A hall is 18 m long and 12 m broad. If the sum of the areas of the floor and the ceiling is equal to the sum of the areas of the four wall, the volume (in `m^(2)`) of the hall is

Options

A. 1175.5
B. 1876.2
C. 1555.2
D. 1376.4

Answer

To solve the problem, we need to find the volume of the hall given its dimensions and a relationship between the areas of the floor, ceiling, and walls. Let's go through the steps systematically.

### Step 1: Identify the dimensions of the hall
- Length (L) = 18 m
- Breadth (B) = 12 m

### Step 2: Calculate the area of the floor and ceiling
The area of the floor (A_floor) is given by the formula:
\[ A_{\text{floor}} = L \times B = 18 \, \text{m} \times 12 \, \text{m} = 216 \, \text{m}^2 \]

Since the ceiling has the same area as the floor:
\[ A_{\text{ceiling}} = A_{\text{floor}} = 216 \, \text{m}^2 \]

### Step 3: Sum of the areas of the floor and ceiling
The total area of the floor and ceiling is:
\[ A_{\text{floor}} + A_{\text{ceiling}} = 216 \, \text{m}^2 + 216 \, \text{m}^2 = 432 \, \text{m}^2 \]

### Step 4: Set up the equation for the area of the four walls
The area of the four walls (A_walls) can be calculated using the formula:
\[ A_{\text{walls}} = 2 \times (L + B) \times H \]
Where H is the height of the hall.

### Step 5: Set the areas equal as per the problem statement
According to the problem, the sum of the areas of the floor and ceiling is equal to the area of the four walls:
\[ A_{\text{floor}} + A_{\text{ceiling}} = A_{\text{walls}} \]
\[ 432 \, \text{m}^2 = 2 \times (L + B) \times H \]

### Step 6: Substitute the known values
Substituting L and B into the equation:
\[ 432 = 2 \times (18 + 12) \times H \]
\[ 432 = 2 \times 30 \times H \]
\[ 432 = 60H \]

### Step 7: Solve for height (H)
To find H, divide both sides by 60:
\[ H = \frac{432}{60} = 7.2 \, \text{m} \]

### Step 8: Calculate the volume of the hall
The volume (V) of the hall can be calculated using the formula:
\[ V = L \times B \times H \]
Substituting the values we have:
\[ V = 18 \, \text{m} \times 12 \, \text{m} \times 7.2 \, \text{m} \]
\[ V = 1555.2 \, \text{m}^3 \]

### Final Answer
The volume of the hall is **1555.2 m³**.