2010 AMC 12A Problem 9

Question

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?

7

8

10

12

15

Answer: A

Concepts:volume inclusion-exclusion cube geometry

Difficulty rating: 1790

Small hint:

Use inclusion-exclusion for the three rectangular holes

Big hint:

The three

2\times2\times3

holes overlap in the central

2\times2\times2

cube

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.

Then the volume of the cutout region is

\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

3

regions.

The remaining volume is then

3^3 - 20 = 27 - 20 = 7.

Thus, A is the correct answer.