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 space (volume) inside a room of the model, if the space inside the corresponding room of the building is `90 m^3`. 

Answer

To solve the problem, we will break it down into two parts as follows:

### Part 1: Finding the Actual Dimensions of the Building

1. **Given Dimensions of the Model:**
- Length = 1 m
- Breadth = 60 cm (which is 0.60 m)
- Height = 1.20 m

2. **Scale Factor:**
- The scale factor is given as 1 : 50. This means that the actual dimensions are 50 times the dimensions of the model.

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

4. **Calculating Actual Breadth:**
\[ \text{Actual Breadth} = \text{Model Breadth} \times \text{Scale Factor} = 0.60 \, \text{m} \times 50 = 30 \, \text{m} \]

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

### Part 2: Finding the Volume Inside a Room of the Model

1. **Given Volume of the Corresponding Room in the Building:**
- Volume = 90 m³

2. **Finding the Volume of the Room in the Model:**
- Since the scale factor is 1 : 50, the volume scale factor will be \( \left(\frac{1}{50}\right)^3 = \frac{1}{125000} \).

3. **Calculating Volume of the Model Room:**
\[ \text{Volume of Model Room} = \text{Volume of Building Room} \times \left(\frac{1}{50}\right)^3 = 90 \, \text{m}^3 \times \frac{1}{125000} \]
\[ = \frac{90}{125000} = 0.00072 \, \text{m}^3 \]

4. **Converting to Centimeter Cubes:**
- Since \(1 \, \text{m}^3 = 1000000 \, \text{cm}^3\),
\[ \text{Volume of Model Room in cm}^3 = 0.00072 \, \text{m}^3 \times 1000000 = 720 \, \text{cm}^3 \]

### Final Results:
- Actual Dimensions of the Building:
- Length = 50 m
- Breadth = 30 m
- Height = 60 m
- Volume Inside the Model Room = 720 cm³