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

Question

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 0.65 of a year since 1900, the population grew by approximately 1,000. 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.

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 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.

Answer 1

To analyze the slope and the y-intercept from the given data, we can derive the following:

1. Y-Intercept: The y-intercept corresponds to the population when \(x = 0\), which is when the year is 1900. According to the data, the population at this point is approximately 20 thousand. Therefore, the y-intercept indicates that the initial population of the city in the year 1900 was 20,000.

2. Slope: The slope represents the rate of change of the population with respect to the years since 1900. The slope can be calculated using two points from the graph. Let's take the points corresponding to year 0 and year 60:

   • At year 0 (1900), population = 20 thousand.
   • At year 60 (1960), population = 57 thousand.

Using the slope formula \( m = \frac{y_2 - y_1}{x_2 - x_1} \):

• \( y_1 = 20 \), \( y_2 = 57 \)
• \( x_1 = 0 \), \( x_2 = 60 \)

Calculating the slope: \[ m = \frac{57 - 20}{60 - 0} = \frac{37}{60} \approx 0.617 \text{ (thousand per year)} \]

This implies that approximately 617 additional people (since the population is in thousands) are added to the city's population each year.

Taking all of this into account, the statements presented can be analyzed:

• For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000. (Correct slope and intercept)

Based on this, the statement that best matches our analysis is:

For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.