Math Snap

STEP 1

Assumptions1. The initial population of Austin, Texas, is494,000.
. The population increases by3% each year.
3. The growth is exponential, meaning the population increases by a fixed percentage each year.
4. We are looking for the population10 years after the beginning of the decade.

STEP 2

The general form of an exponential growth model is $y = a(1 + r)^t$, where- $y$ is the final amount- $a$ is the initial amount- $r$ is the growth rate (expressed as a decimal)
- $t$ is the time (in years)

STEP 3

In this case, the initial amount $a$ is the initial population, which is494,000. The growth rate $r$ is3%, or0.03 when expressed as a decimal. We can substitute these values into the exponential growth model.
$$y =494(1 +0.03)^t$$

STEP 4

implify the equation.
$$y =494(1.03)^t$$This is the exponential growth model that represents the population of Austin, Texas.

STEP 5

To find the population10 years after the beginning of the decade, we substitute $t =10$ into the equation.
$$y =494(1.03)^{10}$$

Calculate the value of $y$.
$$y =494(1.03)^{10} \approx664$$This means that the population of Austin, Texas,10 years after the beginning of the decade is approximately664,000.