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 step by step, we will first find the volumes of the three cubes, then combine those volumes to find the edge length of the new cube formed by melting them together, and finally calculate the diagonal of that new cube.

### Step 1: Calculate the volumes of the individual cubes.
The volume \( V \) of a cube is given by the formula:
\[ V = a^3 \]
where \( a \) is the edge length of the cube.

1. For the cube with edge length 6 cm:
\[ V_1 = 6^3 = 216 \text{ cm}^3 \]

2. For the cube with edge length 8 cm:
\[ V_2 = 8^3 = 512 \text{ cm}^3 \]

3. For the cube with edge length 10 cm:
\[ V_3 = 10^3 = 1000 \text{ cm}^3 \]

### Step 2: Add the volumes of the three cubes.
Now, we will add the volumes of the three cubes to find the total volume:
\[ V_{\text{total}} = V_1 + V_2 + V_3 = 216 + 512 + 1000 \]
Calculating this gives:
\[ V_{\text{total}} = 1728 \text{ cm}^3 \]

### Step 3: Find the edge length of the new cube.
Let \( A \) be the edge length of the new cube. The volume of the new cube can also be expressed as:
\[ V_{\text{total}} = A^3 \]
Setting the two expressions for volume equal to each other, we have:
\[ A^3 = 1728 \]
To find \( A \), we take the cube root:
\[ A = \sqrt[3]{1728} = 12 \text{ cm} \]

### Step 4: Calculate the diagonal of the new cube.
The diagonal \( D \) of a cube with edge length \( A \) is given by the formula:
\[ D = A \sqrt{3} \]
Substituting \( A = 12 \text{ cm} \):
\[ D = 12 \sqrt{3} \text{ cm} \]

### Final Answer:
The diagonal of the new cube is:
\[ \boxed{12 \sqrt{3} \text{ cm}} \]