 In this video, I wanted to do another example of using exponential functions to model growth of a population, but we're going to go a little bit sci-fi in this example. So in start 8, 12.19.6.15.2, the population of the planet Sol3 was 6.1 billion organisms, right? There's 6.1 billion people on Sol3 in that start 8. And the people of Sol3 are currently growing at a rate of 1.47% per year. And what I mean by a year, right? On Sol3, a year is defined to be the time it takes for the planet to revolve once around its star. The star, of course, being Sol, right? Sol3 is the third planet away from Sol, okay? It's claimed that if the growth rate was reduced to just 1% per revolution around the star, that this would make a significant difference in the population growth over a few decades, right? So we want to kind of test this hypothesis. If we decrease the growth rate from 1.47% to 1%, what would be the difference 50 years later? So 50 revolutions around the star later, how that affect the people of Sol3. So we're going to come up with two different growth models. So we're going to have our first model, our first model where our growth rate is going to be 1.47%, okay? And so all of these growth models are going to be based upon the formula that the population will equal the initial population times e to the kt. So we're using this exponential growth to model these things. And so then we see that the growth, well, for both of these models, right? It's the growth rate that's going to change. The initial population is still going to be 6.1 billion. And I'm just going to keep it at 6.1, knowing that the answer will be multiplied by billion in the end. So for the first model, the current growth rate, the population is going to be 6.1e to the .0147 times t. So that's the first growth model. And then I'm going to do it in green, the second growth model, right? If our growth rate is just 1%, 1.0% here, right? Then the model would look like p equals 6.1. The initial population is the same. We're just changing the rate so we get e to the .01t. And so then we wanted to make the test, okay, how different is the population 50 years later? So if we consider 6.1, the initial population, you know, start at 12.19, 0.6, 0.15, 0.2. That's a really awkward way to say the dates, but sure, that's how they do it, okay? We're going to calculate the population. We need to compute p of 50 for these two different models. So we say 6.1 times e to the .0147 times 50. That's the first one. And then for the second one, what we have to do is compute the value p of 50, which is going to be 6.1 times e to the .01 times 50. So it's the growth rate that's different between them. And so putting those together, well, I mean, I'm not putting them together, just actually doing the multiplication here. So we got to take 50 times the 1.47%. That's going to give us 6.1 times e. You just do this on a calculator, right? We're going to have to use the calculator eventually. You don't have to be a hero. When you multiply out the exponent, you're going to get 0.735. You raise e to that exponent, 0.735. You're going to get 6.1 times 2.085, which we times that by 6.1, you're going to get 12.7. And so this is going to be the population of Soul 3. Oops, would be approximately 12.7 billion people, 50 years from the current start date. If we did that with a 1% growth rate, well, we're going to take 6.1 times e. You're going to take 0.01 times 5, of course, which is going to give you 0.5 in that situation. For which then when you raise e to the .5, so we're taking the square root of e in this situation, you end up with 1.649, and then you times that by 6.1, you end up with 10.1 billion in that situation. And so those are the numbers you want to compare with the current growth rate 50 years from now. You have 12.7 billion people on Soul 3 for the smaller growth rate of 1%, you have 10.1 billion. And so that is a difference of about, you know, 2, almost 3 billion right there, right? And so as such, it is fair to say that the growth rate, the decrease of the growth rate would be significant to the population. And if that's, you know, that's the information we get from this. Yes, absolutely. The smaller population growth rate would have effect over a large amount of time. And this is true for exponential growth. Even a small change of growth rate does affect the outcome in the long haul, right? So at the current rate, so at the current rate of K equals 1.47%, at this current growth rate, when will the population reach 122 billion? When does the population equal 122 billion? Right? So we can ask this question about when does it reach a certain threshold, right? Maybe this is significant. 122 billion, of course, is much larger than the 6.1 we started with or even the 12.7. But we can actually use this equation to solve that question, right? We didn't know when the population is 122 billion. The current population is 6.1 billion. They're growing at a rate of 0.0147t. We have to solve for t. So we go about solving this equation right here. Your calculator will be your friend. Divide both sides by 6.1. We end up with, I'm just going to move the exponential to the left-hand side. Now we get our e to the 0.0147t, right? This is going to equal 122 divided by 6.1. Turns out that actually is 20, right there. Then how do you read the exponential on the left-hand side? We're just going to take the natural log. The natural exponential is the inverse of the natural log. So we're going to get 0.0147t is equal to the natural log of 20. For which we do have to approximate the natural log 20, but I'm going to postpone doing that. As much as possible delay any approximations to the variant when you can put all of the data in your calculator to avoid unnecessary rounding errors. Then to solve for t, we're going to divide by its coefficients 0.0147. What's good for the goose is good for the gander. We have to do it to both sides here. And so then we see our estimate is going to be t equals the natural log of 20 divided by 0.0147. Now this number we're going to throw into our calculator and we're going to get 213.98. So I'm going to round that to the nearest year and I'm going to say that approximately 214 years from the current start date is going to be when the population of Sol 3 reaches approximately 122 billion individuals. And so we can use exponentials to compare growth rates. We can use exponential functions to make predictions, to make projections about the future. This is the closest we can get to time travel without like a Borg sphere or a DeLorean or some other type of technology that doesn't necessarily exist, right? We can use mathematical models to make projections of the future. We can also project into the past and make predictions about the past based upon our mathematical models, assuming our mathematical models are correct.