The edges of three cubes of metal are 3cm, 4cm and 5cm. They are melted and formed into a single cube. Find the edge of the new cube.

Answer

To find the edge of the new cube formed by melting three cubes with edges of 3 cm, 4 cm, and 5 cm, we can follow these steps:

### Step 1: Calculate the Volume of Each Cube
1. **Volume of the first cube (edge = 3 cm)**:
\[ V_1 = a^3 = 3^3 = 27 \text{ cm}^3 \]

2. **Volume of the second cube (edge = 4 cm)**:
\[ V_2 = a^3 = 4^3 = 64 \text{ cm}^3 \]

3. **Volume of the third cube (edge = 5 cm)**:
\[ V_3 = a^3 = 5^3 = 125 \text{ cm}^3 \]

### Step 2: Calculate the Total Volume
Now, we add the volumes of the three cubes to find the total volume:
\[ \text{Total Volume} = V_1 + V_2 + V_3 = 27 + 64 + 125 \]
Calculating this gives:
\[ \text{Total Volume} = 216 \text{ cm}^3 \]

### Step 3: Find the Edge of the New Cube
Let the edge of the new cube be \( x \). The volume of the new cube can be expressed as:
\[ \text{Volume of new cube} = x^3 \]
Since the total volume of the new cube is equal to the total volume we calculated:
\[ x^3 = 216 \]

### Step 4: Solve for \( x \)
To find \( x \), we take the cube root of both sides:
\[ x = \sqrt[3]{216} \]
Since \( 216 = 6^3 \), we find:
\[ x = 6 \text{ cm} \]

### Conclusion
The edge of the new cube formed by melting the three cubes is **6 cm**.

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