Question

一个体积为一立方英尺的立方体被三次平行于顶面的切割分成四块。第一刀距离顶面 $\dfrac{1}{2}$ 英尺。第二刀在第一刀下方 $\dfrac{1}{3}$ 英尺处，第三刀在第二刀下方 $\dfrac{1}{17}$ 英尺处。从上到下四块标为 $A$ 、 $B$ 、 $C$ 、 $D$ 。然后按 $C,B,A,D$ 的顺序首尾相接粘在一起，形成如图所示的长立体。这个立体的总表面积是多少平方英尺？

A one-cubic-foot cube is cut into four pieces by three cuts parallel to the top face of the cube. The first cut is $\dfrac{1}{2}$ foot from the top face. The second cut is $\dfrac{1}{3}$ foot below the first cut, and the third cut is $\dfrac{1}{17}$ foot below the second cut. From the top to the bottom the pieces are labeled $A,$ $B,$ $C,$ and $D.$ The pieces are then glued together end to end in the order $C,B,A,D$ to make a long solid as shown. What is the total surface area of this solid in square feet?

$6$

$7$

$\dfrac{419}{51}$

$\dfrac{158}{17}$

$11$

答案：E

知识点：表面积立体几何

难度评级：1620

解答：

从六个坐标方向观察这个立体。

从两个端面看，可见面积各为 $A$ 块的一个面，面积为 $\dfrac{1}{2}$ 平方英尺。

从每个侧面看，四块拼成原单位立方体的侧视图，所以每个侧视图面积为 $1$ 平方英尺。

从上方和下方看，每个视图都显示四个 $1$ 乘 $1$ 的面，所以每个面积为 $4$ 平方英尺。

总表面积为 $\dfrac{1}{2}+\dfrac{1}{2}+1+1+4+4=11$ 平方英尺。

所以正确答案是 E。

Look at the solid from the six coordinate directions.

From the two ends, the visible areas are each the face of piece $A$ , which has area $\dfrac{1}{2}$ square foot.

From each side, the four pieces stack to the side view of the original unit cube, so each side view has area $1$ square foot.

From the top and bottom, each view shows four $1$ -by- $1$ faces, so each has area $4$ square feet.

The total surface area is $\dfrac{1}{2}+\dfrac{1}{2}+1+1+4+4=11$ square feet.

Thus, E is the correct answer.

2009 AMC 8 第 25 题

答案：E

难度评级：1620

解答：

其他年份的第 25 题