Answer :
Let's break down each possible interpretation of the linear model for the population growth of a city starting in the year 1900.
A linear model can be generally represented by the equation:
[tex]\[ P(t) = mt + b \][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at year [tex]\( t \)[/tex],
- [tex]\( m \)[/tex] is the slope of the line, representing the rate of population growth per year,
- [tex]\( t \)[/tex] is the number of years since 1900,
- [tex]\( b \)[/tex] is the y-intercept, representing the population in the year 1900.
Now let's analyze each given interpretation to see which one correctly describes the linear model.
1. First interpretation:
- Slope: For every year since 1900, the population grew by approximately 300.
- Y-intercept: In 1900, the population was 0.
This suggests
[tex]\[ m = 300 \][/tex]
and
[tex]\[ b = 0 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 300t \][/tex]
This means that for each year after 1900, the population increases by 300. In 1900, the population starts at 0. This interpretation seems less likely as population starting at 0 is unusual.
2. Second interpretation:
- Slope: For every year since 1900, the population grew by approximately 1,590.
- Y-intercept: In 1900, the population was 20,000.
This suggests
[tex]\[ m = 1590 \][/tex]
and
[tex]\[ b = 20000 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 1590t + 20000 \][/tex]
This translates to an initial population of 20,000 in the year 1900 and an increase of 1,590 every year thereafter. This is a viable interpretation.
3. Third interpretation:
- Slope: For every year since 1900, the population grew by approximately 650.
- Y-intercept: In 1900, the population was 20,000.
This suggests
[tex]\[ m = 650 \][/tex]
and
[tex]\[ b = 20000 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 650t + 20000 \][/tex]
This means an initial population of 20,000 in 1900 and a yearly increase of 650. This is also a likely interpretation.
4. Fourth interpretation:
- Slope: For every 0.65 of a year since 1900, the population grew by approximately 1,000.
- Y-intercept: In 1900, the population was 20,000.
Converting the slope to a per year basis:
[tex]\[ m = \frac{1000}{0.65} \approx 1538.46 \text{ per year} \][/tex]
and
[tex]\[ b = 20000 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 1538.46t + 20000 \][/tex]
While technically correct with proper conversion, the less common phrasing makes this interpretation less straightforward.
Given these detailed evaluations, the most logical explanation in the context of a clear linear growth model is:
- For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
A linear model can be generally represented by the equation:
[tex]\[ P(t) = mt + b \][/tex]
where:
- [tex]\( P(t) \)[/tex] is the population at year [tex]\( t \)[/tex],
- [tex]\( m \)[/tex] is the slope of the line, representing the rate of population growth per year,
- [tex]\( t \)[/tex] is the number of years since 1900,
- [tex]\( b \)[/tex] is the y-intercept, representing the population in the year 1900.
Now let's analyze each given interpretation to see which one correctly describes the linear model.
1. First interpretation:
- Slope: For every year since 1900, the population grew by approximately 300.
- Y-intercept: In 1900, the population was 0.
This suggests
[tex]\[ m = 300 \][/tex]
and
[tex]\[ b = 0 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 300t \][/tex]
This means that for each year after 1900, the population increases by 300. In 1900, the population starts at 0. This interpretation seems less likely as population starting at 0 is unusual.
2. Second interpretation:
- Slope: For every year since 1900, the population grew by approximately 1,590.
- Y-intercept: In 1900, the population was 20,000.
This suggests
[tex]\[ m = 1590 \][/tex]
and
[tex]\[ b = 20000 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 1590t + 20000 \][/tex]
This translates to an initial population of 20,000 in the year 1900 and an increase of 1,590 every year thereafter. This is a viable interpretation.
3. Third interpretation:
- Slope: For every year since 1900, the population grew by approximately 650.
- Y-intercept: In 1900, the population was 20,000.
This suggests
[tex]\[ m = 650 \][/tex]
and
[tex]\[ b = 20000 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 650t + 20000 \][/tex]
This means an initial population of 20,000 in 1900 and a yearly increase of 650. This is also a likely interpretation.
4. Fourth interpretation:
- Slope: For every 0.65 of a year since 1900, the population grew by approximately 1,000.
- Y-intercept: In 1900, the population was 20,000.
Converting the slope to a per year basis:
[tex]\[ m = \frac{1000}{0.65} \approx 1538.46 \text{ per year} \][/tex]
and
[tex]\[ b = 20000 \][/tex]
Therefore, the model would be
[tex]\[ P(t) = 1538.46t + 20000 \][/tex]
While technically correct with proper conversion, the less common phrasing makes this interpretation less straightforward.
Given these detailed evaluations, the most logical explanation in the context of a clear linear growth model is:
- For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
