The sum of length , breadht and hight of a cuboid is 19 cm and its diagonal is `5sqrt5` cm . Its surface area is

Options

A. 361 `cm^2`
B. 125 `cm^2`
C. 236 `cm^2`
D. 486 `cm^2`

Explanation

To find the surface area of the cuboid given the sum of its dimensions and the length of its diagonal, we can follow these steps:

### Step 1: Define the variables
Let the length, breadth, and height of the cuboid be denoted as \( L \), \( B \), and \( H \) respectively.

### Step 2: Set up the equations
From the problem, we know:
1. \( L + B + H = 19 \) (Equation 1)
2. The diagonal \( D \) of the cuboid is given by the formula \( D = \sqrt{L^2 + B^2 + H^2} \). We know \( D = 5\sqrt{5} \), so squaring both sides gives:
\[ L^2 + B^2 + H^2 = (5\sqrt{5})^2 = 25 \times 5 = 125 \quad \text{(Equation 2)} \]

### Step 3: Use the identity for squares
We can use the identity:
\[ (L + B + H)^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Substituting the values from Equations 1 and 2:
\[ 19^2 = 125 + 2(LB + BH + HL) \]
Calculating \( 19^2 \):
\[ 361 = 125 + 2(LB + BH + HL) \]

### Step 4: Solve for \( LB + BH + HL \)
Rearranging the equation:
\[ 2(LB + BH + HL) = 361 - 125 \]
\[ 2(LB + BH + HL) = 236 \]
Dividing both sides by 2:
\[ LB + BH + HL = 118 \quad \text{(Equation 3)} \]

### Step 5: Calculate the surface area
The surface area \( A \) of the cuboid is given by the formula:
\[ A = 2(LB + BH + HL) \]
Substituting the value from Equation 3:
\[ A = 2 \times 118 = 236 \, \text{cm}^2 \]

### Final Answer
The surface area of the cuboid is \( 236 \, \text{cm}^2 \).

---