A rectangular block that is 12 cm x 8 cm x 5 cm is made by stacking individual centimeter cubes into the formation. The base of the block is 12 cm x 8 cm. Everything on the outside of the block is painted except the base. How many of the centimeter cubes will have no face painted ?......... One face painted ?....... Two faces painted ??....... Three faces painted ?..."..
To find the number of centimeter cubes with a particular number of faces painted, we can analyze the dimensions of the rectangular block.
The rectangular block has dimensions of 12 cm x 8 cm x 5 cm. Let's consider each dimension separately.
1. No face painted:
Since the base of the block is excluded, there are no cubes with all six faces painted. However, there will be cubes with no face painted. These cubes will be the ones forming the interior of the block.
To calculate the number of cubes with no face painted, we need to subtract the cubes on the edges and corners from the total number of cubes.
The number of edge cubes can be found by multiplying the length of each edge by the total number of edges (12 edges for 12 cm x 8 cm x 5 cm block).
For the given dimensions, the number of edge cubes on the 12 cm edges is 12 * 5 = 60 cubes and on the 8 cm edges is 8 * 5 = 40 cubes. Therefore, there are 60 + 40 = 100 edge cubes.
The number of corner cubes can be found by multiplying the total number of corners (8 corners for 12 cm x 8 cm x 5 cm block) by 3 since each corner is shared by three edges.
For the given dimensions, 8 * 3 = 24 corner cubes.
Therefore, the total number of cubes with no face painted will be (12 cm * 8 cm * 5 cm) - (100 edge cubes + 24 corner cubes) = 480 - 124 = 356 cubes.
2. One face painted:
To calculate the number of cubes with one face painted, we need to consider the cubes on the outer surface excluding the base (which is not painted).
The total number of cubes on the outer surface can be found by multiplying the perimeter of each face by the height of the block and then subtracting the cubes on the edges and corners.
For the given dimensions, the perimeter of the 12 cm face is 2 * (12 cm + 5 cm) = 34 cm, and the perimeter of the 8 cm face is 2 * (8 cm + 5 cm) = 26 cm.
Therefore, the number of cubes on the outer surface will be (34 cm * 5 cm) + (26 cm * 5 cm) = 170 cm^2 + 130 cm^2 = 300 cm^2.
Again, subtracting the cubes on the edges and corners, we get 300 cm^2 - (100 edge cubes + 24 corner cubes) = 300 cm^2 - 124 cubes.
Hence, the total number of cubes with one face painted is 176 cubes.
3. Two faces painted:
To calculate the number of cubes with two faces painted, we need to consider the cubes on the corners of the block.
For the given dimensions, there are 8 corners, and each corner cube will have two of its faces painted.
Therefore, the total number of cubes with two faces painted is 8 cubes.
4. Three faces painted:
To calculate the number of cubes with three faces painted, we need to consider the cubes on the corners of the block.
For the given dimensions, there are 8 corners, and each corner cube will have three of its faces painted.
Therefore, the total number of cubes with three faces painted is 8 cubes.
In summary:
- Number of cubes with no face painted: 356 cubes
- Number of cubes with one face painted: 176 cubes
- Number of cubes with two faces painted: 8 cubes
- Number of cubes with three faces painted: 8 cubes