The area of the four walls of a room is `1152 m ^(2)` and its length is ` (5)/(3)` times its breadth. If the height of the room is 12 m then the area of its floor is

Options

A. `580 m^(2)`
B. `520 m^(2)`
C. `540 m^(2)`
D. `560 m^(2)`

Answer

To find the area of the floor of the room, we will follow these steps:

### Step 1: Understand the relationship between length and breadth
Let the breadth of the room be \( B \). According to the problem, the length \( L \) is \( \frac{5}{3} \) times the breadth. Therefore, we can express the length in terms of breadth:
\[ L = \frac{5}{3}B \]

### Step 2: Use the formula for the area of the four walls
The area of the four walls of a room can be calculated using the formula:
\[ \text{Area of four walls} = 2 \times (L + B) \times H \]
where \( H \) is the height of the room. Given that the height \( H = 12 \, m \) and the area of the four walls is \( 1152 \, m^2 \), we can set up the equation:
\[ 1152 = 2 \times (L + B) \times 12 \]

### Step 3: Simplify the equation
First, simplify the equation:
\[ 1152 = 24 \times (L + B) \]
Now, divide both sides by 24:
\[ L + B = \frac{1152}{24} = 48 \]

### Step 4: Substitute \( L \) in terms of \( B \)
Now substitute \( L = \frac{5}{3}B \) into the equation \( L + B = 48 \):
\[ \frac{5}{3}B + B = 48 \]
Combine the terms:
\[ \frac{5}{3}B + \frac{3}{3}B = 48 \]
\[ \frac{8}{3}B = 48 \]

### Step 5: Solve for \( B \)
To find \( B \), multiply both sides by \( \frac{3}{8} \):
\[ B = 48 \times \frac{3}{8} = 18 \]

### Step 6: Find \( L \)
Now substitute \( B = 18 \) back into the equation for \( L \):
\[ L = \frac{5}{3} \times 18 = 30 \]

### Step 7: Calculate the area of the floor
The area of the floor \( A \) can be calculated using the formula:
\[ A = L \times B \]
Substituting the values of \( L \) and \( B \):
\[ A = 30 \times 18 = 540 \, m^2 \]

### Final Answer
The area of the floor is \( 540 \, m^2 \).

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