长 $\dfrac{1}{2}$ 的正方形，即下底面的四分之一，高为 $1$ 。因此体积为 $\dfrac{1}{3} \left(\dfrac{1}{2}\right)^2 (1) = \dfrac{1}{12}\text{。}$

所以正确答案是 A。

The two perpendicular sets of cuts each run from top edges to midlines of the bottom face. Near $W$ they carve out a square pyramid. Its base is the quarter of the bottom face adjacent to $W,$ and its apex lies one unit above the center of the bottom face.

Thus its base is a square of side $\dfrac{1}{2}$ and its altitude is the full height $1.$ Therefore the volume is $\dfrac{1}{3} \left(\dfrac{1}{2}\right)^2 (1) = \dfrac{1}{12}.$

Thus, the correct answer is A.

2005 AMC 12A 第 17 题

答案：A

难度评级：1910

小提示：

大提示：

解答：

其他年份的第 17 题