The sum of length breadth and depth of a cuboid is 19 cm and the length of its diagonal is 11 cm. Find the total surface area of the cuboid.

Options

A. `361" cm"^(2)`
B. `240" cm"^(2)`
C. `209" cm"^(2)`
D. `121" cm"^(2)`

Answer

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

### Step 1: Define Variables
Let:
- Length = L
- Breadth = B
- Height = H

### Step 2: Set Up the Equations
From the problem, we know:
1. The sum of length, breadth, and height is given by:
\[ L + B + H = 19 \quad \text{(Equation 1)} \]
2. The length of the diagonal is given by:
\[ \sqrt{L^2 + B^2 + H^2} = 11 \quad \text{(Equation 2)} \]
Squaring both sides gives:
\[ L^2 + B^2 + H^2 = 121 \quad \text{(Equation 3)} \]

### Step 3: Square Equation 1
Now, we square Equation 1:
\[ (L + B + H)^2 = 19^2 \]
This expands to:
\[ L^2 + B^2 + H^2 + 2(LB + BH + HL) = 361 \quad \text{(Equation 4)} \]

### Step 4: Substitute Equation 3 into Equation 4
From Equation 3, we know that \(L^2 + B^2 + H^2 = 121\). We substitute this into Equation 4:
\[ 121 + 2(LB + BH + HL) = 361 \]

### Step 5: Solve for \(LB + BH + HL\)
Now, we can isolate \(LB + BH + HL\):
\[ 2(LB + BH + HL) = 361 - 121 \]
\[ 2(LB + BH + HL) = 240 \]
\[ LB + BH + HL = 120 \quad \text{(Equation 5)} \]

### Step 6: Calculate Total Surface Area
The formula for the total surface area (TSA) of a cuboid is:
\[ \text{TSA} = 2(LB + BH + HL) \]
Substituting the value from Equation 5:
\[ \text{TSA} = 2 \times 120 = 240 \, \text{cm}^2 \]

### Final Answer
The total surface area of the cuboid is:
\[ \boxed{240 \, \text{cm}^2} \]