The sum of length, breadth and height of a room is 19 m. The length of the diagonal is 11 m. The cost of painting the total surface area of the room at the rate of Rs. 10 per `m^2` is

Options

A. Rs. 240
B. Rs. 2400
C. Rs. 430
D. Rs. 4200

Answer

To solve the problem step by step, we will follow these calculations:

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

### Step 2: Set Up the Equations
We are given two pieces of information:
1. The sum of the dimensions:
\[ L + B + H = 19 \quad \text{(Equation 1)} \]
2. The length of the diagonal:
\[ \sqrt{L^2 + B^2 + H^2} = 11 \quad \text{(Equation 2)} \]
Squaring both sides gives:
\[ L^2 + B^2 + H^2 = 121 \quad \text{(Equation 3)} \]

### Step 3: Use the Identity
We can use the identity:
\[ (L + B + H)^2 = L^2 + B^2 + H^2 + 2(LB + BH + HL) \]
Substituting Equation 1 and Equation 3 into this identity:
\[ 19^2 = 121 + 2(LB + BH + HL) \]
Calculating \( 19^2 \):
\[ 361 = 121 + 2(LB + BH + HL) \]

### Step 4: Solve for \( LB + BH + HL \)
Rearranging the equation:
\[ 361 - 121 = 2(LB + BH + HL) \]
\[ 240 = 2(LB + BH + HL) \]
Dividing by 2:
\[ LB + BH + HL = 120 \quad \text{(Equation 4)} \]

### Step 5: Calculate the Total Surface Area
The total surface area \( A \) of a cuboid is given by:
\[ A = 2(LB + BH + HL) \]
Substituting Equation 4:
\[ A = 2 \times 120 = 240 \, m^2 \]

### Step 6: Calculate the Cost of Painting
The cost of painting per square meter is Rs. 10. Therefore, the total cost \( C \) is:
\[ C = A \times \text{Cost per m}^2 = 240 \times 10 = 2400 \, \text{Rs} \]

### Final Answer
The cost of painting the total surface area of the room is Rs. 2400.

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