Question

as shown below, how many more small cubes are still needed to complete the 3×3×3 cube? a. 7 b. 8 c. 9 d. 10

Explanation:

Step1: Calculate the total number of small cubes in a \(3\times3\times3\) cube

The formula for the volume of a cube is \(V = n\times n\times n\) (where \(n\) is the number of small cubes along one edge). For \(n = 3\), the total number of small cubes \(N_{total}=3\times3\times3=27\).

Step2: Count the number of existing small cubes

By counting the cubes in the given figure:

• In the bottom - layer: \(9\) cubes.
• In the middle - layer: \(7\) cubes.
• In the top - layer: \(6\) cubes.

The number of existing cubes \(N_{existing}=9 + 7+6=22\).

Step3: Calculate the number of additional cubes

The number of additional cubes \(N = N_{total}-N_{existing}\). Substitute \(N_{total}=27\) and \(N_{existing}=22\) into the formula, we get \(N=27 - 22=5\). Wait, this is wrong. Let's count again.
Another way:

• A \(3\times3\times3\) cube has \(3\) layers. Each layer should have \(9\) cubes.
• The bottom layer is complete (\(9\) cubes).
• The middle layer is missing \(2\) cubes (\(9-7 = 2\)).
• The top layer is missing \(3\) cubes (\(9 - 6=3\)).

The total number of missing cubes \(=2 + 3+ 4=9\) (because we can also think in terms of columns. There are \(4\) columns that are incomplete. One column is missing \(2\) cubes (middle and top), one column is missing \(2\) cubes (middle and top), and two columns are missing \(1\) cube (top - layer only). \(2+2 + 1+1+3=9\))

Answer:

C. 9