If the sum of the length, breadth and height of a rectangular parallelopiped is 24 cm and the length of its diagonal is 15 cm, then its total surface area is

Options

A. `256 cm^(2)`
B. `265 cm^(2)`
C. `315 cm^(2)`
D. `351 cm^(2)`

Answer

To solve the problem step by step, we will use the given information to find the total surface area of the rectangular parallelepiped.

### Step 1: Define the variables
Let:
- \( L \) = length
- \( B \) = breadth
- \( H \) = height

### Step 2: Set up the equations
From the problem, we have two key pieces of information:
1. The sum of the length, breadth, and height is 24 cm:
\[ L + B + H = 24 \quad \text{(Equation 1)} \]
2. The length of the diagonal is 15 cm. The formula for the diagonal \( D \) of a rectangular parallelepiped is given by:
\[ D = \sqrt{L^2 + B^2 + H^2} \]
Therefore, we can write:
\[ \sqrt{L^2 + B^2 + H^2} = 15 \]
Squaring both sides gives:
\[ L^2 + B^2 + H^2 = 15^2 = 225 \quad \text{(Equation 2)} \]

### Step 3: Use the equations to find \( L^2 + B^2 + H^2 \)
We can use the identity:
\[ (L + B + H)^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Substituting Equation 1 into this identity:
\[ 24^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Calculating \( 24^2 \):
\[ 576 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Now, substituting Equation 2 into this equation:
\[ 576 = 225 + 2(LB + BH + HL) \]

### Step 4: Solve for \( LB + BH + HL \)
Rearranging gives:
\[ 576 - 225 = 2(LB + BH + HL) \]
\[ 351 = 2(LB + BH + HL) \]
Dividing by 2:
\[ LB + BH + HL = \frac{351}{2} = 175.5 \]

### Step 5: Calculate the total surface area
The formula for the total surface area \( A \) of a rectangular parallelepiped is:
\[ A = 2(LB + BH + HL) \]
Substituting the value we found:
\[ A = 2 \times 175.5 = 351 \, \text{cm}^2 \]

### Final Answer
The total surface area of the rectangular parallelepiped is:
\[ \boxed{351 \, \text{cm}^2} \]