If the surface area of a cuboid is 3328 `m^2` . Its dimensions are in the ratio 4 : 3 : 2, then the volume of the cuboid is

Options

A. 12288 `m^3`
B. 11288 `m^3`
C. 12882 `m^3`
D. 18388 `m^3`

Answer

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

### Step 1: Define the dimensions of the cuboid
Given the ratio of the dimensions is 4:3:2, we can express the length (L), breadth (B), and height (H) in terms of a variable \( x \):
- Length \( L = 4x \)
- Breadth \( B = 3x \)
- Height \( H = 2x \)

### Step 2: Write the formula for the surface area of a cuboid
The formula for the surface area (SA) of a cuboid is:
\[ SA = 2(LB + BH + HL) \]
Substituting the expressions for L, B, and H:
\[ SA = 2((4x)(3x) + (3x)(2x) + (2x)(4x)) \]

### Step 3: Calculate the surface area expression
Now we will calculate each term:
- \( LB = (4x)(3x) = 12x^2 \)
- \( BH = (3x)(2x) = 6x^2 \)
- \( HL = (2x)(4x) = 8x^2 \)

Adding these together:
\[ LB + BH + HL = 12x^2 + 6x^2 + 8x^2 = 26x^2 \]

Thus, the surface area becomes:
\[ SA = 2(26x^2) = 52x^2 \]

### Step 4: Set the surface area equal to the given value
We know the surface area is given as 3328 m², so we set up the equation:
\[ 52x^2 = 3328 \]

### Step 5: Solve for \( x^2 \)
To find \( x^2 \), divide both sides by 52:
\[ x^2 = \frac{3328}{52} \]

Calculating the right side:
\[ x^2 = 64 \]

### Step 6: Find \( x \)
Taking the square root of both sides:
\[ x = 8 \]

### Step 7: Calculate the dimensions of the cuboid
Now that we have \( x \), we can find the dimensions:
- Length \( L = 4x = 4 \times 8 = 32 \, m \)
- Breadth \( B = 3x = 3 \times 8 = 24 \, m \)
- Height \( H = 2x = 2 \times 8 = 16 \, m \)

### Step 8: Calculate the volume of the cuboid
The volume \( V \) of the cuboid is given by:
\[ V = L \times B \times H \]
Substituting the values we found:
\[ V = 32 \times 24 \times 16 \]

Calculating this step-by-step:
1. \( 32 \times 24 = 768 \)
2. \( 768 \times 16 = 12288 \)

Thus, the volume of the cuboid is:
\[ V = 12288 \, m^3 \]

### Final Answer
The volume of the cuboid is \( 12288 \, m^3 \).

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