The ratio of the length and breadth of a rectangular parallelopiped is 5:3. and its height is 6 cm. If the total surface area of the parallelopiped be 558 sq. cm, then its length in dm is

Options

A. 9
B. `1.5`
C. 10
D. 15

Answer

To solve the problem step by step, we will follow the given information and apply the formulas related to the total surface area of a rectangular parallelepiped (cuboid).

### Step 1: Understand the given ratios and dimensions
The ratio of the length (L) and breadth (B) of the rectangular parallelepiped is given as 5:3. We can express this as:
- Length (L) = 5x
- Breadth (B) = 3x
where x is a common multiplier.

The height (H) is given as 6 cm.

### Step 2: Write the formula for Total Surface Area (TSA)
The formula for the total surface area of a rectangular parallelepiped is:
\[ \text{TSA} = 2(LH + BH + LB) \]

### Step 3: Substitute the known values into the TSA formula
We know:
- L = 5x
- B = 3x
- H = 6 cm
- TSA = 558 cm²

Substituting these values into the TSA formula:
\[ 558 = 2[(5x)(6) + (3x)(6) + (5x)(3x)] \]

### Step 4: Simplify the equation
First, simplify the expression inside the brackets:
\[ 558 = 2[30x + 18x + 15x^2] \]
\[ 558 = 2[48x + 15x^2] \]
Now divide both sides by 2:
\[ 279 = 48x + 15x^2 \]

### Step 5: Rearrange the equation
Rearranging gives us:
\[ 15x^2 + 48x - 279 = 0 \]

### Step 6: Solve the quadratic equation
To solve the quadratic equation \(15x^2 + 48x - 279 = 0\), we can use the factorization method or the quadratic formula. Let's factor it.

First, we can simplify by dividing the entire equation by 3:
\[ 5x^2 + 16x - 93 = 0 \]

Now we need to find two numbers that multiply to \(5 \times -93 = -465\) and add to \(16\). The factors of -465 that satisfy this are 31 and -15.

Rewriting the equation:
\[ 5x^2 + 31x - 15x - 93 = 0 \]
Grouping terms:
\[ (5x^2 + 31x) + (-15x - 93) = 0 \]
Factoring by grouping:
\[ x(5x + 31) - 3(5x + 31) = 0 \]
Factoring out the common term:
\[ (5x + 31)(x - 3) = 0 \]

### Step 7: Find the values of x
Setting each factor to zero gives:
1. \(5x + 31 = 0 \Rightarrow x = -\frac{31}{5}\) (not valid since x must be positive)
2. \(x - 3 = 0 \Rightarrow x = 3\)

### Step 8: Calculate the length
Now, substituting \(x = 3\) back to find the length:
\[ L = 5x = 5 \times 3 = 15 \text{ cm} \]

### Step 9: Convert length to decimeters
To convert centimeters to decimeters, we divide by 10:
\[ L = \frac{15}{10} = 1.5 \text{ dm} \]

### Final Answer
The length of the rectangular parallelepiped in decimeters is **1.5 dm**.

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