How Do You Find a Linear Model from a Rate

In another MathFAQ,  I examined how we can find the equation of a line from two data points. In this post I want to look at a closely related problem where we find the equation of the line from a rate.

Problem Assume the growth of the population of Del Webb’s Sun City Hilton Head community was linear from 1996 to 2000, with a population of 198 in 1996 and a rate of growth of 705 people per year.

a. Write an equation for the population P of the community where x is the number of years after 1990.

Solution The population of 198 in 1996 corresponds to the point (6, 198) since the variable x corresponds to years after 1990. We’ll write the slope-intercept form of the line, P = mx + b, and substitute m into the equation. The rate of growth, 705 people per year, is the slope of the function. Therefore, the line describing the population is

P = 705x + b

To find the value of b, we need to substitute x = 6 and P = 198:

198 = 705(6) + b

-4032 = b

This gives us the equation,

P = 705x – 4032

b. Use the function to estimate the population in 2002.

Solution The year 2002 corresponds to x = 12. Substitute this value into the function to yield

P = 705(12) – 4032 = 4428

The population in 2002 will be 4428 people.

c. In what year will the population reach 10,000?

Solution In this part, set P = 10,000 and solve for x.

10000 = 705x – 4032

14032 = 705x

14032/705 = x

19.9 ≈ x

This corresponds to 1990 + 19.9 = 2009.9. So in the year 2009, the population will reach 10000. Since we are asking in what year, we DO NOT round up on the answer.