125 smaller but identical cubes have been put together to form a larger cube. This larger cube is now painted on all 6 faces2.How many of the smaller cubes have exactly one face painted?a.49 b. 54 c. 64 d. 72
Question
125 smaller but identical cubes have been put together to form a larger cube. This larger cube is now painted on all 6 faces2.How many of the smaller cubes have exactly one face painted?a.49 b. 54 c. 64 d. 72
Solution 1
To solve this problem, we need to understand the structure of the larger cube and how the smaller cubes are arranged within it.
Since the larger cube is formed by putting together 125 smaller cubes, we know that it has 5 smaller cubes on each edge (since 5 x 5 x 5 = 125).
Now, let's consider the smaller cubes on the surface of the larger cube. Since all 6 faces of the larger cube are painted, we need to determine how many smaller cubes are on the surface.
Each face of the larger cube has 5 x 5 = 25 smaller cubes on it. Since there are 6 faces, the total number of smaller cubes on the surface is 6 x 25 = 150.
However, we need to find the number of smaller cubes that have exactly one face painted. To do this, we need to subtract the smaller cubes on the edges and corners of the larger cube.
Each edge of the larger cube has 5 smaller cubes, and there are 12 edges in total. So, the number of smaller cubes on the edges is 5 x 12 = 60.
Each corner of the larger cube has 1 smaller cube, and there are 8 corners in total. So, the number of smaller cubes on the corners is 1 x 8 = 8.
To find the number of smaller cubes with exactly one face painted, we subtract the cubes on the edges and corners from the total number of cubes on the surface: 150 - 60 - 8 = 82.
Therefore, the correct answer is not provided in the options given.
Solution 2
To solve this problem, we need to understand the structure of the larger cube and how the smaller cubes are arranged within it.
Since the larger cube is formed by putting together 125 smaller cubes, we know that it has 5 smaller cubes on each edge (since 5 x 5 x 5 = 125).
Now, let's consider the smaller cubes on the surface of the larger cube. Since all 6 faces of the larger cube are painted, we need to determine how many smaller cubes are on the surface.
Each face of the larger cube has 5 x 5 = 25 smaller cubes on it. Since there are 6 faces, the total number of smaller cubes on the surface is 6 x 25 = 150.
However, we need to find the number of smaller cubes that have exactly one face painted. To do this, we need to subtract the smaller cubes on the edges and corners of the larger cube.
Each edge of the larger cube has 5 smaller cubes, and there are 12 edges in total. So, the number of smaller cubes on the edges is 5 x 12 = 60.
Each corner of the larger cube has 1 smaller cube, and there are 8 corners in total. So, the number of smaller cubes on the corners is 1 x 8 = 8.
To find the number of smaller cubes with exactly one face painted, we subtract the cubes on the edges and corners from the total number of cubes on the surface: 150 - 60 - 8 = 82.
Therefore, the correct answer is not provided in the options given.
Solution 3
To solve this problem, we need to understand the structure of the larger cube and how the smaller cubes are arranged within it.
Since the larger cube is formed by putting together 125 smaller cubes, we know that it has 5 smaller cubes on each edge (since 5 x 5 x 5 = 125).
Now, let's consider the smaller cubes on the surface of the larger cube. Since all 6 faces of the larger cube are painted, we need to determine how many smaller cubes are on the surface.
Each face of the larger cube has 5 x 5 = 25 smaller cubes on it. Since there are 6 faces, the total number of smaller cubes on the surface is 6 x 25 = 150.
However, we need to find the number of smaller cubes that have exactly one face painted. To do this, we need to subtract the smaller cubes on the edges and corners of the larger cube.
Each edge of the larger cube has 5 smaller cubes, and there are 12 edges in total. So, the number of smaller cubes on the edges is 5 x 12 = 60.
Each corner of the larger cube has 1 smaller cube, and there are 8 corners in total. So, the number of smaller cubes on the corners is 1 x 8 = 8.
To find the number of smaller cubes with exactly one face painted, we subtract the cubes on the edges and corners from the total number of cubes on the surface: 150 - 60 - 8 = 82.
Therefore, the correct answer is not provided in the options given.
Solution 4
To solve this problem, we need to understand the structure of the larger cube and how the smaller cubes are arranged within it.
Since the larger cube is formed by putting together 125 smaller cubes, we know that it has 5 smaller cubes on each edge (since 5 x 5 x 5 = 125).
Now, let's consider the smaller cubes on the surface of the larger cube. Since all 6 faces of the larger cube are painted, we need to determine how many smaller cubes are on the surface.
Each face of the larger cube has 5 x 5 = 25 smaller cubes on it. Since there are 6 faces, the total number of smaller cubes on the surface is 6 x 25 = 150.
However, we need to find the number of smaller cubes that have exactly one face painted. To do this, we need to subtract the smaller cubes on the edges and corners of the larger cube.
Each edge of the larger cube has 5 smaller cubes, and there are 12 edges in total. So, the number of smaller cubes on the edges is 5 x 12 = 60.
Each corner of the larger cube has 1 smaller cube, and there are 8 corners in total. So, the number of smaller cubes on the corners is 1 x 8 = 8.
To find the number of smaller cubes with exactly one face painted, we subtract the cubes on the edges and corners from the total number of cubes on the surface: 150 - 60 - 8 = 82.
Therefore, the correct answer is not provided in the options given.
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