 Suppose a certain breed of rabbit was introduced onto a small island about eight years ago. The current rapid population on the island is estimated to be 4,100, with a current rate of growth at 55% per year. That is a huge amount of growth. I wish my bank account had a 55% growth per year. That would be a nice APR. So one question to ask is like, okay, the current population is 4,100. What was the first population? So how many rabbits were introduced onto this island? What was the starting factor, right? How many rabbits started this growth, right? This could be like some invasive species that was introduced, right? Australia has a big problem with frogs and things like that. They were introduced, and then like snakes came to eat the frogs, and then like the snakes were too good. And so now they're eating all the birds. Maybe I'm messing up some of those details there, but like these invasive species can be sort of a problem, right? Can we figure out like how many started this whole thing? How many rabbits started this whole growth rate eight years ago? So basically what we know is the following. We're gonna use exponential growth, right? P equals P naught e to the kt. We know the growth rate, right? They're growing at 55% per year. So t, if the growth rate is in per year, that means t is gonna have to be in years, right? Number of years, the number of years since, well, since what? We have to make a choice about that, right? Do we wanna measure since years, how many years since the current timeframe? Since like now, or do we wanna have it from, right? The starting eight years ago. There's a couple of ways you can handle that. So from our perspective, we're gonna take, we're gonna take t to represent the number of years since now, right? So currently, right now means t equals zero. So that's what we're gonna take as an assumption. Therefore, our initial population is gonna be 4100. That's how we're thinking about it. So we have 4100 e to the 0.55 t, okay? But then if we wanna figure out how many bunnies there were at the beginning of this experiment, right? We need to go back in time. And so we'd set t equal to negative eight, right? So t equals negative eight would be, and then eight years ago, okay? For which then we try to solve this. We just wanna solve the right-hand side. Well, I mean, not solve it, just compute it, I guess. We're gonna take 55% times it by eight. That's gonna give us, let me write it down, e to the negative, that's 4.4, like so. We're then going to take, then I'm gonna have to put this in my calculator, of course. I mean, if you want, you could write this as an expo, excuse me, it's already an expo. If you wanted to, you could write this. Negative exponents, of course, the same thing as division. So you get 4100 divided by e to the 4.4. The same thing right here, right? If you estimate e to the 4.4, you end up with 81.45, and then that's been divided by the 4100. This will give you an estimate of about 50. So we would expect, you know, rounding to the nearest bunny, you can't have part of a bunny. So there was approximately 50 bunnies eight years ago that started this whole population growth. Now, the reason I wrote it with the division right here is another approach you could take, right? Because when you decide t equals zero, it doesn't matter what t equals zero is, you just have to make a choice. What if eight years ago, what if we decide that eight years ago is what we meant by t equals zero? In that situation, our model would look like p equals p naught, which we don't know what it is, right? So then we'd be like, hey, okay, the current population is 4100, right? But we don't know the initial population, we get e to the 0.55 times eight, right? So we have a positive eight here, instead of the negative eight we had here, but notice the unknown is the p naught instead of the current population. For which if we wanted to solve for p naught, we'd have to divide by this, 0.55, which we saw earlier, 0.55 times eight was 4.4, right? So we have to divide both sides by 4.4, which gives us that the initial population was 4, 4100 divided by e to the 4.4. That gives us this number we saw right here. But again, dividing by an exponential is just using a negative exponent, so you get 4100 times e to the negative 4.4. So whichever approach you take, it doesn't matter what t equals zero means, does it mean now or eight years ago? Just be consistent, that's all that matters. Pick a model and go with it, all right? So let's say that we have this question right here, so we answer this question. So now let's ask the question, how many rabbits will there be, say, in 12 years? So as we go to the future a little bit, how many rabbits do we expect to have 12 years from now? So again, it matters on what you use as your t equals zero. Where does moment t equals zero in time represent? So with our original example, where t equals zero represented now, right? We then have to compute p, we have to compute p of 12, right? For which we use the current population of 4100, we take e to the 0.55, right, times that by 12, and we compute that. On the other hand though, if we took that t equals zero was eight years ago, notice then our function, what we're trying to compute is we're actually trying to compute p of 20, right? Where does 20 come from? We have to go eight years to get to the present, then we have to get 12 more years to get to the future, right? So that's what we have to compute. And so we take 50 times e to the 0.55 times 20, like so. And if you carry forward that calculation, 0.55 times 20 is gonna give you 11. So you get 50 times e to the 11, e to the 11 is gonna be approximately 5,987, I'm sorry, I'm off by a decimal. That'd be about 59,874.142. You times that by 50, that would give you an estimate of about 2,993,707 bunnies on the island. So wow, that's a lot of bunnies. And there's a reason I'm using rabbits here. They're known for their growth rate. I mean, as Judy Hopps says in the Disney movie, Zootopia, rabbits are known for how good they are at multiplication. Anyways, so we'd estimate approximately 3 million rabbits coming up in the future, of course, right? Now, I'd showed you how to do the model if we had set the initial time back at eight years ago. But what if you wanted to start from the present, right? This calculation is basically the same. You get 4,100 times e, 0.55 times 12 is gonna be 6.6. Right? And then you have to compute e to the 6.6, which is not too hard to do. You put that in your calculator. You end up with 735. Again, we'll keep it to a couple of decimal places there, say like 0,9,5,2 or whatever. And then times that number by 4,100. And you would end up with about the same number, right? There's a little bit of an error though. Like, so when you do it this time, you get about 3,013,890 if I round to the nearest rabbit. So in terms of ballpark, they're both about 3 million. So it's not like one is super better than the other. But why is there a discrepancy here? The discrepancy came down to this observation right here, okay, that how many bunnies did we start off with? About 50, right? That was the estimate we got. So this number 50 was actually approximated using our model as opposed to the 4,100 which was collected by some other method, right? So in terms of population, the current population is actually a more accurate measurement than the population eight years ago. And so this calculation here depended upon an estimate to get another estimate. And it's kind of like if you take a photocopy of a photocopy of a photocopy of a photocopy the quality kind of deteriorates over time. So in my opinion, it's better to stick with the most original numbers as possible when you try to make calculations. So I actually feel that using the current time of T equals now, right? T equals zero is now. This is a better estimate. Now again, rounding to the nearest million these are pretty close to each other. I mean, they're not off by much. There's only a difference of about 20,000 rabbits in this estimate here. So I'd say that both of them are pretty good but if I had to pick one, I think this estimate's a little bit better because I feel that the 4,100 calculation is more reliable than the 50 bunnies we started off with because those small differences like the rounding to the nearest money over here makes a difference on exponential growth. Even a small change in exponential growth can have an effect over a large amount of time. So try to use your most reliable numbers when you're working with these problems so that you have the most reliable answer when you're done.