Using Trend Lines Quick Check
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Question
Use 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
a For every 0.65 of a year since 1900, the population grew by approximately 1,000. In 1900, the population was 20,000..
b For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
c For every year since 1900, the population grew by approximately 1,590. in 1900, the population was 20,000.
d For every year since 1900, the population grew by approximately 300. In 1900, the population was 0.
The correct response is:
b) For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
b For every year since 1900, the population grew by approximately 650. In 1900, the population was 20,000.
To determine the slope and y-intercept of the linear model representing population growth, we need to analyze the given information and the graph.
The graph shows a line that starts at the point (0, 20) and continues to the point (60, 57). We can use these two points to find the slope of the line. The slope is calculated using the formula:
slope = (change in y)/(change in x)
In this case, the "change in y" is the difference between the y-coordinates of the two points, which is 57 - 20 = 37. The "change in x" is the difference between the x-coordinates of the two points, which is 60 - 0 = 60.
Therefore, the slope = 37/60 ≈ 0.6167.
Interpretation of the slope: For every 1 unit increase in x (representing 1 year since 1900), the population grows by approximately 0.6167 units.
Next, we need to find the y-intercept of the line, which is the point where the line intersects the y-axis. From the given information, we know that in 1900 (when x = 0), the population was 20 (per thousand). Therefore, the y-intercept is 20.
Interpretation of the y-intercept: In 1900, the population was 20 (per thousand).
Based on this analysis, the correct interpretation of the slope and y-intercept is:
For every 1 year since 1900, the population grew by approximately 0.6167 (or 616.7 per thousand). In 1900, the population was 20 (per thousand).
Therefore, the correct answer is option a:
For every 0.65 of a year since 1900, the population grew by approximately 1,000. In 1900, the population was 20,000.