Question

Will give medal to first answer

Iris has been studying an invasive population of snails. This particular snail has no local predators so the population grows wildly. She has observed that the population follows an exponential rate of growth for fifteen years.

Create your own exponential function, f(x), which models the snail population. You will need to identify the principal population of the snails and the rate of growth each year. Explain to Iris how your function shows the principal population and the rate of growth, in complete sentences.

Answer 1

First consider that the formula f(x) = P (1 + r) ^x, is the formula used for compound interest.
In a bank if you invest $1 and the APR (annual percent rate) is 5% will you will end up with
$1,05 at the end of the year
The first consideration in this case is that you have to get at least your dollar back.
That is where the "1 +" comes in to play... now for you extra money (the interest) that is the "r"
It would be 5% (.05)... so if you keep the money in there for one year you get 1(1+.05)^1
or 1.05 ... now the compounding ... if you keep it there for a second yeat then the total is
1(1.05)(1.05) OR .... 1(1.05)^2

Now if the compounding is more than once a year (lets say quarterly) then the compounding periods has to be multiplied by 4 and the rate has to be divided by 4

1(1 + .05/4)^ 2*4 for two years...

The number of compounding can be any number...
Now there is a special situation where there is a infinite compoundings per year.
when done the equation produces a special number "e" (2.71...)
this is called continuous compounding... this actually will model things like radioactive decay
f(x)= P e^r

bottom line here is if the snails reproduce continuously you would use the "e" formula...
f(x) = P (1 + r) ^x.