Question

g The population of Little Townville in 2005 was 13,625 people. In 2009 the population grew to 15,325 people. Assume that this trend is linear and the population has been growing like this since 2000. a. Write a function for the population, P, in terms of the number of years since 2000, ????. (6 points) b. What is the slope of the function and what does it represent in the context of the problem. Include units. (5 points) c. What is the vertical intercept of the function and what does it represent? Include units (5 points) d. What do you believe the population was in 2015?

Answer 1

a)
`P(t) = 11500 + 425t`

b) The slope is of 425 people, which is by how much the population increases each year.

c) The vertical intercept is of 11500 people, and it represents the population of Little Townville in the year 2000.

d) 17,875 people.

Explanation:

A linear function that measures population after t years has the following format:

`P(t) = P(0) + at`

In which P(0) is the vertical intercept, which represents the initial population, and a is the slope, which represents the yearly population.

a. Write a function for the population, P, in terms of the number of years since 2000.

The first step is finding the slope.

Finding the slope:

The slope is the change in the output divided by the change in the input, during a period of time.

The population of Little Townville in 2005 was 13,625 people. In 2009 the population grew to 15,325 people.

This means that in 2009 - 2005 = 4 years, the population increased by 15325 - 13625 = 1700.

So the slope is

`a = (1700)/(4) = 425`

So

`P(t) = P(0) + 425t`

Finding the initial population:

We want to find P(0), which is the population in 2000. The population of Little Townville in 2005 was 13,625 people, which means that
`P(5) = 13625`. So

`P(t) = P(0) + 425t`

`13625 = P(0) + 425*5`

`P(0) = 13625 - 425*5`

`P(0) = 11500`

So

`P(t) = 11500 + 425t`

b. What is the slope of the function and what does it represent in the context of the problem. Include units.

The slope is of 425 people, which is by how much the population increases each year.

c. What is the vertical intercept of the function and what does it represent? Include units

The vertical intercept is of 11500 people, and it represents the population of Little Townville in the year 2000.

d. What do you believe the population was in 2015?

2015 is 2015 - 2000 = 15 years after 2000, so this is P(15).

`P(15) = 11500 + 425*15 = 17875`

17,875 people.