Three cubes of a metal whose edge are in the ratio 3:4:5 are melted and converted into a single cube whose diagonal is `12 sqrt3` cm. Find the edge of three cubes

Answer

To solve the problem, we need to find the edges of three cubes whose edges are in the ratio 3:4:5, given that they are melted into a single cube with a diagonal of \(12\sqrt{3}\) cm.

### Step-by-Step Solution:

1. **Understanding the Diagonal of a Cube:**
The diagonal \(d\) of a cube with edge length \(A\) can be calculated using the formula:
\[ d = A\sqrt{3} \]
Given that the diagonal of the new cube is \(12\sqrt{3}\) cm, we can set up the equation:
\[ A\sqrt{3} = 12\sqrt{3} \]

2. **Finding the Edge Length of the New Cube:**
To find \(A\), we can divide both sides of the equation by \(\sqrt{3}\):
\[ A = 12 \text{ cm} \]

3. **Calculating the Volume of the New Cube:**
The volume \(V\) of a cube is given by:
\[ V = A^3 \]
Substituting \(A = 12\):
\[ V = 12^3 = 1728 \text{ cm}^3 \]

4. **Setting Up the Ratio of the Edges of the Smaller Cubes:**
Let the edges of the three smaller cubes be \(3x\), \(4x\), and \(5x\). The volume of each cube can be calculated as follows:
- Volume of the first cube: \((3x)^3 = 27x^3\)
- Volume of the second cube: \((4x)^3 = 64x^3\)
- Volume of the third cube: \((5x)^3 = 125x^3\)

5. **Finding the Total Volume of the Smaller Cubes:**
The total volume of the three cubes is:
\[ 27x^3 + 64x^3 + 125x^3 = 216x^3 \]

6. **Setting the Total Volume Equal to the Volume of the New Cube:**
Since the total volume of the smaller cubes equals the volume of the new cube, we have:
\[ 216x^3 = 1728 \]

7. **Solving for \(x^3\):**
Dividing both sides by 216:
\[ x^3 = \frac{1728}{216} = 8 \]

8. **Finding \(x\):**
Taking the cube root of both sides:
\[ x = 2 \text{ cm} \]

9. **Calculating the Edges of the Smaller Cubes:**
Now we can find the edges of the three cubes:
- Edge of the first cube: \(3x = 3 \times 2 = 6 \text{ cm}\)
- Edge of the second cube: \(4x = 4 \times 2 = 8 \text{ cm}\)
- Edge of the third cube: \(5x = 5 \times 2 = 10 \text{ cm}\)

### Final Answer:
The edges of the three cubes are:
- First cube: \(6 \text{ cm}\)
- Second cube: \(8 \text{ cm}\)
- Third cube: \(10 \text{ cm}\)