Question

In a scale model of a building, 1 ft represents 60 ft. The volume of the scale model is 18 ft3.

What is the volume of the building?
The roof of the scale model has a surface area of 4 ft^2. What is the surface area?

Answer 1

`\bf \qquad \qquad \textit{ratio relations} \\\\ \begin{array}{ccccllll} &Sides&Area&Volume\\ &-----&-----&-----\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}&\cfrac{s}{s}&\cfrac{s^2}{s^2}&\cfrac{s^3}{s^3} \end{array} \\\\ -----------------------------\\\\ \cfrac{\textit{similar shape}}{\textit{similar shape}}\qquad \cfrac{s}{s}=\cfrac{√(s^2)}{√(s^2)}=\cfrac{\sqrt[3]{s^3}}{\sqrt[3]{s^3}}\\\\ -------------------------------\\\\`


`\bf \textit{using the volume ratio}\\\\ \cfrac{model}{building}\qquad \cfrac{1}{60}=\cfrac{\sqrt[3]{18}}{\sqrt[3]{v}}\implies \cfrac{1}{60}=\sqrt[3]{\cfrac{18}{v}}\implies \left( \cfrac{1}{60} \right)^3=\cfrac{18}{v} \\\\\\ \cfrac{1^3}{60^3}=\cfrac{18}{v}\implies v=\cfrac{60^3\cdot 18}{1^3}\\\\ -------------------------------\\\\`


`\bf \textit{using the area ratio}\\\\ \cfrac{model}{building}\qquad \cfrac{1}{60}=\cfrac{√(4)}{√(a)}\implies \cfrac{1}{60}=\sqrt{\cfrac{4}{a}}\implies \left( \cfrac{1}{60} \right)^2=\cfrac{4}{a} \\\\\\ \cfrac{1^2}{60^2}=\cfrac{4}{a}\implies a=\cfrac{60^2\cdot 4}{1^2}`