AMC8 2000

AMC8 2000 · Q22

AMC8 2000 · Q22. It mainly tests 3D geometry (surface area).

A cube has edge length 2. Suppose that we glue a cube of edge length 1 on top of the big cube so that one of its faces rests entirely on the top face of the larger cube. The percent increase in the surface area (sides, top, and bottom) from the original cube to the new solid formed is closest to:

一个边长为2的立方体。我们将一个边长为1的立方体粘在大立方体的顶部，使其一个面完全贴在大立方体顶面上。从原立方体到新形成的固体的表面积（侧面、顶部和底部）的百分比增加最接近：

(A) 10 10

(B) 15 15

(D) 21 21

(E) 25 25

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): The area of each face of the larger cube is $2^2=4$. There are six faces of the cube, so its surface area is $6(4)=24$. When we add the smaller cube, we decrease the original surface area by $1$, but we add $5(1^2)=5$ units of area (1 unit for each of the five unglued faces of the smaller cube). This is a net increase of $4$ from the original surface area, and $4$ is $\frac{4}{24}=\frac{1}{6}\approx16.7\%$ of $24$. The closest value given is $17$.

答案（C）：较大立方体每个面的面积为 $2^2=4$。立方体有 6 个面，所以其表面积为 $6(4)=24$。当我们加上较小的立方体时，原来的表面积减少 $1$，但会新增 $5(1^2)=5$ 个面积单位（小立方体有 5 个未粘合的面，每个贡献 1 个单位面积）。因此相对于原表面积净增加 $4$，而 $4$ 占 $24$ 的比例为 $\frac{4}{24}=\frac{1}{6}\approx16.7\%$。所给选项中最接近的是 $17$。

Topics

GE.014 3D geometry (surface area)