2007 AMC 10A Problem 21

Problem 21 of 25HarderGeometry

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?

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Solution

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.