Answer :
To find the surface area of the exposed faces when three cubes are placed in a row with no gaps in between them, and each cube has a side length of 4.6 cm, follow these steps:
### Step 1: Calculate the surface area of a single cube
The total surface area of a single cube is calculated by using the formula for the surface area of a cube:
[tex]\[ \text{Surface area of a cube} = 6 \times (\text{side length})^2 \][/tex]
Given that the side length of the cube is 4.6 cm:
[tex]\[ \text{Surface area of one cube} = 6 \times (4.6)^2 \][/tex]
### Step 2: Analyze the exposure of faces when cubes are placed in a row
- Middle cube:
The middle cube in a row of three cubes has two faces that are adjacent to other cubes, which means these two faces are not exposed. Therefore, 4 faces of the middle cube are exposed.
- End cubes:
Each of the cubes placed at the ends have one face adjacent to another cube, leaving 5 faces exposed each.
### Step 3: Calculate the total number of exposed faces
When considering all three cubes together:
- The two end cubes each have 5 exposed faces.
- The middle cube has 4 exposed faces.
So, the total number of exposed faces is:
[tex]\[ 2 \text{ (end cubes)} \times 5 \text{ (faces each)} + 4 \text{ (faces of middle cube)} = 10 + 4 = 14 \text{ (faces)} \][/tex]
### Step 4: Calculate the exposed surface area
Each face of the cube has an area of:
[tex]\[ \text{Area of one face} = (\text{side length})^2 \][/tex]
Given the side length of 4.6 cm:
[tex]\[ \text{Area of one face} = (4.6)^2 \][/tex]
So, the total exposed surface area of the cubes, considering the 14 exposed faces, is:
[tex]\[ \text{Exposed surface area} = 14 \times (4.6)^2 \][/tex]
### Final Result
The total surface area of one cube, which has 6 faces each of (4.6 cm)^2, is approximately 126.96 cm^2. The total exposed surface area of the three cubes when placed in a row, considering 14 faces each of (4.6 cm)^2, is approximately 296.24 cm^2.
### Step 1: Calculate the surface area of a single cube
The total surface area of a single cube is calculated by using the formula for the surface area of a cube:
[tex]\[ \text{Surface area of a cube} = 6 \times (\text{side length})^2 \][/tex]
Given that the side length of the cube is 4.6 cm:
[tex]\[ \text{Surface area of one cube} = 6 \times (4.6)^2 \][/tex]
### Step 2: Analyze the exposure of faces when cubes are placed in a row
- Middle cube:
The middle cube in a row of three cubes has two faces that are adjacent to other cubes, which means these two faces are not exposed. Therefore, 4 faces of the middle cube are exposed.
- End cubes:
Each of the cubes placed at the ends have one face adjacent to another cube, leaving 5 faces exposed each.
### Step 3: Calculate the total number of exposed faces
When considering all three cubes together:
- The two end cubes each have 5 exposed faces.
- The middle cube has 4 exposed faces.
So, the total number of exposed faces is:
[tex]\[ 2 \text{ (end cubes)} \times 5 \text{ (faces each)} + 4 \text{ (faces of middle cube)} = 10 + 4 = 14 \text{ (faces)} \][/tex]
### Step 4: Calculate the exposed surface area
Each face of the cube has an area of:
[tex]\[ \text{Area of one face} = (\text{side length})^2 \][/tex]
Given the side length of 4.6 cm:
[tex]\[ \text{Area of one face} = (4.6)^2 \][/tex]
So, the total exposed surface area of the cubes, considering the 14 exposed faces, is:
[tex]\[ \text{Exposed surface area} = 14 \times (4.6)^2 \][/tex]
### Final Result
The total surface area of one cube, which has 6 faces each of (4.6 cm)^2, is approximately 126.96 cm^2. The total exposed surface area of the three cubes when placed in a row, considering 14 faces each of (4.6 cm)^2, is approximately 296.24 cm^2.
