The population of a small town is growing at a constant rate. (Please show me step by step)

Question

The population of a small town is growing at a constant rate. In 2007 the population was 15,400 and in 2013 the population was 22,600.

Part A: Use the data points (7, 15400) and (13, 22600) to write a linear model of the form: f(t) = mt + b, where t is the number of years after 2000 and b is the population in 2000.

Part B: Use the model from Part B to approximate the population in 2020.

Answer 1

Part A: Use the data points (7, 15400) and (13, 22600) to write a linear model of the form: f(t) = mt + b, where t is the number of years after 2000 and b is the population in 2000.

f(t) = mt + b, where t is the time / number of years after 2000, m is the slope, and b is the y-intercept.

Let's call f(t) = f(x) = y and t = x.

m = (y2 - y1) / (x2 - x1), where (x1, y1) = (7, 15400), the 1st set of x and y values, and (x2, y2) = (13, 22600) is the 2nd set of x and y values.

m = (22600 - 15400) / (13 - 7) = 7200 / 6 = 1200 is the slope of the data points, which says that the population grew by 7200 in 6 years, 2007 to 2013.

For the y-intercept, b, just plug in your slope (m = 7200) and then your choice of data points (x1, y1) OR (x2, y2) into the f(t) = mt + b formula, where f(t) = y and t = x.

Then f(x1) = mx1 + b => 15400 = 7200*(7) = b => 15400 = 50400 + b

Solving for b... 15400 - 50400 = 50400 - 54000 + b => -35000 = 0 + b = b

Thus f(t) = mt + b is f(t) = 7200*t - 35000 upon substitution of m and b into the equation.

Part B: Use the model from Part B to approximate the population in 2020.

Using the model you found in part (a), f(t) = 7200*t - 35000, just know that t is the number of years after 2020...which is 2020 - 2000 = 20 years, so t = 20

Now f(20) = 7200*20 - 35000 = 144,000 - 35,000 = 109,000! A fast growing town!

Answer 2

This will fit the form of a straight line y=mx +b so start by finding the slope using the 2 given points

slope(m) = 22600-15400/ 13-7 = 1200

Now you have y= 1200x +b. calculate b using either of the two data points

15400= 1200(7) +b

15400=8400 +b

b= 7000 so your equation is

y = 1200b +7000

2020 is an additional 7 years from 2013 so your x value increases from 13 to 20

so population in 2020 is 1200(20) +7000 =31000