Question

In 2005, the population of a town was 17,500. Suppose the population grows exponentially at a rate of 3% per year. Write an equation to model this situation, let x represent the number of years since 2005.

Answer 1

This situation represents exponential growth

The rate of growth or decay, r, is equal to 0.10

So each year the number of residents in the town is 1.10 times the number in the previous year.

There were around 14,641 residents in the town 4 years after 2005.

Explanation:

Since the number of residents increases by a constant percentage each year, this situation represents exponential growth.

The number of residents increases by 10% per year. So the rate of growth, r, is 10%, or 0.1.

Use the equation to find b, the growth factor by which the population increases each year.

So the population increases by a factor of 1.1 each year. This growth factor means that the number of residents in the town in any year is 1.1 times the number in the previous year.

Write the exponential expression to model the town’s growth by substituting 10,000 for A and 1.1 for b in the general form of the exponential expression.

Set this expression equal to 14,641, and isolate the exponential term.

Use a table of values to rewrite 1.4641 as a value with a base of 1.1.

So there were around 14,641 residents in the town 4 years after 2005.

Answer 2

The expression is:
`p(x) = 17500*(1.03)^x`

Explanation:

Since we're starting our count in 2005, then let's call that year 0 and every year after that will be 2005 + x, where t grows from 0 as an integer.

In 2005 the population is:

`p(0) = 17500`

In 2006 the population is:

`p(1) = 17500*(1 + (3)/(100)) = 17500*(1.03)`

In 2007 the population is:

`p(2) = p(1)*(1.03) = 17500*(1.03)*(1.03) = 17500*(1.03)^2`

In 2008 the population is:

`p(3) = p(2)*(1.03) = 17500*(1.03)^2*(1.03) = 17500*(1.03)^3`

We can notice a pattern that the population pf the city is equal to the number of the starting population multiplied by the rate of change powered by the number of years after 2005. Therefore a generic expression for the population of that town can be written as:

`p(x) = 17500*(1.03)^x`