2007 AMC 10A 第 21 题

Question

一个球内切于表面积为

24

平方米的立方体。再将第二个立方体内接于这个球。内部立方体的表面积是多少平方米？

A sphere is inscribed in a cube that has a surface area of

24

square meters. A second cube is then inscribed within the sphere. What is the surface area in square meters of the inner cube?

3

6

8

9

12

球的直径等于外部立方体的边长，也等于内部立方体的空间对角线。

The sphere’s diameter equals the outer cube’s side and the inner cube’s space diagonal

若内部立方体边长为

l

，则空间对角线为

l\sqrt3 = 2

。

For inner cube side

l,

the space diagonal is

l\sqrt3 = 2

外部立方体每个面的面积为

24 \div 6 = 4

，所以边长为

2

，球的直径为

2

。

这个直径是内部立方体的空间对角线，所以

l\sqrt3 = 2

，得到

l^2 = \tfrac43

。

内部立方体表面积为

6 l^2 = 6 \cdot \tfrac43 = 8

。

所以正确答案是 C。

Each face of the outer cube has area

24 \div 6 = 4,

so its side is

2,

and the sphere has diameter

2.

This diameter is the space diagonal of the inner cube, so

l\sqrt3 = 2,

giving

l^2 = \tfrac43.

The inner cube’s surface area is

6 l^2 = 6 \cdot \tfrac43 = 8.

Thus, the correct answer is C.

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