College Algebra Question

Question

The population of a small town in central Florida has shown a linear decline in the years 2001-2011. In 2001 the population was 25500 people. In 2011 it was 18200 people.

A)  Write a linear equation expressing the population of the town, , as a function of , the number of years since 2001. Include the entire equation as your answer.

Answer ?

B) If the town is still experiencing a linear decline, what will the population be in 2016?

Answer 1

Hi, Gabrielle:

The question specifies a linear decline. This means the population falls in a straight line from year to year.

So, we need an equation of the from: P(t) = mt + b [where P(t) is the population at year t after 2001]

This equation has 2 unknowns we must find, namely, m and b

From the information given, P(0) = 25,500: 25,500 = m x 0 +b . . . . . . . eqn (1)

and P(10) = 18,200: 18,200 = m x 10 + b . . . . . . . eqn (2)

From eqn (1), we see that: b = 25,500

Plug this into eqn (2): 18,200 = 10m + 25,500

-7,300 = 10m

so m = -730 [the number of residents lost per year]

Therefore, answer A: the population equation is: P(t) = -730t + 25,500

Part B: In 2016, t = 2016 - 2001 = 15 years. Plug this into eqn (3): P(15) = -730 x 15 + 25,500

= 14,550

Answer 2

Remember that in order to determine a linear equation, we need two points (or one point and a given slope). In this case, we have two points: (years since 2001, pop) = (0,25500), (10,18200). To find the linear equation, we use the point-slope form:

P - P₀ = m(t - t₀)

P - 25500 = m(t-0).

The slope, m, is given by the change in population divided by the change in time:

m = (18200-25500)/(10-0).

For part b, once you have the equation P = mt+b, you can compute P(t=15) to find population in 2016.