Total surface area of a cuboid is `846 cm^2`. If ratio of its length, breadth and height is 5.4:3, then find the lenght, breadth and height of cuboid. (In cm.)

Options

A. 15,12,9
B. 15,9,12
C. 12,9,8
D. 16,12,8

Answer

To find the length, breadth, and height of the cuboid given the total surface area and the ratio of its dimensions, we will follow these steps:

### Step 1: Understand the given information
The total surface area (TSA) of a cuboid is given as \( 846 \, \text{cm}^2 \). The ratio of its length (L), breadth (B), and height (H) is given as \( 5.4 : 3 : 1 \).

### Step 2: Set up the ratio
Let the length, breadth, and height be represented as:
- Length \( L = 5.4x \)
- Breadth \( B = 3x \)
- Height \( H = x \)

### Step 3: Write the formula for total surface area
The formula for the total surface area of a cuboid is:
\[ \text{TSA} = 2(LB + BH + HL) \]
Substituting the expressions for L, B, and H:
\[ \text{TSA} = 2((5.4x)(3x) + (3x)(x) + (x)(5.4x)) \]

### Step 4: Simplify the expression
Calculating each term:
- \( LB = (5.4x)(3x) = 16.2x^2 \)
- \( BH = (3x)(x) = 3x^2 \)
- \( HL = (x)(5.4x) = 5.4x^2 \)

Now, substituting back into the TSA formula:
\[ \text{TSA} = 2(16.2x^2 + 3x^2 + 5.4x^2) \]
\[ = 2(24.6x^2) \]
\[ = 49.2x^2 \]

### Step 5: Set the TSA equal to the given value
Now, we set the total surface area equal to the given value:
\[ 49.2x^2 = 846 \]

### Step 6: Solve for \( x^2 \)
Dividing both sides by 49.2:
\[ x^2 = \frac{846}{49.2} \]
Calculating the right side:
\[ x^2 = 17.2 \]

### Step 7: Solve for \( x \)
Taking the square root of both sides:
\[ x = \sqrt{17.2} \approx 4.14 \]

### Step 8: Find the dimensions of the cuboid
Now substituting \( x \) back to find L, B, and H:
- Length \( L = 5.4x = 5.4 \times 4.14 \approx 22.34 \, \text{cm} \)
- Breadth \( B = 3x = 3 \times 4.14 \approx 12.42 \, \text{cm} \)
- Height \( H = x \approx 4.14 \, \text{cm} \)

### Final Answer
Thus, the dimensions of the cuboid are approximately:
- Length \( L \approx 22.34 \, \text{cm} \)
- Breadth \( B \approx 12.42 \, \text{cm} \)
- Height \( H \approx 4.14 \, \text{cm} \)

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