Question

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?

Answer 1

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

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

Explanation:

Area of rectangle is

`A=length* width`

Dimensions of paver in model:

`Length=(1)/(3)in`

`width=(1)/(6)in`

Area of model

`A=length* width`

`A=(1)/(3)* (1)/(6)=(1)/(18)`

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

We know that 1 ft = 12 inches

Actual dimensions of paver:

`Length=(1)/(4)ft=3 in`

`width=(1)/(8)ft =1.5in`

Actual area is

`A=length* width`

`A=3* 1.5=4.5`

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

`\text{Constant of proportionality of length}=\frac{\text{Length of model}}{\text{Actual length}}=(1/3)/(3)=(1)/(9)`

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

`\text{Constant of proportionality of area}=\frac{\text{Area of model}}{\text{Actual area}}=(1/18)/(4.5)=(1)/(81)`

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

Answer 2

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

The proportionality constant for the area is 81.

Explanation:

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

The actual dimensions of the pavers are:

`Width = (1)/(8) ft * (12in)/(1 ft) = (3)/(2) in\\\\ Length = (1)/(4) ft * (12in)/(1 ft) = 3in`

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

Width:

`((3)/(2))/((1)/(6))= 9`

Long =

`(3)/((1)/(3))= 9`

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) =
`9 ^ 2` * (Model width) * (model length)

(real area) = 81 * (Model area)

The proportionality constant for the area is 81.