AMC 8 2009 Problem 25
AMC8 2009 年第 25 题
立体几何★★★★☆
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 1/2 foot from the top face. The second cut is 1/3 foot below the first cut, and the third cut is 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 as shown in the second diagram. What is the total surface area of this solid in square feet?
一个 1 立方英尺的正方体被平行于立方体顶面的三个切口切成四块。第一个切口距离顶面 英尺。第二个切口比第一个切口低 英尺，第三个切口比第二个切口低 英尺。从上到下，切出的这 4 块立体图形被标记为 A、B、C 和 D。接着这 4 块立体图形被一个接一个地粘合在一起，如第二张图所示。那么新形成的立体图形的总表面积是多少平方英尺？
- A.6
- B.7
- C.
- D.
- E.11
正确答案 · Correct Answer
E
解析 · Solution
解法1
常规解法 · 逐面累加
按官方拼法，拼成的立体里四块各露出一个 1×1 的顶面和一个 1×1 的底面，顶、底共 4+4=8。两个长侧面各把四段高度 a+b+c+d=1 拼齐、长为 1，面积各 1，共 2。两个端面各为 ，共 1。总表面积 8+2+1=11。
巧解 · 从原立方体加减
原立方体表面积为 6。三刀切开各暴露两张单位截面：6+2+2+2=12。再扣去拼接处的重叠 2(b+c+d)；因 ，故 ，扣 ，得 12-1=11。
Solution 1
Elementary · add face by face
In the assembled solid each of the four pieces shows a 1×1 top and a 1×1 bottom, contributing 4+4=8. The two long sides each span the combined heights a+b+c+d=1 over length 1, adding 2. The two end faces contribute each, adding 1. The total surface area is 8+2+1=11.
Slick · add and subtract from the original cube
Start from the cube's surface area 6. The three cuts each expose two new unit faces: 6+2+2+2=12. Re-gluing end to end hides an overlap of 2(b+c+d); since , , so subtract , giving 12-1=11.
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