Question

home value in ohio and kentucky (adjusted for inflation) are shown below:

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$$\begin{tabular}{|c|c|} \\hline ohio & kentucky \\\\ \\hline 50300 & 32000 \\\\ \\hline 103700 & 86700 \\\\ \\hline \\end{tabular}$$

ume that the house values are changing linearly,

ich state have home values increased at a higher rate? select an answer

ese trends were to continue, what would be the median home value in ohio in 2010?

we assume the linear trend existed before 1950 and continues after 2000, the two states median se values will be (or were) equal in what year? (the answer might be absurd)
e year

Explanation:

Based on the standard context of this problem (which typically compares median home values from the years 1950 to 2000), we have the following data points:

• Year 1950 (\(t = 0\)):
• Ohio: \(50,300\)
• Kentucky: \(32,000\)
• Year 2000 (\(t = 50\)):
• Ohio: \(103,700\)
• Kentucky: \(86,700\)

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⚡ Using what you learned: Average Rate of Change

Step 1: Calculate the rate of change for each state

The rate of change (slope) represents how much the home value increases per year over the 50-year period from 1950 to 2000.

For Ohio:

$$ m_{\text{Ohio}} = \frac{103700 - 50300}{2000 - 1950} = \frac{53400}{50} = 1068 \text{ dollars/year} $$

For Kentucky:

$$ m_{\text{Kentucky}} = \frac{86700 - 32000}{2000 - 1950} = \frac{54700}{50} = 1094 \text{ dollars/year} $$

Comparing the rates:

$$ 1094 > 1068 $$

Kentucky's home values increased at a higher rate.

Step 2: Find the median home value in Ohio in 2010

Using the linear model for Ohio, where \(t\) is the number of years after 1950:

$$ V_{\text{Ohio}}(t) = 1068t + 50300 $$

For the year 2010, \(t = 2010 - 1950 = 60\):

$$ V_{\text{Ohio}}(60) = 1068(60) + 50300 = 64080 + 50300 = 114380 $$

Step 3: Find the year when the median home values are equal

Set the linear equation for Ohio equal to the linear equation for Kentucky.

The linear model for Kentucky is:

$$ V_{\text{Kentucky}}(t) = 1094t + 32000 $$

Set them equal to find \(t\):

$$ 1068t + 50300 = 1094t + 32000 $$

$$ 50300 - 32000 = 1094t - 1068t $$

$$ 18300 = 26t $$

$$ t = \frac{18300}{26} \approx 703.85 \text{ years} $$

Find the calendar year by adding \(t\) to 1950:

$$ \text{Year} = 1950 + 703.85 \approx 2653.85 $$

Rounding to the nearest whole year gives 2654.

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Answer:

1. Which state has home values increased at a higher rate? Kentucky
2. Median home value in Ohio in 2010: 114380
3. Year when the values are equal: 2654