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 1/2 in. long and 1/6 in. wide. The actual dimensions of the pavers are shown. Complete parts a and b.

a. What is the constant of proportionality that relates the length of a paver in the model and the actual length? Write an equation that represents this relationship. The constant of proportionality is __________. (Type the ratio as a simplified fraction.)

b. What is the constant of proportionality that relates the area of a paver in the model and the area of an actual paver? Explain your reasoning. The area of a rectangle is _______ and the constant of proportionality is _______. (Type an integer or a simplified fraction.)

Answer 1

The constant of proportionality that relates the length of a paver in the model and the actual length is 1/4. The constant of proportionality that relates the area of a paver in the model and the area of an actual paver is 1/16.

Step-by-step explanation:

The constant of proportionality that relates the length of a paver in the model and the actual length can be found by setting the two length ratios equal to one another. The given unit is inches.

Using the information provided, we have the proportion:

1/20 = 1/2L/5

Cross multiplying and solving for L, we get:

L = (1/20) * 5

L = 1/4

Therefore, the constant of proportionality is 1/4.

To find the constant of proportionality relating the area of a paver in the model and the area of an actual paver, we need to remember that the area of a rectangle is given by the product of its length and width. Thus, if the length and width are scaled by the same factor, the area will be scaled by the square of that factor.

Since the length and width of the paver are scaled by the same factor of 1/4, the area of the paver in the model will be 1/4 * 1/4 = 1/16 times the area of the actual paver.

Therefore, the constant of proportionality is 1/16.