The dimensions of the model of a multistoreyed 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 :

(i) the floor area of a room of the building, if the floor area of the corresponding room in the model is 50 sq. cm.

(ii) the space (volume) inside a room of the model, if the space inside the corresponding room of the building is 90 m³.

Given,

Scale factor (k) = 1 : 50.

Dimensions of model = l × b × h = 1 m × 0.6 m × 1.2 m.

By formula,

$\phantom{\Rightarrow} \dfrac{\text{Length of building's model}}{\text{Length of building}} = k \\[1em] \Rightarrow \dfrac{1}{\text{Length of building}} = \dfrac{1}{50} \\[1em] \Rightarrow \text{Length of building} = 50 \text{ m}. \\[2em] \phantom{\Rightarrow} \dfrac{\text{Breadth of building's model}}{\text{Breadth of building}} = k \\[1em] \Rightarrow \dfrac{0.6}{\text{Breadth of building}} = \dfrac{1}{50} \\[1em] \Rightarrow \text{Breadth of building} = 50 \times 0.6 = 30 \text{ m}. \\[2em] \phantom{\Rightarrow} \dfrac{\text{Height of building's model}}{\text{Height of building}} = k \\[1em] \Rightarrow \dfrac{1.2}{\text{Height of building}} = \dfrac{1}{50} \\[1em] \Rightarrow \text{Height of building} = 50 \times 1.2 = 60 \text{ m}.$

Dimensions of building = 50 m × 30 m × 60 m.

(i) By formula,

Floor area of model room = k² × Floor area of building room

50 = $\dfrac{1}{50} \times \dfrac{1}{50}$ × Floor area of building room

Floor area of building room = 50 × 50 × 50 = 125000 cm² = $\dfrac{125000}{100 \times 100}$ m² = 12.5 m².

Hence, floor area of a room of the building = 12.5 m².

(ii) By formula,

Volume of a room of model = k³ × Volume of a room of building

Volume of a room of model = $\dfrac{1}{50} \times \dfrac{1}{50} \times \dfrac{1}{50} \times 90$ = 0.00072 m³ = 0.00072 x 100 x 100 x 100 cm³ = 720 cm³.

Hence, the space inside a room of the model = 720 cm³.