Four identical cubes are joined end to end to form a cuboid. If the total surface area of the resulting cuboid is 648 `cm^(2)`, find the length of edge of each cube.
Also, find the ratio between the surface area of the resulting cuboid and the surface area of a cube.

Answer

To solve the problem step by step, we can follow these instructions:

### Step 1: Define the side length of the cube
Let the length of the edge of each cube be \( a \).

### Step 2: Determine the dimensions of the cuboid
Since four identical cubes are joined end to end, the dimensions of the resulting cuboid will be:
- Length \( L = 4a \) (since there are 4 cubes)
- Breadth \( B = a \) (same as the side of the cube)
- Height \( H = a \) (same as the side of the cube)

### Step 3: Write the formula for the surface area of the cuboid
The total surface area \( S \) of a cuboid is given by the formula:
\[ S = 2(LB + BH + HL) \]
Substituting the dimensions of the cuboid into the formula:
\[ S = 2(4a \cdot a + a \cdot a + a \cdot 4a) \]
This simplifies to:
\[ S = 2(4a^2 + a^2 + 4a^2) = 2(9a^2) = 18a^2 \]

### Step 4: Set the surface area equal to the given value
We know from the problem that the total surface area of the cuboid is 648 cm². Therefore, we can set up the equation:
\[ 18a^2 = 648 \]

### Step 5: Solve for \( a^2 \)
To find \( a^2 \), divide both sides by 18:
\[ a^2 = \frac{648}{18} = 36 \]

### Step 6: Find \( a \)
Now, take the square root of both sides to find \( a \):
\[ a = \sqrt{36} = 6 \text{ cm} \]

### Step 7: Calculate the surface area of one cube
The surface area \( S_c \) of one cube is given by:
\[ S_c = 6a^2 \]
Substituting \( a = 6 \):
\[ S_c = 6 \cdot (6^2) = 6 \cdot 36 = 216 \text{ cm}^2 \]

### Step 8: Find the ratio of the surface area of the cuboid to the surface area of the cube
Now, we can find the ratio of the surface area of the cuboid to the surface area of the cube:
\[ \text{Ratio} = \frac{648}{216} = 3 \]
Thus, the ratio is \( 3:1 \).

### Final Answers
- The length of the edge of each cube is \( 6 \text{ cm} \).
- The ratio between the surface area of the resulting cuboid and the surface area of a cube is \( 3:1 \).