Three solid iron cubes of edges 4 cm, 5 cm and 6 cm are melted together to make a new cube. `62 cm^(3)` of the melted material is lost area (in `cm^(2)`) of the whole surface of the newly formed cube is

Options

A. 294
B. 343
C. 125
D. 216

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Calculate the volumes of the individual cubes
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
For the cubes with edges 4 cm, 5 cm, and 6 cm:
- Volume of the cube with edge 4 cm:
\[ V_1 = 4^3 = 64 \, \text{cm}^3 \]
- Volume of the cube with edge 5 cm:
\[ V_2 = 5^3 = 125 \, \text{cm}^3 \]
- Volume of the cube with edge 6 cm:
\[ V_3 = 6^3 = 216 \, \text{cm}^3 \]

### Step 2: Calculate the total volume of the three cubes
Now, we add the volumes of the three cubes:
\[ \text{Total Volume} = V_1 + V_2 + V_3 = 64 + 125 + 216 = 405 \, \text{cm}^3 \]

### Step 3: Account for the lost volume
According to the problem, 62 cm³ of the melted material is lost. Therefore, the volume of the new cube will be:
\[ \text{Volume of new cube} = \text{Total Volume} - \text{Lost Volume} = 405 - 62 = 343 \, \text{cm}^3 \]

### Step 4: Find the side length of the new cube
Let \( s \) be the side length of the new cube. The volume of the cube is given by:
\[ s^3 = 343 \]
To find \( s \), we take the cube root of 343:
\[ s = \sqrt[3]{343} = 7 \, \text{cm} \]

### Step 5: Calculate the surface area of the new cube
The surface area \( A \) of a cube is given by the formula:
\[ A = 6 \times s^2 \]
Substituting the side length we found:
\[ A = 6 \times (7^2) = 6 \times 49 = 294 \, \text{cm}^2 \]

### Final Answer
The area of the whole surface of the newly formed cube is:
\[ \boxed{294 \, \text{cm}^2} \]