AMC10 2010 A

AMC10 2010 A · Q17

AMC10 2010 A · Q17. It mainly tests 3D geometry (volume), 3D geometry (surface area).

A solid cube has side length 3 inches. A 2-inch by 2-inch square hole is cut into the center of each face. The edges of each cut are parallel to the edges of the cube, and each hole goes all the way through the cube. What is the volume, in cubic inches, of the remaining solid?

一个边长3英寸的实心立方体。在每个面上中心切出一个2英寸×2英寸的正方形孔。每个切口的边平行于立方体的边，每个孔贯穿整个立方体。剩余固体的体积是多少立方英寸？

(A) 7 7

(B) 8 8

(D) 12 12

(E) 15 15

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): The volume of the solid cube is $27\ \text{in}^3$. The first hole to be cut removes $2\times2\times3=12\ \text{in}^3$ from the volume. The other holes remove $2\times2\times0.5=2\ \text{in}^3$ from each of the four remaining faces. The volume of the remaining solid is $27-12-4(2)=7\ \text{in}^3$.

答案（A）：实心立方体的体积是 $27\ \text{in}^3$。切出的第一个孔从总体积中移除 $2\times2\times3=12\ \text{in}^3$。其余的孔从剩下的四个面中每个面移除 $2\times2\times0.5=2\ \text{in}^3$。剩余实体的体积为 $27-12-4(2)=7\ \text{in}^3$。

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