Question

In what they call “Year Zero," a group of 26 people started a settlement. Every yearthe population changes as babies are born, people move in, and people move out.Generally, the population increases by an average of 2.6 people a year.At the same time, a nearby established community discovered that their populationcould be described by the following function, where f(x) is the population, inpeople, and x is the time, in years, from "year zero."f(x) = -5.3x + 256Part AUsing your knowledge of functions, explain specifically why both communities' waysof expressing their populations represent functions. Provide evidence to supportyour answerPart BAnalyze the functions and compare the populations for the two communities overtime by describing in detail under what conditions one community's population isgreater than the other's population. Provide evidence to support your answer.

Answer 1

Solution

In the first paragraph,

It is given that a group of 26 people started a settlement and the population increased by an average of 2.6 people a year.

We can represent the population function as ;

g(x) = 26 + 2.6 x

Where x denotes the number of years and g(x) is the population after some certain years.

At a nearby community, it was discovered that the population can be written as;

f(x) = -5.3x + 256

Part A.

The population can be expressed as a function because the population at a particular time depends on the number of years x.

Specifically, it can be represented as a linear function because the rate at which the population increases per year is constant.

Part B.

Equating the functions

-5.3x + 256 = 26 + 2.6x

=> 5.3x + 2.6x = 256 - 26

=> 7.9x = 230

=> x = 29

Therefore, if the number of years is less than 29

The population of the first community will be less than the population of the second community

If the number of years is greater than 29

The population of the first community will be greater than the population of the second community