How do you write an exponential function to model the situation, then find the value of the function after 5 years to the nearest whole number. A population of 200 people that decreases at an annual rate of 6%?

Aug 8, 2016

We can now write this is a function of time in years leading to $f \left(x\right) = 200 {\left(.94\right)}^{x}$

$f \left(5\right) \approx 147$

Explanation:

Firs you should consider that if its exponential then it has some form similar to ${3}^{x}$. Where we have some known part (3) and the unknown part ($x$) that we are trying to find out. Mathematically we can say it has the form ${a}^{x}$. In this case we know what the $a$ is and that is the entire population that was given to us. We know over time it will change but why are we even using the exponential model.

Well it turns out that if you multiply some value over and over again it has this form. Example we multiply $2 \cdot 2 \cdot 2 \cdot 2 = {2}^{4}$. So if something doubled over time then this is what would happen.

Now the problem is that it will continue to decrease at the same rate leading to $a \cdot \left(.94\right)$ because only 94\% of the population remains each year. After two years $a \cdot \left(.94\right) \cdot \left(.94\right)$ . We can now write this is a function of time in years leading to $f \left(x\right) = 200 \cdot {\left(.94\right)}^{x}$
$f \left(5\right) \approx 147$