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2010 AMC 12A Problem 9

Problem 9 of 25EasierGeometryCombinatorics

A solid cube has side length 33 inches. A 22-inch by 22-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?

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Solution

Note that all the cut out solids intersect in the middle of the cube. This region of intersection is a cube with side length 2.2. Then the volume of the cutout region is 3⋅2⋅2⋅3−2⋅23=36−16=20. \begin{align*}3 \cdot 2 \cdot 2 \cdot 3 - 2 \cdot 2^3 &= 36 - 16 \\&= 20.\end{align*} We have to subtract out the center region twice since it is included in all 33 regions. The remaining volume is then 33−20=27−20=7. 3^3 - 20 = 27 - 20 = 7. Thus, A is the correct answer.
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Tagged: volume · inclusion-exclusion · cube geometry

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