 Suppose a culture of bacteria starts with 10,000 bacteria inside that petri dish and the number doubles every 40 minutes. So we don't necessarily have the growth rate. What we know is how quickly it's doubling, right? Which gives us information. So the first question to ask ourselves here is can we find a formula, some model of the population for the population of this? I guess it's not really a question, it's a statement. You're being told to do it, it's really what it is, command. So find a formula for this population growth, right? And so there's a couple of ways you could approach it. So we've seen in the past that the populations can equal some initial population times e to the kt, right? But we don't know what the growth rate is. How does one actually compute the growth rate? So what we can do with the following. We know the initial population is 10,000, right? This e to the k times t, right? Now we also know that it doubles every 40 minutes. So if we measure time in, say, minutes, because this thing grows pretty rapidly, what we could say is 40 minutes later the population will be doubled. It's gonna be 20,000, right? And then we could solve for k in this situation. Divide both sides by 10,000, you're gonna get two is equal to e to the 40k, right? So notice that it actually doesn't matter what the initial population is. If we know it doubles every 40 minutes, if we have three bacteria, then it doubles, it would be six. If we take six divided by three, it's still two. So the growth rate doesn't actually depend on the population size in this situation, right? Cause we're growing, we're doubling every 40 minutes here. So then if we wanna keep on going to solve for k, we probably have to take the natural log of both sides, take the natural log of two, cause the natural log will cancel out the base e you have right there, you get 40k. And so therefore k is equal to the natural log of two divided by 40. Which if we estimate that, we get an estimate of, well, let's see, this would be 0.017333. I'm just using a calculator to help us out right here. In which case, then we could say that the growth rate of the model, I should say, will look something like p equals 10,000 times e to the, our growth rate k here, 0.0173332, right? But there's some things, there's some things that have to caution you about. First of all, some of us might be tempted to round, right? Because after all, that's a lot of decimal places to keep track of a lot of luggage to put in the plane, right? But when we start rounding, especially when we start rounding exponents, they can have a huge difference to the calculation. So it's kind of best that we keep things as exact as possible. But it turns out even if we keep all these decimal places, this is not the most exact answer. The most exact answer is this friend right here, right? K equals the natural log of two over 40. Now, I wanna show you something that's slightly different, which actually gives me kind of a more, a different model for the same growth right here that I think is more preferable in this situation. So I guess what I'm saying is if you wanna use a growth rate, I would try to keep it as the natural log of two over 40 as much as possible. Hesitate approximate until the very end. But on the other hand, if we took p equals p naught, e to the natural log of two over 40, e multiplied that by t, right? Well, let me show you a nice little trick here. So I'm not even gonna worry about what the initial population is. By exponent laws, right, I could factor the exponent as p naught, we're gonna get e times the natural log of two times t over 40, right? Just kind of refactoring things like that. But remember, when you have like a to the m to the n, if you have an exponent and an exponent, you multiply them together. This means that if you factor the exponents, you can actually break them up in a manner like this. So we can actually rewrite this thing as p naught times e to the natural log of two to the t over 40 right here. And so what you're gonna notice is that e to the natural log of two, these are inverse operations, the natural log of two is the power of e that gives you two. If I raise e to the power of e that gives you two, this is gonna equal two. And so this then simplifies to be p to the, p is equal to p naught times two to the t over 40. And so this right here is a much simpler form to use I think because it avoids a lot of the approximation. There's no ease that necessary here. And the way you wanna read this is that this number 40 right here, if we kind of erase it for a moment, we could call this our period. Okay, what do I mean by period? Well, if we know that the population doubles every period p, then we can calculate the population as the initial population times two to the t over p where that's the period and you can go for there. And again, this will lead to a lot easier calculations than this model over here, which is perfectly good. But like I said, this one's gonna be a little bit easier to use and we have far less rounding error I think than the previous model. So p equals our initial population was 10,000. And then it doubles, excuse me, it doubles t over 40. So it doubles every 40 minutes. So can we ask ourselves, okay, ask ourselves, what's the population after an hour, right? After one hour, what would the population be? Well, so we're basically asking what happens when t equals 60, okay? t equals 60. So we can look at the first model that we have, we have to compute p of 60. So we're gonna get 10,000 times e to the point zero one, seven, three, three times 60. And if you compute that exponent, you're gonna get e to the 1.0398 times that by 10,000, right? Raise e to the 1.0398 power. That'll give you approximately, so I'm just gonna write down the 10,000 again, you're gonna get approximately 2.8287. You wanna have a lot of decimal places because you're moving by, you're times it by 10,000 which is gonna move your decimal places over a bunch, right? And so then our estimate would look something like 28,287. That's our estimate of the population of bacteria after 60 minutes using the first model. On the other hand though, right? If we used our doubling model, which is honestly the one I kind of prefer personally, in that situation, we still have to compute p of 60. So we're gonna get 10,000 times two to the 60 power, right? And so that fraction actually simplifies to be three halves, three halves right there. So basic or 1.5 if you prefer, doesn't really matter. Basically what we're trying to say is we, time has elapsed one and a half periods, okay? So we have to take the square root of two and then cube it. And so if we consult our calculator, right? What are we gonna get in that situation? Well, whoops, sorry about that. We need to start off with the 10,000 and then we need to raise two to the 1.25 power. And I'm sorry, I can't really show you the calculator. I mean, I wish I could show you like what the calculator says right now, but you would get about 2.82847, you know, something like that. You notice it's really similar to the number we had over here. And so when you times it by 10,000, then just move the decimal places over a couple of spots, right? And so you're gonna end up with 28,200 and let's say 240, we'll round that up five, like so. And so this is the estimate. I'd say the one on the right is a little bit more reliable in terms, I think it has a little bit less error there. But honestly, both of them are kind of in a reasonable ballpark. We'd say about 28, you know, maybe like 2850. That seems like a pretty good estimate. 28,000, excuse me, 28,250, we'll say that. We'll say that's a pretty good estimate. You know, somewhere in that ballpark, 28,300, 28,200. We're only off by a couple tens of back to here. That's not gonna be a huge, huge error. And that's the thing is different models can give different predictions because they have some slightly different assumptions in them. They also may have slightly different error inside of them. It becomes more pertinent as you start studying models more and more to decide which one is more accurate, which one's more, well, has less error. And that oftentimes comes with your assumptions but also there are rounding errors for which calculators can usually avoid those considerations. And so we'd say there's somewhere close about 28,250 bacteria after 60 minutes of growth.