An architect

An architect makes a model of a new house. The model shows a tile patio in the backyard. In the model, each tile has a length of 1/2 inch and a width of 1/6 inch. The actual tiles have a length of 2/3 feet and a width of 2/9 feet. What is the ratio of the length of a tile in the model to the length of the actual tile? What is the ratio of the area of a tile in the model to the area of an actual tile?

Correct answer:

k₁ =  1:16
k₂ =  1:288

Step-by-step explanation:

$$\begin{gathered} a=1/2=\frac{1}{2}=0.5\text{inch} \\ b=1/6=\frac{1}{6}\doteq 0.1667\text{inch} \\ {a}_2=2/3=\frac{2}{3}\doteq 0.6667\text{ft} \\ {b}_2=2/8=\frac{1}{4}=0.25\text{ft} \\ x=a_2\to \text{inch}=a_2\cdot 12\text{inch}=0.6667\cdot 12\text{inch}=8\text{inch} \\ y=b_2\to \text{inch}=b_2\cdot 12\text{inch}=0.25\cdot 12\text{inch}=3\text{inch} \\ {k}_1=a/x=0.5/8=\frac{1}{16}=0.0625=0.0625=1:16 \\ \text{ Verifying Solution: } \\ {K}_1=b/y=0.1667/3=\frac{1}{18}\doteq 0.0556 \\ {K}_1\ne k_1 \end{gathered}$$

$$\begin{gathered} S_1=a\cdot b=0.5\cdot 0.1667=\frac{1}{12}\doteq 0.0833{\text{inch}}^2 \\ {S}_2=x\cdot y=8\cdot 3=24{\text{inch}}^2 \\ {k}_2=S_1/S_2=0.0833/24\approx 0.0035=1:288 \end{gathered}$$

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