Three cubes of volumes , `1 cm^(3), 216 cm^(3) and 512 cm^(3)` are melted to form a new cube. What is the diagonal of the new cube ?

Answer

To find the diagonal of the new cube formed by melting three smaller cubes, we can follow these steps:

### Step 1: Calculate the total volume of the new cube
The volumes of the three cubes are given as:
- Volume of Cube 1 = \(1 \, cm^3\)
- Volume of Cube 2 = \(216 \, cm^3\)
- Volume of Cube 3 = \(512 \, cm^3\)

To find the total volume of the new cube, we sum the volumes of the three cubes:
\[ \text{Total Volume} = 1 \, cm^3 + 216 \, cm^3 + 512 \, cm^3 \]
\[ \text{Total Volume} = 729 \, cm^3 \]

### Step 2: Find the side length of the new cube
The volume \(V\) of a cube is given by the formula:
\[ V = s^3 \]
where \(s\) is the side length of the cube. We can find the side length of the new cube by taking the cube root of the total volume:
\[ s = \sqrt[3]{729} \]
Calculating the cube root:
\[ s = 9 \, cm \]

### Step 3: Calculate the diagonal of the new cube
The diagonal \(d\) of a cube can be calculated using the formula:
\[ d = s\sqrt{3} \]
Substituting the value of \(s\):
\[ d = 9 \sqrt{3} \, cm \]

### Final Answer
The diagonal of the new cube is \(9\sqrt{3} \, cm\).

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