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

Options

A. 243 sq cm
B. 463 sq cm
C. 486 sq cm
D. 293 sq cm

Answer

To solve the problem of finding half of the surface area of a new cube formed by melting three cubes of sides 1 cm, 6 cm, and 8 cm, we can follow these steps:

### 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 cube with side 1 cm:
\[ V_1 = 1^3 = 1 \text{ cm}^3 \]
- For the cube with side 6 cm:
\[ V_2 = 6^3 = 216 \text{ cm}^3 \]
- For the cube with side 8 cm:
\[ V_3 = 8^3 = 512 \text{ cm}^3 \]

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

### Step 3: Find the side length of the new cube
Since the total volume of the new cube is equal to the total volume of the three melted cubes, we can find the side length \( s \) of the new cube using the formula for the volume of a cube:
\[ V = s^3 \]
Setting the total volume equal to \( s^3 \):
\[ s^3 = 729 \]
To find \( s \), we take the cube root:
\[ s = \sqrt[3]{729} = 9 \text{ cm} \]

### Step 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 of the new cube:
\[ A = 6 \times 9^2 = 6 \times 81 = 486 \text{ cm}^2 \]

### Step 5: Find half of the surface area
To find half of the surface area:
\[ \text{Half of Surface Area} = \frac{A}{2} = \frac{486}{2} = 243 \text{ cm}^2 \]

### Final Answer
Thus, half of the surface area of the new cube is:
\[ \boxed{243 \text{ cm}^2} \]