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 $\frac{1}{4}$ in. long and $\frac{1}{6}$ 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 $square$
(type the ratio as a simplified fraction.)

Explanation:

Step1: Identify model and actual lengths

Model length (let's take length first, say the length in model is $\frac{1}{4}$ in, actual length is $\frac{2}{9}$ ft. Wait, need to convert units? Wait, maybe the width? Wait, the problem says "relates the length of a paver in the model and the actual length". Wait, maybe the model length is $\frac{1}{4}$ in and actual length is $\frac{2}{9}$ ft? Wait, no, maybe the other way. Wait, let's check the actual dimensions: the actual paver has length $\frac{2}{9}$ ft and width $\frac{1}{3}$ ft? Wait, the model paver is $\frac{1}{4}$ in long and $\frac{1}{6}$ in wide? Wait, the problem says "In the model, each paver in the patio is $\frac{1}{4}$ in. long and $\frac{1}{6}$ in. wide. The actual dimensions of the pavers are shown." The actual length is $\frac{2}{9}$ ft? Wait, let's convert inches to feet or feet to inches. 1 foot = 12 inches. So model length: $\frac{1}{4}$ in = $\frac{1}{4\times12}$ ft = $\frac{1}{48}$ ft? No, wait, actual length is $\frac{2}{9}$ ft, model length is $\frac{1}{4}$ in. Let's convert model length to feet: $\frac{1}{4}$ in = $\frac{1}{4} \div 12$ ft = $\frac{1}{48}$ ft? No, that can't be. Wait, maybe the actual length is $\frac{2}{9}$ ft and model length is $\frac{1}{4}$ in. So constant of proportionality $k$ is actual length / model length. So $k = \frac{\frac{2}{9} \text{ ft}}{\frac{1}{4} \text{ in}}$. But we need to have same units. Let's convert feet to inches: $\frac{2}{9}$ ft = $\frac{2}{9} \times 12$ in = $\frac{24}{9}$ in = $\frac{8}{3}$ in. Then $k = \frac{\frac{8}{3} \text{ in}}{\frac{1}{4} \text{ in}} = \frac{8}{3} \div \frac{1}{4} = \frac{8}{3} \times 4 = \frac{32}{3}$? Wait, no, maybe I got the model and actual reversed. Wait, the constant of proportionality for direct variation is $y = kx$, where $y$ is actual, $x$ is model. So $k = y/x$. Let's check the width: model width is $\frac{1}{6}$ in, actual width is $\frac{1}{3}$ ft. Convert $\frac{1}{3}$ ft to inches: $\frac{1}{3} \times 12 = 4$ in. Then $k = 4 / (\frac{1}{6}) = 24$. Wait, that's different. Wait, maybe the length: model length $\frac{1}{4}$ in, actual length $\frac{2}{9}$ ft. Convert $\frac{2}{9}$ ft to inches: $\frac{2}{9} \times 12 = \frac{24}{9} = \frac{8}{3}$ in. Then $k = \frac{8}{3} / (\frac{1}{4}) = \frac{32}{3}$? No, that's not matching. Wait, maybe the problem is about the length ratio. Wait, the user's image shows actual length as $\frac{2}{9}$ ft and model length as $\frac{1}{4}$ in. Wait, maybe the units are same? No, inches and feet. Wait, maybe the problem has a typo, but let's re-express. Wait, the constant of proportionality is actual length / model length. Let's take length: model length $m = \frac{1}{4}$ in, actual length $a = \frac{2}{9}$ ft. Convert $a$ to inches: $\frac{2}{9} \times 12 = \frac{24}{9} = \frac{8}{3}$ in. Then $k = \frac{8}{3} \div \frac{1}{4} = \frac{32}{3}$? No, that seems off. Wait, maybe the width: model width $\frac{1}{6}$ in, actual width $\frac{1}{3}$ ft = 4 in. Then $k = 4 / (\frac{1}{6}) = 24$. Ah, that's a whole number. Maybe the length is a mistake, and the actual length is $\frac{2}{9}$ ft? Wait, no, $\frac{2}{9}$ ft is about 2.666 inches, and model length is 0.25 inches. Then 2.666 / 0.25 = 10.666, which is $\frac{32}{3}$. But the width: 4 inches / 0.1666 inches (1/6 in) = 24. So maybe the problem is about the width? Wait, the question says "the length of a paver in the model and the actual length". Wait, maybe the actual length is $\frac{2}{9}$ ft, model length is $\frac{1}{4}$ in. Let's convert both to feet: model length $\frac{1}{4}$ in = $\frac{1}{48}$ ft. Then $k = (\frac{2}{9}) / (\frac{1}{48}) = \frac{2}{9} \times 48 = \frac{96}{9} = \frac{32}{3}$. But that's a fraction. Wait, maybe the problem is using the width. Wait, model width $\frac{1}{6}$ in, actual width $\frac{1}{3}$ ft = 4 in. Then $k = 4 / (\frac{1}{6}) = 24$. But the question says length. Wait, maybe the actual length is $\frac{2}{9}$ ft, model length is $\frac{1}{4}$ in. Let's do the calculation again. Model length: $\frac{1}{4}$ in = $\frac{1}{4} \div 12$ ft = $\frac{1}{48}$ ft. Actual length: $\frac{2}{9}$ ft. Then $k = \frac{2}{9} \div \frac{1}{48} = \frac{2}{9} \times 48 = \frac{96}{9} = \frac{32}{3}$. But that's $\frac{32}{3}$. Wait, maybe the problem is about the ratio of actual to model, so $k = \frac{\text{actual}}{\text{model}}$. Let's check the numbers again. Wait, the user's image: "the actual dimensions of the pavers are shown" with $\frac{2}{9}$ ft and $\frac{1}{3}$ ft. Model dimensions: $\frac{1}{4}$ in and $\frac{1}{6}$ in. So let's take length: model length $l_m = \frac{1}{4}$ in, actual length $l_a = \frac{2}{9}$ ft. Convert $l_a$ to inches: $\frac{2}{9} \times 12 = \frac{24}{9} = \frac{8}{3}$ in. Then $k = \frac{l_a}{l_m} = \frac{8}{3} \div \frac{1}{4} = \frac{8}{3} \times 4 = \frac{32}{3}$. But that's a fraction. Wait, maybe the problem is using the width: model width $w_m = \frac{1}{6}$ in, actual width $w_a = \frac{1}{3}$ ft = 4 in. Then $k = \frac{w_a}{w_m} = 4 \div \frac{1}{6} = 24$. But the question says length. Maybe there's a mistake, but let's proceed with the length. So $k = \frac{32}{3}$? Wait, no, maybe I messed up the units. Wait, 1 foot is 12 inches, so to convert inches to feet, divide by 12. To convert feet to inches, multiply by 12. So model length: $\frac{1}{4}$ in = $\frac{1}{4} \times \frac{1}{12}$ ft = $\frac{1}{48}$ ft. Actual length: $\frac{2}{9}$ ft. Then $k = \frac{2}{9} \div \frac{1}{48} = \frac{2}{9} \times 48 = \frac{96}{9} = \frac{32}{3}$. So the constant of proportionality is $\frac{32}{3}$. But that seems complicated. Wait, maybe the problem is using the ratio of actual to model without converting units? No, units must be same. Wait, maybe the model length is $\frac{1}{4}$ ft and actual is $\frac{2}{9}$ in? No, that doesn't make sense. Wait, maybe the problem has a typo, and the model length is $\frac{1}{4}$ ft, actual is $\frac{2}{9}$ ft? No, that would be $k = \frac{2}{9} \div \frac{1}{4} = \frac{8}{9}$. No, that's not. Wait, I think I made a mistake. Let's start over.

The constant of proportionality for direct variation is $y = kx$, where $y$ is actual length, $x$ is model length. So $k = y/x$.

Model length (x) = $\frac{1}{4}$ in.

Actual length (y) = $\frac{2}{9}$ ft. Convert $\frac{2}{9}$ ft to inches: $\frac{2}{9} \times 12 = \frac{24}{9} = \frac{8}{3}$ in.

So $k = \frac{8}{3} \div \frac{1}{4} = \frac{8}{3} \times 4 = \frac{32}{3}$.

But that's $\frac{32}{3}$. Wait, maybe the problem is using the width. Model width: $\frac{1}{6}$ in, actual width: $\frac{1}{3}$ ft = 4 in. Then $k = 4 \div \frac{1}{6} = 24$. But the question says length. Maybe the actual length is $\frac{2}{9}$ ft, model length is $\frac{1}{4}$ in. So the calculation is correct. So the constant of proportionality is $\frac{32}{3}$.

Wait, but let's check the arithmetic. $\frac{2}{9}$ ft is 2/9 of a foot, which is (2/9)*12 = 24/9 = 8/3 inches. Model length is 1/4 inch. So 8/3 divided by 1/4 is (8/3)*(4/1) = 32/3. Yes, that's correct. So the constant of proportionality is 32/3.

Step2: Write the equation

The relationship is $y = kx$, where $y$ is actual length (in feet), $x$ is model length (in inches). So $y = \frac{32}{3}x$. But since we converted model length to feet, maybe $x$ is in inches, so we need to adjust. Wait, no, if $x$ is in inches, and $y$ is in feet, then $k$ is in feet per inch. So $k = \frac{32}{3}$ feet per inch. But the problem says "the constant of proportionality that relates the length of a paver in the model and the actual length". So the equation is $y = \frac{32}{3}x$, where $x$ is model length (in inches), $y$ is actual length (in feet). Or if we use the same unit, convert model to feet: $x = \frac{1}{48}$ ft (since 1/4 in = 1/48 ft), then $k = \frac{2}{9} \div \frac{1}{48} = \frac{32}{3}$, so $y = \frac{32}{3}x$, where $x$ is model length in feet. But the model length is given in inches, so probably better to use inches for model and feet for actual, with $k = \frac{32}{3}$ ft/in.

But the question says "the constant of proportionality" as a simplified fraction. So $\frac{32}{3}$? Wait, no, maybe I got the model and actual reversed. If $x$ is actual and $y$ is model, then $k = y/x$, but that would be $\frac{1}{4} \div \frac{8}{3} = \frac{1}{4} \times \frac{3}{8} = \frac{3}{32}$, but that's the inverse. The constant of proportionality for direct variation is $y = kx$, so $k$ is actual over model. So I think it's $\frac{32}{3}$.

Answer:

$\frac{32}{3}$