A room is half as long as it is broad. The cost of carpeting the room at Rs 3.25 sq.m is Rs 175.50 and the cost of preparing the walls at Rs1.40 per sq.m is Rs240.80. If 1 door and 2 windows occupy 8sq.m , then find the dimensions of the room ?

Answer

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

### Step 1: Define the dimensions of the room
Let the breadth (width) of the room be \( A \) meters. Since the room is half as long as it is broad, the length \( L \) can be expressed as:
\[ L = \frac{3A}{2} \]

### Step 2: Calculate the area of the room
The area \( A_r \) of the room can be calculated using the formula for the area of a rectangle:
\[ A_r = L \times A = \frac{3A}{2} \times A = \frac{3A^2}{2} \]

### Step 3: Set up the equation for carpeting cost
The cost of carpeting the room is given as Rs 3.25 per square meter, and the total cost is Rs 175.50. Therefore, we can set up the equation:
\[ 3.25 \times A_r = 175.50 \]
Substituting the area of the room:
\[ 3.25 \times \frac{3A^2}{2} = 175.50 \]

### Step 4: Solve for \( A \)
To solve for \( A \), first simplify the equation:
\[ \frac{9.75A^2}{2} = 175.50 \]
Multiplying both sides by 2:
\[ 9.75A^2 = 351 \]
Now, divide by 9.75:
\[ A^2 = \frac{351}{9.75} = 36 \]
Taking the square root:
\[ A = 6 \text{ meters} \]

### Step 5: Calculate the length of the room
Now that we have the breadth, we can find the length:
\[ L = \frac{3A}{2} = \frac{3 \times 6}{2} = 9 \text{ meters} \]

### Step 6: Determine the height of the room
Next, we need to find the height of the room using the cost of preparing the walls. The cost of preparing the walls is Rs 1.40 per square meter, and the total cost is Rs 240.80. The area of the walls can be calculated as:
\[ \text{Area of walls} = 2 \times (L + A) \times h \]
Substituting the values:
\[ \text{Area of walls} = 2 \times (9 + 6) \times h = 30h \]
Now, we subtract the area occupied by the door and windows (8 sq.m):
\[ \text{Effective area} = 30h - 8 \]
Setting up the equation for the cost of preparing the walls:
\[ 1.40 \times (30h - 8) = 240.80 \]
Dividing both sides by 1.40:
\[ 30h - 8 = \frac{240.80}{1.40} = 172 \]
Adding 8 to both sides:
\[ 30h = 180 \]
Dividing by 30:
\[ h = 6 \text{ meters} \]

### Final Dimensions
Thus, the dimensions of the room are:
- Length = 9 meters
- Width = 6 meters
- Height = 6 meters