Question

A city’s population has been growing linearly. In 2008, the population was 28,200. By 2012, the population was 36,800. Assuming this trend continues, which of the following is a linear function that properly models the relationship between the town's population, P, and the years that have passed since 2008, t. Select the correct answer below: P(t)=2,150t+28,200 P(t)=28,200t+2,150 P(t)=2,150t+2,008 P(t)=4t+28,200

Answer 1

P(t) = 2150t + 28200

Explanation:

• Since population changes with time, population (P) is a function of time (t).

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General form of the slope-intercept form:

• Since the 2008 population was 28200 and the 2012 population was 36800, we have the points (2008, 28200) and (2012, 36800).

Since we have multiple points, we can model the function in slope-intercept form, whose general equation is given by:

y = mx + b, where:

• m is the slope,
• and b is the y-intercept.

In terms of P and t, our general equation is given by:

P(t) = mt + b, where:

• P(t) is the population after t years,
• m is the change in population per year,
• and b is the initial population size.

Finding the slope:

We can find the slope using the slope formula, which is given by:

m = (y2 - y1) / (x2 - x1) , where:

• m is the slope,
• (x1, y1) is one point on the line,
• and (x2, y2) is another point.

Thus, we can find the slope by substituting (2008, 28200) for (x1, y1) and (2012, 36800) for (x1, y1) in the slope formula:

m = (36800 - 28200) / (2012 - 2008)

m = 8600 / 4

m = 2150

Thus, the slope (m) of the line is 2150.

Identifying the y-intercept and determining the correct function:

• Since we want the linear function to model the population since 2007, 28200 is the y--intercept.

Therefore, P(t) = 2150t + 28200 is a linear function that properly models the relationship between the town's population, P, and the years that have passed since 2008, t.