In 1980 the population of alligators in a particular region...

In 1980 the population of alligators in a particular region was estimated to be 1400. In 2004 the population had grown to an estimated 5000. Using the Malthusian law for population growth, estimate the alligator population in this region in the year 2020. The alligator population in this region in the year 2020 is estimated to be . (Round to the nearest whole number as needed.)

Transcript

00:01 In this question, we're told that the population of alligators in 1980 is 1400, and its population in 2004 is 5 ,000.

00:12 So assuming a malthusian growth rate, we want to know how many alligators there will likely be in 2020.

00:21 So the malthusian growth rate is defined as the population at time t will be equal to p times e to the power of r.

00:31 Whereas p is the initial population at time t zero and r is a growth rate.

00:39 So to solve this problem we want to plug these numbers into this equation to get the known values.

00:54 So at time t equals 180 we know the population is 1400 so plugging those values into this equation gives us this equation and likewise with the other data and then we want to solve this equation at time t equals to 2020 what will the population of alligators be so to solve this we first want to solve for the values of p and r and not doing that will allow us to solve the equation for 2020 so to solve for the value of r we're going to divide this second equation by the first equation that will eliminate the value of p and then we can solve for r so dividing the second equation by the first will give us 1500 over 5 ,000 over 1 ,400 on the one side and this expression over this expression on the other side and so then to simplify using a lot of exponents we get this and then if we take the natural logarithm of both sides we'll get this and then simplifying and solving for r will get this.

02:07 And so now we've solved for r.

02:09 So now we want to find the value of p.

02:13 And to do so, we can use the value we computed for r and plug it back into one of these equations.

02:19 Let's say we'll plug it back into the second equation.

02:22 So if we plug r into the second equation, we're going to get this.

02:27 And then from there, we can just divide out the exponential portion on the lab.

02:32 To solve for p.

02:35 So now we've solved for r and p, and now we can use these values of r and p and plug them back in to this third expression here in order to derive the desired value, which is the population at time 2020...

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