Three cubes of metal , whose edges are 3cm, 4cm and 5 cm respectively are melted to form a new cube. What is the surface area of the new cube ?

Options

A. `216 cm^(2) `
B. `56 cm^(2) `
C. `36 cm^(2) `
D. none of these

Answer

To solve the problem of finding the surface area of a new cube formed by melting three smaller cubes with edges of 3 cm, 4 cm, and 5 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 (edge = 3 cm):
\[ V_1 = 3^3 = 27 \text{ cm}^3 \]
- For the second cube (edge = 4 cm):
\[ V_2 = 4^3 = 64 \text{ cm}^3 \]
- For the third cube (edge = 5 cm):
\[ V_3 = 5^3 = 125 \text{ cm}^3 \]

2. **Calculate the Total Volume of the Three Cubes**:
- The total volume \( V_{total} \) is the sum of the volumes of the three cubes:
\[ V_{total} = V_1 + V_2 + V_3 = 27 + 64 + 125 = 216 \text{ cm}^3 \]

3. **Determine the Side Length of the New Cube**:
- The volume of the new cube (which is formed by melting the three cubes) is equal to the total volume calculated above. Let the side length of the new cube be \( s \). Therefore:
\[ s^3 = V_{total} = 216 \text{ cm}^3 \]
- To find \( s \), we take the cube root of 216:
\[ s = \sqrt[3]{216} = 6 \text{ cm} \]

4. **Calculate the Surface Area of the New Cube**:
- The surface area \( A \) of a cube is given by the formula:
\[ A = 6 \times \text{side}^2 \]
- Substituting the side length we found:
\[ A = 6 \times 6^2 = 6 \times 36 = 216 \text{ cm}^2 \]

### Final Answer:
The surface area of the new cube is **216 cm²**.