The sum of length,breadth and depth of cuboid is 12 cm and its diagonal is `5sqrt2` cm . Its surface area is

Options

A. `152 cm^2`
B. `94 cm^2`
C. `108 cm^2`
D. `60sqrt2 cm^2`

Answer

To find the surface area of the cuboid given the sum of its dimensions and 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
We know from the problem statement:
1. The sum of the dimensions:
\[ l + b + h = 12 \quad \text{(Equation 1)} \]
2. The diagonal of the cuboid:
\[ \sqrt{l^2 + b^2 + h^2} = 5\sqrt{2} \quad \text{(Equation 2)} \]
Squaring both sides gives:
\[ l^2 + b^2 + h^2 = (5\sqrt{2})^2 = 50 \quad \text{(Equation 3)} \]

### Step 3: Use the Identity
We can use the identity for the square of a sum:
\[ (l + b + h)^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) \]
Substituting Equation 1 into this identity:
\[ 12^2 = l^2 + b^2 + h^2 + 2(lb + bh + hl) \]
This simplifies to:
\[ 144 = l^2 + b^2 + h^2 + 2(lb + bh + hl) \]
Now substituting Equation 3 into this:
\[ 144 = 50 + 2(lb + bh + hl) \]

### Step 4: Solve for \( lb + bh + hl \)
Rearranging gives:
\[ 2(lb + bh + hl) = 144 - 50 \]
\[ 2(lb + bh + hl) = 94 \]
Dividing by 2:
\[ lb + bh + hl = 47 \quad \text{(Equation 4)} \]

### Step 5: Calculate the Surface Area
The surface area \( A \) of the cuboid is given by:
\[ A = 2(lb + bh + hl) \]
Substituting the value from Equation 4:
\[ A = 2 \times 47 = 94 \, \text{cm}^2 \]

### Conclusion
The surface area of the cuboid is \( 94 \, \text{cm}^2 \).

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