So here we have a 3 part example problem where we can apply the equations of the exponential population growth model. And so part number 1 says, to calculate the population size at t equals 3 years using the initial population size of 100 and a constant per capita population growth rate, or r, of 2.303 individuals per year per individual. Now, of course, recall from our previous lesson videos that any population will grow exponentially as long as the per capita population growth rate, or r, is both positive and constant. And so 2.303 is positive and we're told by the problem that it's also constant. So this population will be growing exponentially.
Let's go ahead and pull up the equations for the exponential population growth model. So how do I know which of these three equations I need to use for this first part? Well, we need to look for clues in the problem, and so notice this problem wants us to calculate the population size, more specifically, the final population size after 3 years. Recall that the final population size is abbreviated with the variable nt, and notice that the only equation that allows us to calculate the final population size or nt directly is this first equation on the far left where nt is already isolated. So that's the equation that we're going to use here for this problem, so let's go ahead and get rid of these other equations.
All we need to do now is just go ahead and plug in some variables and type them into our calculators. So nt, the final population size we're trying to calculate, is going to be equal to n0, which is the initial population size given to us as 100, and this is going to be times ert. R is given to us as 2.303, and t is given to us as 3 years. And when we type all this into our calculators, we get an answer of about 100,125, and the units are going to be individuals. And so this is the correct answer to part number 1, the final population size under these conditions.
Let's go ahead and move on to part number 2, and in part number 2, it says given the slope of the line between two points (1,100) and (2,1000), on an exponential growth curve described by the equation, calculate the average per capita population growth rate or rΔt between t equals 1 and t equals 2. Again, let's pull up our 3 equations of the exponential population growth model. This problem is asking us to calculate the average per capita growth rate or rΔt, which is only found in this equation over here on the far right. And so that's the equation that we're going to use for this part of the problem, so let's go ahead and get rid of these other equations and shift this over to the left, so that we have a little bit more room. We need to isolate rΔt, and so we can do that by rearranging this equation algebraically by dividing both sides of the equation by n.
What we get is rΔt is equal to Δn/Δt all divided by n. Now we just plug in values into this equation. For Δn, we take the difference in population size, which is going to be 1000 minus 100, and for Δt, we take the difference in the time, which is 2 minus 1. This is all going to be over n, which is the initial population size, which is 100 here. So, 1000 minus 100 gives 900, and divided by 100, we get 9. And the units are going to be individuals per year per individual, which we will just abbreviate. This answer of 9 individuals per year per individual is the correct answer to part number 2.
Let's move on to part number 3, which is the last part of this problem. Part number 3 says, given the slopes of the lines tangent to the exponential growth curves at the points (1,100) and (2,1000), which are described by the equation dndt = rn, calculate the instantaneous population growth rate at t equals 1 and t equals 2. This problem makes it a little bit easy for us to decide which equation we need to use since it tells us, basically to use dndt = rn, and that's exactly what we want to use since dndt is the instantaneous population growth rate.
So, at t equals 1, with r still at 2.303 and n as 100, the calculation gives 230.3 individuals per year. For t equals 2, with r still at 2.303 and n now 1,000, the calculation results in 2,303 individuals per year. These are the two answers to part number 3, concluding this three-part practice problem. Hopefully, this was helpful for you, and I'll see you all in our next video.