Three metal cubes with edges 6cm , 8cm and 10cm respectively are melted together and formed in to a single cube. Find the diagonal of this cube.

Answer

To solve the problem of finding the diagonal of a new cube formed by melting three smaller cubes with edges of 6 cm, 8 cm, and 10 cm, we can follow these steps:

### Step-by-Step Solution:

1. **Calculate the Volume of Each Cube:**
- The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
- For the first cube with edge 6 cm:
\[ V_1 = 6^3 = 216 \, \text{cm}^3 \]
- For the second cube with edge 8 cm:
\[ V_2 = 8^3 = 512 \, \text{cm}^3 \]
- For the third cube with edge 10 cm:
\[ V_3 = 10^3 = 1000 \, \text{cm}^3 \]

2. **Calculate the Total Volume of the New Cube:**
- The total volume \( V \) of the new cube formed by melting the three smaller cubes is the sum of their volumes:
\[ V = V_1 + V_2 + V_3 = 216 + 512 + 1000 = 1728 \, \text{cm}^3 \]

3. **Determine the Side Length of the New Cube:**
- Let \( A \) be the side length of the new cube. Since the volume of a cube is also given by \( A^3 \), we can set up the equation:
\[ A^3 = 1728 \]
- To find \( A \), take the cube root of 1728:
\[ A = \sqrt[3]{1728} = 12 \, \text{cm} \]

4. **Calculate the Diagonal of the New Cube:**
- The diagonal \( D \) of a cube can be calculated using the formula:
\[ D = A \sqrt{3} \]
- Substituting the value of \( A \):
\[ D = 12 \sqrt{3} \, \text{cm} \]

### Final Answer:
The diagonal of the new cube is \( 12 \sqrt{3} \, \text{cm} \).

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