The length, breadth and height of a cuboid are in the raio 3:4:6 and its volume is `576 cm^(3)`. The whole surface of the cuboid is

Options

A. `216 cm^(2)`
B. `324 cm^(2)`
C. `432 cm^(2)`
D. `460 cm^(2)`

Answer

To find the total surface area of a cuboid given the ratio of its dimensions and its volume, we can follow these steps:

### Step 1: Understand the given ratio
The length (L), breadth (B), and height (H) of the cuboid are in the ratio 3:4:6. We can express these dimensions in terms of a variable \( x \):
- Length \( L = 3x \)
- Breadth \( B = 4x \)
- Height \( H = 6x \)

### Step 2: Use the volume formula
The volume \( V \) of a cuboid is given by the formula:
\[ V = L \times B \times H \]
Substituting the expressions for L, B, and H:
\[ V = (3x) \times (4x) \times (6x) = 72x^3 \]
We know the volume is \( 576 \, \text{cm}^3 \), so we set up the equation:
\[ 72x^3 = 576 \]

### Step 3: Solve for \( x \)
To find \( x \), we divide both sides by 72:
\[ x^3 = \frac{576}{72} \]
Calculating the right side:
\[ x^3 = 8 \]
Taking the cube root of both sides gives:
\[ x = 2 \]

### Step 4: Calculate the dimensions
Now we can find the actual dimensions of the cuboid:
- Length \( L = 3x = 3 \times 2 = 6 \, \text{cm} \)
- Breadth \( B = 4x = 4 \times 2 = 8 \, \text{cm} \)
- Height \( H = 6x = 6 \times 2 = 12 \, \text{cm} \)

### Step 5: Calculate the total surface area
The total surface area (TSA) of a cuboid is given by the formula:
\[ \text{TSA} = 2(LB + BH + HL) \]
Substituting the values of L, B, and H:
\[ \text{TSA} = 2(6 \times 8 + 8 \times 12 + 12 \times 6) \]
Calculating each term:
- \( LB = 6 \times 8 = 48 \)
- \( BH = 8 \times 12 = 96 \)
- \( HL = 12 \times 6 = 72 \)

Now, substituting back into the TSA formula:
\[ \text{TSA} = 2(48 + 96 + 72) = 2(216) = 432 \, \text{cm}^2 \]

### Final Answer
The total surface area of the cuboid is \( 432 \, \text{cm}^2 \).

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