Question

The median home value in Minnesota and California (adjusted for inflation) are shown below:

Year Minnesota California
1950 47300 57900
2000 122400 211500


If we assume that the house values are changing linearly,

a) In which state have home values increased at a higher rate?
the rates are equal
Minnesota
California
b) If these trends were to continue, what would be the median home value in Minnesota in 2010?
$


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

Answer 1

To determine the answers to the questions, we will first calculate the annual rate of change for home values in both Minnesota and California between the given years.

### a) Rate of Change Calculation

**Minnesota:**
- Home Value in 1950: $47,300
- Home Value in 2000: $122,400
- Time Period: 2000 - 1950 = 50 years

Rate of change in Minnesota:
\[
\text{Rate} = \frac{\text{New Value} - \text{Old Value}}{\text{Years}} = \frac{122400 - 47300}{50} = \frac{75000}{50} = 1500 \text{ per year}
\]

**California:**
- Home Value in 1950: $57,900
- Home Value in 2000: $211,500
- Time Period: 2000 - 1950 = 50 years

Rate of change in California:
\[
\text{Rate} = \frac{211500 - 57900}{50} = \frac{153600}{50} = 3072 \text{ per year}
\]

### Conclusion for a):
California's home values are increasing at a higher rate than Minnesota's.

**Answer: California**

---

### b) Median Home Value in Minnesota in 2010

To find the median home value in Minnesota in 2010, we will use the rate of change we calculated and apply it for an additional 10 years after 2000.

Home Value in 2000: $122,400
Years from 2000 to 2010: 10 years

Using the calculated rate of change:
\[
\text{Value in 2010} = \text{Value in 2000} + (\text{Rate} \times \text{Years})
\]
\[
\text{Value in 2010} = 122400 + (1500 \times 10) = 122400 + 15000 = 137400
\]

**Answer: $137,400**

---

### c) Year when Median House Values are Equal

Next, we want to find when the home values in Minnesota and California would be the same, assuming a linear trend before 1950 and after 2000.

Let \( x \) be the number of years after 1950. The equations for the states' home values would be:

**Minnesota:**
\[
V_m = 47300 + 1500x
\]

**California:**
\[
V_c = 57900 + 3072x
\]

Set \( V_m = V_c \):

\[
47300 + 1500x = 57900 + 3072x
\]

Now, rearranging the equation:

\[
47300 - 57900 = 3072x - 1500x
\]
\[
-10600 = 1572x
\]
\[
x = \frac{-10600}{1572} \approx -6.74 \text{ years}
\]

Since \( x \) represents the number of years after 1950, we convert this to a year:

\[
1950 - 6.74 \approx 1943.26
\]

This means that approximately in **1943**, the two states' median home values would have been equal, which is before the given data point of 1950.

**Answer: The year 1943** (note that this is an absurd result, as it is prior to the data given).