Three cubes of sides 6 cm, 8 cm and 1 cm are melted to form a new cube. The surface area of the new cube is

Options

A. `486 cm^(2)`
B. `496 cm^(2)`
C. `586 cm^(2)`
D. `658 cm^(2)`

Answer

To find the surface area of the new cube formed by melting three smaller cubes with sides of 6 cm, 8 cm, and 1 cm, we will follow these steps:

### Step 1: Calculate the volume of each cube
The volume \( V \) of a cube is given by the formula:
\[ V = \text{side}^3 \]
- For the cube with side 6 cm:
\[ V_1 = 6^3 = 216 \text{ cm}^3 \]
- For the cube with side 8 cm:
\[ V_2 = 8^3 = 512 \text{ cm}^3 \]
- For the cube with side 1 cm:
\[ V_3 = 1^3 = 1 \text{ cm}^3 \]

### Step 2: Calculate the total volume of the three cubes
Now, we sum the volumes of the three cubes:
\[ \text{Total Volume} = V_1 + V_2 + V_3 = 216 + 512 + 1 = 729 \text{ cm}^3 \]

### Step 3: Find the side length of the new cube
Let the side length of the new cube be \( a \). The volume of the new cube can be expressed as:
\[ a^3 = 729 \]
To find \( a \), we take the cube root of 729:
\[ a = \sqrt[3]{729} = 9 \text{ cm} \]

### Step 4: Calculate the surface area of the new cube
The surface area \( S \) of a cube is given by the formula:
\[ S = 6a^2 \]
Substituting \( a = 9 \) cm into the formula:
\[ S = 6 \times (9^2) = 6 \times 81 = 486 \text{ cm}^2 \]

### Final Answer
The surface area of the new cube is \( 486 \text{ cm}^2 \).

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