The volume of a cuboid is twice the volume of a cube. If the dimensions of the cuboid are 9 cm, 8 c and 6 cm, the total surface are of the cube is :

Options

A. `72 cm^(2)`
B. `216 cm^(2)`
C. `432 cm^(2)`
D. `108 cm^(2)`

Answer

To solve the problem, we need to find the total surface area of a cube given that the volume of a cuboid is twice the volume of that cube. The dimensions of the cuboid are provided as 9 cm, 8 cm, and 6 cm.

### Step-by-Step Solution:

1. **Calculate the Volume of the Cuboid:**
The formula for the volume of a cuboid is given by:
\[ \text{Volume of Cuboid} = \text{Length} \times \text{Breadth} \times \text{Height} \]
Substituting the given dimensions:
\[ \text{Volume of Cuboid} = 9 \, \text{cm} \times 8 \, \text{cm} \times 6 \, \text{cm} \]
\[ = 432 \, \text{cm}^3 \]

2. **Relate the Volume of the Cuboid to the Volume of the Cube:**
According to the problem, the volume of the cuboid is twice the volume of the cube:
\[ \text{Volume of Cuboid} = 2 \times \text{Volume of Cube} \]
Therefore, we can express the volume of the cube as:
\[ \text{Volume of Cube} = \frac{\text{Volume of Cuboid}}{2} = \frac{432 \, \text{cm}^3}{2} = 216 \, \text{cm}^3 \]

3. **Find the Side Length of the Cube:**
The volume of a cube is given by:
\[ \text{Volume of Cube} = \text{side}^3 \]
Setting this equal to the volume we found:
\[ \text{side}^3 = 216 \, \text{cm}^3 \]
To find the side length, we take the cube root:
\[ \text{side} = \sqrt[3]{216} \]
Since \(6 \times 6 \times 6 = 216\), we find:
\[ \text{side} = 6 \, \text{cm} \]

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

### Final Answer:
The total surface area of the cube is:
\[ \text{TSA} = 216 \, \text{cm}^2 \]