2007 AMC 10A Problem 21

Below is the professionally curated solution for Problem 21 of the 2007 AMC 10A, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2007 AMC 10A solutions, or check the answer key.

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Concepts:cube geometry sphere surface area

Difficulty rating: 1580

21.

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?

$3$

$6$

$8$

$9$

$12$

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.

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