Three cubes of metal whose edges are in the ratio 3 : 4 : 5, are melted and one cube is formed. If the diagonal of the cube is 12√3 cm, then find the edge of the largest among three cubes.

Solution

Given:
– The ratio of the edges of the three cubes is 3:4:5.
– The diagonal of the new cube formed by melting the three cubes is \(12\sqrt{3}\) cm.

Step 1: Find the edge of the new cube

The diagonal of a cube is related to its edge (\(a\)) by the formula:
\[ \text{Diagonal} = a\sqrt{3} \]

So, the edge of the new cube is:
\[ a = \frac{\text{Diagonal}}{\sqrt{3}} = \frac{12\sqrt{3}}{\sqrt{3}} = 12 \text{ cm} \]

Step 2: Find the volume of the new cube

The volume of the new cube is:
\[ V = a^3 = 12^3 = 1728 \text{ cm}^3 \]

Step 3: Find the edge of the largest original cube

Let the edges of the three original cubes be \(3x\), \(4x\), and \(5x\) respectively. The volume of the largest cube is:
\[ V_{\text{largest}} = (5x)^3 = 125x^3 \]

The total volume of the three cubes is equal to the volume of the new cube:
\[ 3^3x^3 + 4^3x^3 + 5^3x^3 = 1728 \]
\[ 27x^3 + 64x^3 + 125x^3 = 1728 \]
\[ 216x^3 = 1728 \]
\[ x^3 = 8 \]
\[ x = 2 \]

So, the edge of the largest cube is:
\[ 5x = 5 \times 2 = 10 \text{ cm} \]

Conclusion

The edge of the largest among the three cubes is 10 cm.