Problem 42 - College Algebra

Question

For the following exercises, use the median home values in Indiana and Alabama (adjusted for inflation) shown in Table \(3 .\) Assume that the house values are changing linearly. \(\begin{array}{lll}{\text { Year }} & {\text { Indiana }} & {\text { Alabama }} \\ {1950} & {\$ 37,700} & {\$ 27,100} \\ {2000} & {\$ 94,300} & {\$ 85,100}\end{array}\) In which state have home values increased at a higher rate?

Step-by-Step Solution

Answer

Home values increased at a higher rate in Alabama.

Step 1: Calculate the Rate of Change for Indiana

To find the rate of change for Indiana, we need to calculate the slope of the line connecting the two data points given for 1950 and 2000.The formula for the slope of a line is:\[\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\]For Indiana:\[y_1 = 37700, \ y_2 = 94300, \ x_1 = 1950, \ x_2 = 2000\]Plug the values into the formula:\[\text{slope}_{\text{Indiana}} = \frac{94300 - 37700}{2000 - 1950} = \frac{56600}{50} = 1132\]This means that the median home value in Indiana increased by $1132 each year.

Step 2: Calculate the Rate of Change for Alabama

Next, we calculate the slope for Alabama using the same method.For Alabama:\[y_1 = 27100, \ y_2 = 85100, \ x_1 = 1950, \ x_2 = 2000\]Plug these values into the slope formula:\[\text{slope}_{\text{Alabama}} = \frac{85100 - 27100}{2000 - 1950} = \frac{58000}{50} = 1160\]This implies that the median home value in Alabama increased by $1160 each year.

Step 3: Compare the Rates of Change

Now we compare the annual increase in median home value for both states: - Indiana: $1132 per year - Alabama: $1160 per year Since $1160 (Alabama) is greater than $1132 (Indiana), the home values in Alabama increased at a higher rate.

Key Concepts

Median Home Value

Median home value is a crucial statistic used in real estate to understand the midpoint of home prices in a specific area. This means half of the homes are priced above this value, and the other half are priced below. It offers a more accurate picture than the average because it is less affected by extreme values, like very high or very low prices. In our exercise, we looked at median home values in Indiana and Alabama over two years: 1950 and 2000. This metric helps us analyze how home prices have evolved over time. These values, adjusted for inflation, give us insight into the economic and housing market trends of those periods. Understanding median home value is essential for:

• Evaluating market trends
• Informing both buyers and sellers
• Providing data for policy-making

When analyzing such data over time, it is crucial to also consider how prices change and what factors might cause these variations.

Rate of Change

The rate of change in this context refers to how rapidly the median home prices increase over time. This rate is essential when examining economic indicators and making predictions about future market conditions. To compute this, we use the concept of the slope from mathematics. In the given exercise, it is assumed that the change in home values between two years (1950 and 2000) is linear. This means the values increase steadily over time. The rate of change lets us find out:

• How quickly the housing market is growing
• The effectiveness of economic conditions in each state
• The potential future trends in housing costs

By examining the rate of change for Indiana and Alabama, we gain insights into which state had a more rapidly increasing housing market.

Slope Calculation

Slope calculation is a method used to find the rate at which a line rises or falls—essentially, how steep the line is. In the context of our exercise, it tells us how quickly the home values are changing.To find the slope between two points, use the formula:\[\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}\]where

• \(y_1\) and \(y_2\) are the values of one variable (e.g., home values)
• \(x_1\) and \(x_2\) are the values of the other variable (e.g., years)

In our exercise, Indiana's slope was \(1132\) dollars per year, while Alabama's was \(1160\) dollars per year. This tells us that Alabama's median home values increased at a faster rate.Understanding slope calculation is fundamental when dealing with linear models, as it aids in making:

• Comparisons between different data sets
• Predictions about future trends
• Assessments of economic growth rates over time

By leading with this information, you can develop a deeper understanding of how home values progressed and different factors influencing them.