The length, breadth and height of a rectangular solid are in the ratio `5: 4, 2`. If the total surface area is `1216cm^(2)`, find the length, the breadth and the height of the solid.

Answer

To solve the problem, we need to find the dimensions of a rectangular solid (cuboid) given the ratio of its length, breadth, and height, as well as its total surface area.

### Step-by-Step Solution:

1. **Define the Variables:**
Let the length (L), breadth (B), and height (H) of the rectangular solid be represented in terms of a variable \( x \):
- Length \( L = 5x \)
- Breadth \( B = 4x \)
- Height \( H = 2x \)

2. **Write the Formula for Total Surface Area:**
The formula for the total surface area (TSA) of a rectangular solid is given by:
\[ \text{TSA} = 2(LB + BH + HL) \]

3. **Substitute the Values:**
Substitute \( L \), \( B \), and \( H \) into the TSA formula:
\[ \text{TSA} = 2((5x)(4x) + (4x)(2x) + (2x)(5x)) \]
Simplifying the terms inside the parentheses:
- \( LB = 5x \cdot 4x = 20x^2 \)
- \( BH = 4x \cdot 2x = 8x^2 \)
- \( HL = 2x \cdot 5x = 10x^2 \)

Therefore:
\[ \text{TSA} = 2(20x^2 + 8x^2 + 10x^2) = 2(38x^2) = 76x^2 \]

4. **Set the TSA Equal to the Given Value:**
We know from the problem that the total surface area is \( 1216 \, \text{cm}^2 \):
\[ 76x^2 = 1216 \]

5. **Solve for \( x^2 \):**
Divide both sides by 76:
\[ x^2 = \frac{1216}{76} \]
Simplifying this gives:
\[ x^2 = 16 \]

6. **Find \( x \):**
Taking the square root of both sides:
\[ x = 4 \]

7. **Calculate the Dimensions:**
Now substitute \( x \) back into the expressions for length, breadth, and height:
- Length \( L = 5x = 5 \cdot 4 = 20 \, \text{cm} \)
- Breadth \( B = 4x = 4 \cdot 4 = 16 \, \text{cm} \)
- Height \( H = 2x = 2 \cdot 4 = 8 \, \text{cm} \)

### Final Answer:
- Length = \( 20 \, \text{cm} \)
- Breadth = \( 16 \, \text{cm} \)
- Height = \( 8 \, \text{cm} \)