A room 5 m long 4.5 m wide and 3.6m high has one door 1.5 m by 2.4m and two windows each 1m by 0.75m . Find:
(i) the area of its walls ,excluding door and windows.
(ii) the cost of distempering its walls at the rate of rupees 4.50 ` per m ^(2)`
(iii) The cost of painting its roof at the rate of rupees 9 per `m ^(2)`

Answer

To solve the problem step by step, we will break it down into three parts as per the question.

### Step 1: Calculate the Area of the Walls Excluding Door and Windows

1. **Identify the dimensions of the room:**
- Length (L) = 5 m
- Width (B) = 4.5 m
- Height (H) = 3.6 m

2. **Calculate the total surface area of the four walls:**
The formula for the area of the four walls of a cuboid is:
\[ \text{Area of walls} = 2H(L + B) \]
Substituting the values:
\[ \text{Area of walls} = 2 \times 3.6 \times (5 + 4.5) = 2 \times 3.6 \times 9.5 \]
\[ = 68.4 \, \text{m}^2 \]

3. **Calculate the area of the door:**
- Dimensions of the door = 1.5 m by 2.4 m
\[ \text{Area of door} = 1.5 \times 2.4 = 3.6 \, \text{m}^2 \]

4. **Calculate the area of the two windows:**
- Dimensions of each window = 1 m by 0.75 m
\[ \text{Area of one window} = 1 \times 0.75 = 0.75 \, \text{m}^2 \]
\[ \text{Area of two windows} = 2 \times 0.75 = 1.5 \, \text{m}^2 \]

5. **Calculate the area of walls excluding the door and windows:**
\[ \text{Area excluding door and windows} = \text{Area of walls} - \text{Area of door} - \text{Area of windows} \]
\[ = 68.4 - 3.6 - 1.5 = 63.3 \, \text{m}^2 \]

### Step 2: Calculate the Cost of Distempering the Walls

1. **Cost per square meter for distempering:**
- Cost = 4.50 rupees/m²

2. **Calculate the total cost:**
\[ \text{Total cost} = \text{Area excluding door and windows} \times \text{Cost per m}^2 \]
\[ = 63.3 \times 4.50 = 284.85 \, \text{rupees} \]

### Step 3: Calculate the Cost of Painting the Roof

1. **Calculate the area of the roof:**
The area of the roof can be calculated using the formula:
\[ \text{Area of roof} = L \times B \]
\[ = 5 \times 4.5 = 22.5 \, \text{m}^2 \]

2. **Cost per square meter for painting:**
- Cost = 9 rupees/m²

3. **Calculate the total cost of painting the roof:**
\[ \text{Total cost} = \text{Area of roof} \times \text{Cost per m}^2 \]
\[ = 22.5 \times 9 = 202.50 \, \text{rupees} \]

### Summary of Results
- (i) Area of walls excluding door and windows = **63.3 m²**
- (ii) Cost of distempering the walls = **284.85 rupees**
- (iii) Cost of painting the roof = **202.50 rupees**