An illustration shows the first quadrant of a coordinate plane titled Population of a City Since 1900.

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 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 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 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.

Answer 1

Bot GPT 4-o mini answered
7 days ago

To interpret the slope and the y-intercept of the linear model represented by the graph, we start with the coordinates and the information given:

The y-intercept represents the population in the year 1900. According to the graph, the population at (0, 20) means that in 1900, the population was 20 (where "20" refers to thousands, so it corresponds to 20,000).
The slope of the line indicates the change in population over time. In this case, the change in population can be calculated using two points on the line. We can use the points (0, 20) and (10, 28).

The slope (m) can be calculated with the formula:

\[ m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} \]

Using the two points:

\( y_1 = 20 \) (in thousands)
\( y_2 = 28 \) (in thousands)
\( x_1 = 0 \)
\( x_2 = 10 \)

So:

\[ m = \frac{28 - 20}{10 - 0} = \frac{8}{10} = 0.8 \]

This slope of 0.8 means that for every year since 1900, the population grew by approximately 0.8 thousand, or 800 people.

Now interpreting this in the context of the options provided:

The first option indicates a growth of approximately 650 people, which is not consistent with our calculation.
The second option states that the population grew by 300, which is also not correct.
The third option states a growth of approximately 1,590 in 1900 population of 20,000, which is incorrect.
The fourth option states approximately 1,000 growth for 0.65 years, which relates to a non-integer portion of years, not matching our previous computations.

Thus, the correct interpretation of the slope and the y-intercept is:

In 1900, the population was 20,000, and the population grew by approximately 800 each year since 1900.

Since this option is not given directly, we need to conclude based on the corrected understanding of the data from the graph. Please check the answer choices once more for any rephrasing that may represent this data accurately.