Four solid cubes, each of volume 1728 `cm^(3)`, are kept in two rows having two cubes in each row. They form a rectangular solid with square base. The total surface area (in `cm^(2)`) of the resulting solid is:

Options

A. 1440
B. 2304
C. 576
D. 1152

Answer

To find the total surface area of the rectangular solid formed by the four cubes, we will follow these steps:

### Step 1: Calculate the side length of one cube
Given the volume of one cube is \( 1728 \, \text{cm}^3 \).

The formula for the volume of a cube is:
\[ V = a^3 \]
where \( a \) is the side length of the cube.

To find \( a \), we take the cube root of the volume:
\[ a = \sqrt[3]{1728} \]

Calculating this gives:
\[ a = 12 \, \text{cm} \]

### Step 2: Determine the dimensions of the rectangular solid
Since there are 4 cubes arranged in 2 rows with 2 cubes in each row, the dimensions of the rectangular solid will be:
- Length (L) = 2 cubes × side length of one cube = \( 2 \times 12 = 24 \, \text{cm} \)
- Breadth (B) = 2 cubes × side length of one cube = \( 2 \times 12 = 24 \, \text{cm} \)
- Height (H) = height of one cube = \( 12 \, \text{cm} \)

Thus, the dimensions of the rectangular solid are:
- Length = 24 cm
- Breadth = 24 cm
- Height = 12 cm

### Step 3: Calculate the total surface area of the rectangular solid
The formula for the total surface area (SA) of a rectangular solid is:
\[ SA = 2(LB + BH + HL) \]

Substituting the values:
\[ SA = 2(24 \times 24 + 24 \times 12 + 12 \times 24) \]

Calculating each term:
1. \( LB = 24 \times 24 = 576 \)
2. \( BH = 24 \times 12 = 288 \)
3. \( HL = 12 \times 24 = 288 \)

Now substituting back into the surface area formula:
\[ SA = 2(576 + 288 + 288) \]
\[ SA = 2(1152) \]
\[ SA = 2304 \, \text{cm}^2 \]

### Final Answer
The total surface area of the resulting solid is \( 2304 \, \text{cm}^2 \).

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