The length, breadth and height of a.cuboid are in the ratio of 6:5:4. If the total surface area is 5328 cm", then the length, breadth and height of the cuboid will be

Options

A. 20 cm, 22 cm, 23 cm
B. 38 cm, 25 cm, 23 cm
C. 36 cm, 30 cm, 24 cm
D. 22 cm, 23 cm, 21 cm

Answer

To solve the problem step-by-step, we follow these instructions:

### Step 1: Understand the Ratio
The length (L), breadth (B), and height (H) of the cuboid are in the ratio of 6:5:4. We can express them in terms of a variable \( x \):
- Length \( L = 6x \)
- Breadth \( B = 5x \)
- Height \( H = 4x \)

### Step 2: Write the Formula for Total Surface Area
The total surface area (TSA) of a cuboid is given by the formula:
\[ TSA = 2(LB + BH + HL) \]
Substituting the expressions for L, B, and H:
\[ TSA = 2(6x \cdot 5x + 5x \cdot 4x + 4x \cdot 6x) \]

### Step 3: Simplify the Expression
Calculating each term:
- \( LB = 6x \cdot 5x = 30x^2 \)
- \( BH = 5x \cdot 4x = 20x^2 \)
- \( HL = 4x \cdot 6x = 24x^2 \)

Adding these together:
\[ LB + BH + HL = 30x^2 + 20x^2 + 24x^2 = 74x^2 \]

Thus, the TSA becomes:
\[ TSA = 2(74x^2) = 148x^2 \]

### Step 4: Set Up the Equation
We know the total surface area is 5328 cm², so we set up the equation:
\[ 148x^2 = 5328 \]

### Step 5: Solve for \( x^2 \)
To find \( x^2 \), divide both sides by 148:
\[ x^2 = \frac{5328}{148} \]
Calculating this gives:
\[ x^2 = 36 \]

### Step 6: Find \( x \)
Taking the square root of both sides:
\[ x = \sqrt{36} = 6 \]

### Step 7: Calculate Length, Breadth, and Height
Now we can find the dimensions:
- Length \( L = 6x = 6 \cdot 6 = 36 \) cm
- Breadth \( B = 5x = 5 \cdot 6 = 30 \) cm
- Height \( H = 4x = 4 \cdot 6 = 24 \) cm

### Final Answer
The dimensions of the cuboid are:
- Length: 36 cm
- Breadth: 30 cm
- Height: 24 cm

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