Question
The median home value in Virginia and New Jersey (adjusted for inflation) are shown below:
Year Virginia New Jersey
1950 39900 63100
2000 125400 170800
If we assume that the house values are changing linearly,
a) In which state have home values increased at a higher rate?
New Jersey
Virginia
the rates equal
b) If these trends were to continue, what would be the median home value in Virginia 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
Year Virginia New Jersey
1950 39900 63100
2000 125400 170800
If we assume that the house values are changing linearly,
a) In which state have home values increased at a higher rate?
New Jersey
Virginia
the rates equal
b) If these trends were to continue, what would be the median home value in Virginia 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
Answers
GPT-4o mini
To solve the problem, we will analyze the growth rates of home values and perform calculations based on the data provided.
**a) Calculate the rate of increase for each state.**
1. **Virginia:**
- Value in 1950: $39,900
- Value in 2000: $125,400
- Change in value: \(125,400 - 39,900 = 85,500\)
- Time period: \(2000 - 1950 = 50 \text{ years}\)
- Rate of increase: \(\frac{85,500}{50} = 1,710\)
2. **New Jersey:**
- Value in 1950: $63,100
- Value in 2000: $170,800
- Change in value: \(170,800 - 63,100 = 107,700\)
- Time period: \(2000 - 1950 = 50 \text{ years}\)
- Rate of increase: \(\frac{107,700}{50} = 2,154\)
**Comparison of rates:**
- Virginia: $1,710 per year
- New Jersey: $2,154 per year
**Answer for part (a):**
**New Jersey** increased at a higher rate.
---
**b) Calculate the median home value in Virginia in 2010, assuming the trend continues.**
- The rate of increase in Virginia (from part a) is $1,710 per year.
- The time from 2000 to 2010 is 10 years.
- Increase over 10 years: \(1,710 \times 10 = 17,100\)
- Median home value in Virginia in 2000: $125,400
- Estimated median home value in 2010: \(125,400 + 17,100 = 142,500\)
**Answer for part (b):**
**$142,500**
---
**c) Find the year when home values in both states would be equal if the trends continued.**
1. **Virginia Home Value Formula:**
\[ V_V(t) = 39,900 + 1,710 \times (t - 1950) \]
2. **New Jersey Home Value Formula:**
\[ V_N(t) = 63,100 + 2,154 \times (t - 1950) \]
Set the two equations equal to find \(t\):
\[
39,900 + 1,710(t - 1950) = 63,100 + 2,154(t - 1950)
\]
Expanding both sides:
\[
39,900 + 1,710t - 3,334,500 = 63,100 + 2,154t - 4,209,300
\]
Simplifying:
\[
1,710t - 2,154t = 63,100 - 39,900 + 4,209,300 - 3,334,500
\]
\[
-444t = 23,200 + 874,800
\]
\[
-444t = 898,000
\]
\[
t = \frac{898,000}{-444} \approx -2,018.02
\]
Since the value of \(t\) is negative and approximately -2018, it suggests that the home values were equal long before available data (in the context given) or implies an absurd finding.
**Answer for part (c):**
**The year -2018 (absurd).**
**a) Calculate the rate of increase for each state.**
1. **Virginia:**
- Value in 1950: $39,900
- Value in 2000: $125,400
- Change in value: \(125,400 - 39,900 = 85,500\)
- Time period: \(2000 - 1950 = 50 \text{ years}\)
- Rate of increase: \(\frac{85,500}{50} = 1,710\)
2. **New Jersey:**
- Value in 1950: $63,100
- Value in 2000: $170,800
- Change in value: \(170,800 - 63,100 = 107,700\)
- Time period: \(2000 - 1950 = 50 \text{ years}\)
- Rate of increase: \(\frac{107,700}{50} = 2,154\)
**Comparison of rates:**
- Virginia: $1,710 per year
- New Jersey: $2,154 per year
**Answer for part (a):**
**New Jersey** increased at a higher rate.
---
**b) Calculate the median home value in Virginia in 2010, assuming the trend continues.**
- The rate of increase in Virginia (from part a) is $1,710 per year.
- The time from 2000 to 2010 is 10 years.
- Increase over 10 years: \(1,710 \times 10 = 17,100\)
- Median home value in Virginia in 2000: $125,400
- Estimated median home value in 2010: \(125,400 + 17,100 = 142,500\)
**Answer for part (b):**
**$142,500**
---
**c) Find the year when home values in both states would be equal if the trends continued.**
1. **Virginia Home Value Formula:**
\[ V_V(t) = 39,900 + 1,710 \times (t - 1950) \]
2. **New Jersey Home Value Formula:**
\[ V_N(t) = 63,100 + 2,154 \times (t - 1950) \]
Set the two equations equal to find \(t\):
\[
39,900 + 1,710(t - 1950) = 63,100 + 2,154(t - 1950)
\]
Expanding both sides:
\[
39,900 + 1,710t - 3,334,500 = 63,100 + 2,154t - 4,209,300
\]
Simplifying:
\[
1,710t - 2,154t = 63,100 - 39,900 + 4,209,300 - 3,334,500
\]
\[
-444t = 23,200 + 874,800
\]
\[
-444t = 898,000
\]
\[
t = \frac{898,000}{-444} \approx -2,018.02
\]
Since the value of \(t\) is negative and approximately -2018, it suggests that the home values were equal long before available data (in the context given) or implies an absurd finding.
**Answer for part (c):**
**The year -2018 (absurd).**