# The population of Australia in x years after 1980 can be modeled by the function y=14.6...

## Question:

The population of Australia in {eq}x {/eq} years after 1980 can be modeled by the function {eq}y=14.6 (1.014)^x {/eq}. Estimate the population of Australia for each year.

a) 1976.

b) 1980.

c) 1972.

## Exponential Growth: Population Modeling

Suppose that an exponential growth model is given in order to model the population of a particular area by {eq}P_t = P_o(1 + \frac{p}{100})^t {/eq}. In this equation:

• {eq}P_t {/eq} is the size of our population at time {eq}t {/eq} years after the year of interest
• {eq}P_o {/eq} is the original population at {eq}t = 0 {/eq}, or the year of interest
• {eq}p {/eq} is the percentage increase of our population yearly

In order to evaluate the population size, {eq}P_t {/eq}, at a particular value {eq}t {/eq}, the model must be used and evaluated numerically substituting the value of time of interest.

Given an exponential model:

{eq}y = 14.6(1.014)^x {/eq}

a. The year of 1976 corresponds to the {eq}x {/eq} value of:

{eq}x = 1976 - 1980 = -4 {/eq}

Therefore, the population in that year would be:

{eq}y(-4) = 14.6(1.014)^{-4}\approx 13.8 {/eq}

b. The year of 1980 corresponds to:

{eq}x = 1980 - 1980 = 0 {/eq}

So that:

{eq}y(0) = 14.6 {/eq}

c. The year of 1972 can be identified by:

{eq}x = 1972 - 1980 = -8 {/eq}

And the population at that time would be:

{eq}y(-8) = 14.6(1.014)^{-8}\approx 13.1 {/eq}

Thus:

a. The population in 1976 was approximately {eq}\boxed{13.8~\rm{units}} {/eq}

b. The population in 1980 was {eq}\boxed{14.6~\rm{units}} {/eq}

c. The population in 1972 was about {eq}\boxed{13.1~\rm{units}} {/eq} 