Question
se the image to answer the question.
An illustration shows the first quadrant of a coordinate plane titled Population of a City Since 1900.
The x-axis shows years since 1900 and ranges from 0 to 80 in increments of 10. The y-axis shows population per thousand and ranges from 0 to 80 in increments of 5. Nine points are plotted on the graph. The points are plotted at approximate coordinates left parenthesis 0 comma 20 right parenthesis, left parenthesis 10 comma 28 right parenthesis, left parenthesis 20 comma 30 right parenthesis, left parenthesis 30 comma 35 right parenthesis, left parenthesis 40 comma 47 right parenthesis, left parenthesis 50 comma 55 right parenthesis, left parenthesis 60 comma 57 right parenthesis, left parenthesis 70 comma 68 right parenthesis, and left parenthesis 80 comma 66 right parenthesis. An upward slanting line starts at left parenthesis 0 comma 20 right parenthesis and continues to left parenthesis 60 comma 57 right parenthesis.
The population growth of a city since 1900 is represented by a linear model. Interpret the slope and the y
-intercept.
(1 point)
Responses
For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
For every 0.65 of a year since 1900, the population grew by approximately 1,000. In 1900, the population was 20,000.
An illustration shows the first quadrant of a coordinate plane titled Population of a City Since 1900.
The x-axis shows years since 1900 and ranges from 0 to 80 in increments of 10. The y-axis shows population per thousand and ranges from 0 to 80 in increments of 5. Nine points are plotted on the graph. The points are plotted at approximate coordinates left parenthesis 0 comma 20 right parenthesis, left parenthesis 10 comma 28 right parenthesis, left parenthesis 20 comma 30 right parenthesis, left parenthesis 30 comma 35 right parenthesis, left parenthesis 40 comma 47 right parenthesis, left parenthesis 50 comma 55 right parenthesis, left parenthesis 60 comma 57 right parenthesis, left parenthesis 70 comma 68 right parenthesis, and left parenthesis 80 comma 66 right parenthesis. An upward slanting line starts at left parenthesis 0 comma 20 right parenthesis and continues to left parenthesis 60 comma 57 right parenthesis.
The population growth of a city since 1900 is represented by a linear model. Interpret the slope and the y
-intercept.
(1 point)
Responses
For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
For every 0.65 of a year since 1900, the population grew by approximately 1,000. In 1900, the population was 20,000.
Answers
GPT-4o mini
To interpret the slope and the y-intercept from the given graph, let’s consider the coordinates of the graph.
The y-intercept is the value of the population at the year 0 (which corresponds to the year 1900 in this case). From the graph, it appears that at (0, 20), the population is 20 (implying 20,000 people when considering 'per thousand').
The slope measures the change in population per year. Between two points, such as (0, 20) and (10, 28), we can calculate the slope as:
\[
\text{slope} = \frac{\text{change in population}}{\text{change in years}} = \frac{28 - 20}{10 - 0} = \frac{8}{10} = 0.8
\]
This slope means that the population grows by approximately 0.8 per thousand per year, which when scaled up corresponds to approximately 800 people per year (since each unit on the y-axis represents 1,000).
Now looking at the choices provided in the question, the most fitting interpretation of the slope and the y-intercept seems to be:
**For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.**
Even though the slope was calculated to be approximately 800 (0.8 per year indicates an average increase), a rounding or representation in the options led to this choice. It represents a close approximation of growth based on the linear model presented in the graph, with the intercept correctly defined around 20,000.
So, the correct response is:
**For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.**
The y-intercept is the value of the population at the year 0 (which corresponds to the year 1900 in this case). From the graph, it appears that at (0, 20), the population is 20 (implying 20,000 people when considering 'per thousand').
The slope measures the change in population per year. Between two points, such as (0, 20) and (10, 28), we can calculate the slope as:
\[
\text{slope} = \frac{\text{change in population}}{\text{change in years}} = \frac{28 - 20}{10 - 0} = \frac{8}{10} = 0.8
\]
This slope means that the population grows by approximately 0.8 per thousand per year, which when scaled up corresponds to approximately 800 people per year (since each unit on the y-axis represents 1,000).
Now looking at the choices provided in the question, the most fitting interpretation of the slope and the y-intercept seems to be:
**For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.**
Even though the slope was calculated to be approximately 800 (0.8 per year indicates an average increase), a rounding or representation in the options led to this choice. It represents a close approximation of growth based on the linear model presented in the graph, with the intercept correctly defined around 20,000.
So, the correct response is:
**For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.**