This week you will be graphing the function from Project 3. To find the equation for this function, you need to utilize the initial population and doubling time of the population. The goal of this post is to help you to find the rate r in the function $latex A(t)=P{{e}^{rt}}$. You will need to use the doubling time assigned to you in the project letter to do this.
Let’s take a look at an increasing exponential function.
This graph starts at an initial amount of 25 and doubles to 50 in 100 units of time. Additionally, that amount doubles to 200 in another 100 units of time. This tells us that the doubling time of this exponential function is 100 units. We can use this information to find the rate r in the exponential function.
In general, if the initial amount is P, double that amount is 2P. If we know that this happens in 100 units of time, we can set A(t) = 2P and t = 100 and solve for the rate r.
$latex 2P=P{{e}^{r(100)}}$
Divide both sides by P to give
$latex 2={{e}^{r(100)}}$
Now convert this equation to logarithm form:
$latex 100r=ln ( 2)$
Divide both sides by 100 to give
$latex r=\frac{ln ( 2 )}{100}\approx 0.00693$
This means that we can now graph the function using the formula
$latex A(t)=P{{e}^{0.00693t}}$
with an appropriate value put in place of P.
Each of you has a different initial number for your population on the island and a different doubling time. So you will have a unique P and r for your function.
Once you have graphed your function in Tech 8, check the graph to make sure it has the proper initial amount and that the initial amount doubles in the proper amount of time.