Question

The median home value in Pennsylvania and Rhode Island (adjusted for inflation) are shown below:

Year Pennsylvania Rhode Island
1950 42400 59200
2000 97000 133000


If we assume that the house values are changing linearly,

a) In which state have home values increased at a higher rate?
pennyslvania
the rates are equal
Rhode Islands

b) If these trends were to continue, what would be the median home value in Pennsylvania 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 analyze the home value trends in Pennsylvania and Rhode Island, we can compute the annual increase (rate of change) for each state between the years 1950 and 2000.

### a) Rate of increase for each state

**Pennsylvania:**
- Home value in 1950: $42,400
- Home value in 2000: $97,000
- Number of years: 2000 - 1950 = 50 years

Rate of change for Pennsylvania:
\[
\text{Rate of Increase (PA)} = \frac{97000 - 42400}{50} = \frac{54600}{50} = 1092 \text{ dollars per year}
\]

**Rhode Island:**
- Home value in 1950: $59,200
- Home value in 2000: $133,000
- Number of years: 2000 - 1950 = 50 years

Rate of change for Rhode Island:
\[
\text{Rate of Increase (RI)} = \frac{133000 - 59200}{50} = \frac{73800}{50} = 1476 \text{ dollars per year}
\]

### Comparison:
- Pennsylvania: $1,092 per year
- Rhode Island: $1,476 per year

**Answer for part (a):** Rhode Island

---

### b) Median home value in Pennsylvania in 2010

To find the median home value in Pennsylvania in 2010, we can extend the trend established from the previous calculation.

**Years from 2000 to 2010:** 10 years

Using the 2000 value (97,000) and the rate of increase (1,092), we can calculate the 2010 value:
\[
\text{Value in 2010} = 97000 + (10 \times 1092) = 97000 + 10920 = 107920
\]

**Answer for part (b):** $107,920

---

### c) Determine the year when home values were equal

We can use the linear equations we derived for each state to find out when the values would be equal.

Let \( t \) be the number of years since 1950.

**For Pennsylvania:**
\[
Value_{PA}(t) = 42400 + 1092t
\]

**For Rhode Island:**
\[
Value_{RI}(t) = 59200 + 1476t
\]

Setting them equal:
\[
42400 + 1092t = 59200 + 1476t
\]

Rearranging gives:
\[
42400 - 59200 = 1476t - 1092t
\]
\[
-16800 = 384t
\]
\[
t = \frac{-16800}{384} \approx -43.75
\]

Now, we convert \( t \) back to the actual year:
\[
1950 - 43.75 \approx 1906.25
\]

Since years must be whole numbers, we can round to the nearest year.

**Answer for part (c):** The year: 1906 (this is indeed absurd, as it precedes the data given).