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An architect makes a model of a new house with a patio made with pavers. In the model, each paver in the patio is 1/3 in. long and 1/6 wide. The actual dimensions of pavers are shown: 1/8 ft and 1/4ft. What is the constant of proportionality that relates the length of a paver in the model and the length of an actual paver? What is the constant of proportionality that relates the area of an actual paver?

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Answer:

The constant of proportionality between the actual dimensions of the pavers and the model is 9.


The proportionality constant for the area is 81.

Step-by-step explanation:

To solve this problem, let's transform all quantities to the same units (inches)


The actual dimensions of the pavers are:


[tex]Width = \frac{1}{8} ft * \frac{12in}{1 ft} = \frac{3}{2} in\\\\ Length = \frac{1}{4} ft * \frac{12in}{1 ft} = 3in[/tex]


Then we divide the real dimensions between those of the model:


Width:


[tex]\frac{\frac{3}{2}}{\frac{1}{6}}= 9[/tex]

Long =


[tex]\frac{3}{\frac{1}{3}}= 9[/tex]



Then, the constant of proportionality between the actual dimensions of the pavers and the model is 9.


Actual length = model length * (9)


The "A" area of a paver is the product of its width multiplied by its length.


So:


(real width) * (real length) = ((9) Model width) * ((9) model length)


(real width) * (real length) = [tex]9 ^ 2[/tex] * (Model width) * (model length)


(real area) = 81 * (Model area)


The proportionality constant for the area is 81.

Answer:

The length of a paver in the model and the length is 1/9.

The constant of proportionality that relates the area 1/81.

Step-by-step explanation:

Area of rectangle is

[tex]A=length\times width[/tex]

Dimensions of paver in model:

[tex]Length=\frac{1}{3}in[/tex]

[tex]width=\frac{1}{6}in[/tex]

Area of model

[tex]A=length\times width[/tex]

[tex]A=\frac{1}{3}\times \frac{1}{6}=\frac{1}{18}[/tex]

The area of the model is 1/18 square inches.

We know that 1 ft = 12 inches

Actual dimensions of paver:

[tex]Length=\frac{1}{4}ft=3 in[/tex]

[tex]width=\frac{1}{8}ft =1.5in[/tex]

Actual area is

[tex]A=length\times width[/tex]

[tex]A=3\times 1.5=4.5[/tex]

The actual area is 4.5 square inches.

The constant of proportionality that relates the length of a paver in the model and the length of an actual paver is

[tex]\text{Constant of proportionality of length}=\frac{\text{Length of model}}{\text{Actual length}}=\frac{1/3}{3}=\frac{1}{9}[/tex]

The length of a paver in the model and the length is 1/9.

The constant of proportionality that relates the area of an actual paver is

[tex]\text{Constant of proportionality of area}=\frac{\text{Area of model}}{\text{Actual area}}=\frac{1/18}{4.5}=\frac{1}{81}[/tex]

The constant of proportionality that relates the area 1/81.

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