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

Answer

To solve the problem step by step, we will follow the reasoning presented in the video transcript.

### Step 1: Define the edges of the cubes
Let the edges of the three cubes be represented as:
- First cube: \(3x\)
- Second cube: \(4x\)
- Third cube: \(5x\)

### Step 2: Calculate the volume of each cube
The volume \(V\) of a cube is given by the formula \(V = a^3\), where \(a\) is the length of an edge. Therefore, the volumes of the three cubes can be calculated as follows:
- Volume of the first cube: \(V_1 = (3x)^3 = 27x^3\)
- Volume of the second cube: \(V_2 = (4x)^3 = 64x^3\)
- Volume of the third cube: \(V_3 = (5x)^3 = 125x^3\)

### Step 3: Total volume of the three cubes
The total volume of the three cubes when melted down is:
\[ V_{total} = V_1 + V_2 + V_3 = 27x^3 + 64x^3 + 125x^3 = (27 + 64 + 125)x^3 = 216x^3 \]

### Step 4: Volume of the new cube
The volume of the new cube formed from the melted metal is also given by the formula for the volume of a cube:
\[ V_{new} = a^3 \]
where \(a\) is the edge of the new cube.

### Step 5: Relate the edge of the new cube to its diagonal
We know the diagonal \(d\) of the cube is given as \(12\sqrt{3}\) cm. The diagonal of a cube can be calculated using the formula:
\[ d = a\sqrt{3} \]
Setting this equal to the given diagonal:
\[ a\sqrt{3} = 12\sqrt{3} \]
Dividing both sides by \(\sqrt{3}\):
\[ a = 12 \text{ cm} \]

### Step 6: Calculate the volume of the new cube
Now we can find the volume of the new cube:
\[ V_{new} = a^3 = 12^3 = 1728 \text{ cm}^3 \]

### Step 7: Set the volumes equal
Since the total volume of the three cubes equals the volume of the new cube:
\[ 216x^3 = 1728 \]

### Step 8: Solve for \(x\)
Dividing both sides by 216:
\[ x^3 = \frac{1728}{216} = 8 \]
Taking the cube root:
\[ x = 2 \]

### Step 9: Find the edges of the original 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: \(10 \text{ cm}\)

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