The dimensions of the model of a multi storeyed building are 1 m by 60 cm by 1.20 m. If the scale factor is 1 : 50, find the actual dimensions of the building. Also, find :
the floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq. cm. 

Answer

To solve the problem step by step, we will find the actual dimensions of the building based on the model dimensions and the scale factor, and then we will calculate the floor area of a room in the building.

### Step 1: Convert the model dimensions to meters
The dimensions of the model are given as:
- Length = 1 m
- Width = 60 cm = 0.6 m (since 100 cm = 1 m)
- Height = 1.20 m

### Step 2: Calculate the actual dimensions using the scale factor
The scale factor is given as 1:50. This means that 1 unit in the model corresponds to 50 units in reality.

1. **Actual Length**:
\[ \text{Actual Length} = \text{Model Length} \times \text{Scale Factor} = 1 \, \text{m} \times 50 = 50 \, \text{m} \]

2. **Actual Width**:
\[ \text{Actual Width} = \text{Model Width} \times \text{Scale Factor} = 0.6 \, \text{m} \times 50 = 30 \, \text{m} \]

3. **Actual Height**:
\[ \text{Actual Height} = \text{Model Height} \times \text{Scale Factor} = 1.20 \, \text{m} \times 50 = 60 \, \text{m} \]

### Step 3: State the actual dimensions of the building
The actual dimensions of the building are:
- Length = 50 m
- Width = 30 m
- Height = 60 m

### Step 4: Calculate the floor area of a room in the building
The floor area of the corresponding room in the model is given as 50 sq. cm. To find the actual floor area, we need to apply the square of the scale factor.

1. **Convert the floor area from model to actual**:
\[ \text{Actual Floor Area} = \text{Model Floor Area} \times \left(\text{Scale Factor}\right)^2 \]
\[ = 50 \, \text{cm}^2 \times (50)^2 \]
\[ = 50 \, \text{cm}^2 \times 2500 = 125000 \, \text{cm}^2 \]

2. **Convert the floor area from cm² to m²**:
Since \(1 \, \text{m}^2 = 10000 \, \text{cm}^2\):
\[ \text{Actual Floor Area in m}^2 = \frac{125000 \, \text{cm}^2}{10000} = 12.5 \, \text{m}^2 \]

### Final Answers:
- Actual dimensions of the building: 50 m x 30 m x 60 m
- Actual floor area of a room: 12.5 m²