The dimensions of a model of multi-storied building are $1 \; m$ by $60 \; cm$ by $1.20 \; m$. If the scale factor is $1 : 50$, find:
- the floor area of a room of the building in $m^2$, if the floor area of the corresponding room in the model is $50 \; cm^2$;
- the space inside a room of the model in $cm^3$, if the space inside the corresponding room of the building is $90 \; m^3$.
Scale factor $= k = 1 : 50 = \dfrac{1}{50}$
- Floor area of room in the model $= 50 \; cm^2 = 50 \times 10^{-4} \; m^2$
Floor area of room in model $= k^2 \times$ Floor area of room of building
i.e. $\;$ $50 \times 10^{-4} = \left(\dfrac{1}{50}\right)^2 \times $ Floor area of room of building
i.e. $\;$ Floor area of room of building $= 50 \times 10^{-4} \times 2500 = 7.5 \; m^2$ - Volume of a room of the model $= k^3 \times $ Volume of a room of the building
$\begin{aligned} i.e. \; \text{Volume of a room of the model} & = \left(\dfrac{1}{50}\right)^3 \times 90 \\\\ & = \dfrac{90}{125 \times 10^3} \\\\ & = 0.72 \times 10^{-3} \; m^3 \\\\ & = 0.72 \times 10^{-3} \times 10^6 \; cm^3 \\\\ & = 720 \; cm^3 \end{aligned}$