Problem 42 - Elementary and Intermediate Algebra

Question

Solve each problem by writing a variation model. The following table shows the listing price for three homes in the same general locality. Write the variation model (direct or inverse) that describes the relationship between the listing price and the number of square feet of a house in this area. $$ \begin{array}{|c|c|} \hline \text { Number of square feet } & \text { Listing price } \\ \hline 1,720 & \$ 180,600 \\ \hline 1,205 & \$ 126,525 \\ \hline 1,080 & \$ 113,400 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The variation model is a direct variation: \( P = 105S \).

Step 1: Understanding Variation Types

Identify whether the relationship between the house size (square feet) and price is direct or inverse. In this context, a direct variation means when the square footage increases, the price increases. An inverse variation would mean that as square footage increases, the price decreases.

Step 2: Formulating Direct Variation Model

Since generally, larger homes have higher listing prices, we assume a direct variation model. In a direct variation, price \( P \) is directly proportional to square feet \( S \), i.e., \( P = kS \) where \( k \) is a constant.

Step 3: Calculating the Proportional Constant

Calculate the constant \( k \) using one of the data points. Take \( P = 180,600 \) and \( S = 1,720 \): \( 180,600 = k \times 1,720 \). Solve for \( k \) to get \( k = \frac{180,600}{1,720} = 105 \).

Step 4: Verifying the Model

Use the other data points to verify the model: For \( S = 1,205, P = k \times 1,205 = 105 \times 1,205 = 126,525 \) and for \( S = 1,080, P = k \times 1,080 = 105 \times 1,080 = 113,400 \). Both match the listing prices.

Key Concepts

Direct Variation

In mathematics, direct variation describes a simple yet powerful relationship between two variables. Imagine two quantities that move together in harmony. When one rises, the other follows suit, and when one falls, the other dwindles too. This is what we call direct variation. In a direct variation, one variable is completely dependent on another. You see this frequently in real-life scenarios. For example, the total cost of apples is directly proportional to the number of apples bought. More apples, more cost, and vice versa.

In the context of our real estate problem, we hypothesize that the house listing price varies directly with the square footage. This is an intuitive conclusion because, typically, larger houses (more square feet) are priced higher. In mathematical terms, this relationship is expressed as:\[ P = kS \]where \( P \) is the price, \( S \) is the square footage, and \( k \) is the constant of proportionality that we calculate using the data provided.

Direct variation is foundational in helping us understand and create models that describe real-world situations succinctly.

Proportional Relationships

Proportional relationships are the cornerstone of understanding variation models like direct variation. When two quantities maintain a constant ratio, they're in a proportional relationship. As one variable increases or decreases, the other changes in a way that keeps this ratio unaltered.

In our exercise, the relationship between square feet and listing price is direct and proportional. This means that for every square foot increase in size, the cost increases by a consistent amount. This is important because:

• It helps in predicting prices for different home sizes.
• Makes communication between buyers and sellers clearer.
• Provides a straightforward way to set and compare prices across similar properties.

To see this in action, remember our equation:\[ P = kS \]Here, \( k \) is the constant of proportionality. We found this constant through calculation, and it serves as the fixed amount by which price increases per square foot. This consistent ratio is what makes proportional relationships so vital in direct variation scenarios.

Mathematical Modeling

Mathematical modeling is a step-by-step process that transforms real-world problems into mathematical expressions. It serves as a tool to analyze, understand, and predict scenarios, drawing from data to derive meaningful conclusions.

In our house pricing exercise, creating a mathematical model involves identifying the type of variation (direct or inverse) and deriving the equation that defines the relationship between listing price and square footage. By using the direct variation model:\[ P = kS \]we set up a framework that allows us to compute price based on size.

This model isn't merely theoretical. We tested it by using different data points, ensuring that our model holds true in each case. Testing the model with multiple data inputs confirms its reliability and accuracy, making it a practical tool for evaluations.

• Modeling simplifies complex relationships into understandable forms.
• Helps in predicting changes and trends.
• Facilitates decision-making in diverse fields like economics, engineering, and real estate.

By converting abstract numerical relationships into tangible, usable models, mathematical modeling empowers individuals to make insights-driven decisions.