Three cubes of metal whose edges are in the ratio 3:4:5 are melted down in to a single cube whose diagonal is `12(sqrt(3))` cm. Find the edges of the three cubes.

Explanation

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

### Step 1: Define the edges of the cubes
Let the edges of the three cubes be represented as \( A \), \( B \), and \( C \). According to the problem, the edges are in the ratio \( 3:4:5 \). We can express the edges in terms of a variable \( x \):
- \( A = 3x \)
- \( B = 4x \)
- \( C = 5x \)

### Step 2: Calculate the volume of the cubes
The volume \( V \) of a cube with edge length \( a \) is given by the formula:
\[ V = a^3 \]
Thus, the volumes of the three cubes are:
- Volume of cube A: \( V_A = (3x)^3 = 27x^3 \)
- Volume of cube B: \( V_B = (4x)^3 = 64x^3 \)
- Volume of cube C: \( V_C = (5x)^3 = 125x^3 \)

### Step 3: Find the total volume of the three cubes
The total volume \( V_{total} \) of the three cubes is:
\[ V_{total} = V_A + V_B + V_C = 27x^3 + 64x^3 + 125x^3 \]
Combining these volumes gives:
\[ V_{total} = (27 + 64 + 125)x^3 = 216x^3 \]

### Step 4: Relate the total volume to the volume of the larger cube
The problem states that the three cubes are melted to form a single larger cube. The diagonal \( d \) of the larger cube is given as \( 12\sqrt{3} \) cm. The relationship between the side length \( s \) of a cube and its diagonal is given by:
\[ d = s\sqrt{3} \]
Setting this equal to the given diagonal:
\[ 12\sqrt{3} = s\sqrt{3} \]
Dividing both sides by \( \sqrt{3} \) gives:
\[ s = 12 \]

### Step 5: Calculate the volume of the larger cube
The volume \( V_{large} \) of the larger cube is:
\[ V_{large} = s^3 = 12^3 = 1728 \text{ cm}^3 \]

### Step 6: Set the total volume equal to the volume of the larger cube
Since the total volume of the three smaller cubes equals the volume of the larger cube, we set:
\[ 216x^3 = 1728 \]

### Step 7: Solve for \( x \)
To find \( x \), divide both sides by 216:
\[ x^3 = \frac{1728}{216} \]
Calculating the right side:
\[ x^3 = 8 \]
Taking the cube root of both sides gives:
\[ x = 2 \]

### Step 8: Calculate the edges of the three cubes
Now we can find the lengths of the edges:
- \( A = 3x = 3 \times 2 = 6 \) cm
- \( B = 4x = 4 \times 2 = 8 \) cm
- \( C = 5x = 5 \times 2 = 10 \) cm

Thus, the edges of the three cubes are:
- Cube A: 6 cm
- Cube B: 8 cm
- Cube C: 10 cm

### Final Answer
The edges of the three cubes are 6 cm, 8 cm, and 10 cm.

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