A 3 cm cube is painted on all its faces.It is then cut into 1 cm cubes.How many cubes will have paint on exactly 2 faces?
To determine the number of 1 cm cubes that will have paint on exactly 2 faces, we need to break down the problem step by step.
Step 1: Calculate the number of 1 cm cubes that make up the original 3 cm cube.
Since the 3 cm cube is made up of smaller 1 cm cubes, we can calculate the total number of 1 cm cubes by multiplying the length, width, and height of the larger cube.
In this case, each side of the 3 cm cube is 3 cm long, so we have:
Number of 1 cm cubes = (3 cm) x (3 cm) x (3 cm) = 27 cubes
Therefore, the original 3 cm cube is composed of 27 smaller 1 cm cubes.
Step 2: Determine the number of 1 cm cubes that will have paint on exactly 2 faces.
To visualize this, let's think about the original 3 cm cube and focus on the cubes on the edge. These edge cubes are the ones that are most likely to have paint on exactly 2 faces since they have exposed side faces.
Each edge of the 3 cm cube consists of 3 smaller 1 cm cubes (since each side is 3 cm long). There are 12 edges on a cube (each side has 4 edges), so we have:
Number of edge 1 cm cubes = 12 edges x 3 cubes per edge = 36 edge cubes
Now, all the remaining 1 cm cubes in the interior of the 3 cm cube will have paint on all three exposed faces (top, bottom, and sides). These interior cubes have no exposed faces where paint could be present on exactly 2 faces.
So, the number of 1 cm cubes that will have paint on exactly 2 faces is:
Number of cubes with paint on 2 faces = Total number of cubes - Number of interior cubes
= 27 cubes - 36 edge cubes
= -9 cubes
However, we cannot have a negative number of cubes, so in this case, there are no 1 cm cubes that will have paint on exactly 2 faces.
Thus, the answer to the question is 0 cubes.