 Welcome back to our lecture series math 1050 college algebra for students at Southern Utah University. As usual be your professor today, Dr. Andrew Misseldine and this lecture 43 also in lecture 44. I want to talk about well basically story problems involving exponential functions. How do we model things using exponentials and why should we model things using exponential functions? It turns out that many natural phenomenon have been found to follow the law that an amount a varies with time according to the following formula you see right here. The amount a is equal to some initial amount a not times e to the k t where t is some measurement of time and k is going to be some growth rate so different things might grow at different rates. So we have that initial value and so this right here is often this formula is often referred to as exponential growth or sometimes called uninhibited growth in contrast to some other growth models we'll see a little bit later exponential growth. It turns out we've seen this formula before we wrote it in a slightly different way a equals p e to the r t pert right continuously compounded interest is an example of this type of exponential growth that interest grows exponentially. Okay, and when we talked about compounded interest, we gave some idea of where this number e came from as the number of compounds was allowed to go towards infinity, we are able to pull out this number e. That is there was no break in growth, right. So when it comes to like population growth that often is a good model right that they're when you have a certain species that you're watching whether it's humans or bacteria or bunnies or whatever. If you know if there's like seasonal type birth right like you know bears only have their babies in the spring or something like that right then this wouldn't be an appropriate model at least not from a month to month but maybe from year to year that would be appropriate. But and so the idea is if there's no like noticeable break in the growth right there's no like seasonal growth or something like that then this idea of continuously compounded growth is appropriate and that's where this exponential growth comes from the basic idea behind exponential growth. Is that the growth rate is proportional to the current population the bigger the population is the faster the population will grow and that's that's the main idea behind this exponential growth. We can use that to grow talk about population which we'll talk about we've already done some financial examples, and we'll see also later this example that we can use this this growth model also to represent decay models right. So when you have this expression e to the K, like we see in this formula right here, if K is a positive number that means e to the K will be a positive base that is a base greater than one, and therefore it will represent a growth model when K is positive. On the other hand when K is negative, you're actually going to be taking e to a negative power right that to give you a reciprocal your exponential will have a base less than one and that causes it to be a decay model. So when K is positive, this means growth. When K is negative, this means decay. And so sometimes it's appropriate to have a decay model we'll see that if you have a hot object and you put it in a cold environment, the temperature actually will decay over time it gets cooler that we can see that with Newton's law of cooling. We'll also see later in this lecture, the idea of radioactive decay that the mass of an object goes down over time as atoms fall apart in the with the radiation there. Again, those are all some examples we'll see in this lecture so for this current video. Let's consider a very basic exponential population growth problem. So we have an initial population count of 500 in a culture so culture just means a collection of bacteria right here. So we have 500 the initial bacteria count in a culture is 500. A biologist later makes a sample count of the bacteria and the culture and finds out the growth rate is 40% per hour. Okay, so our first question is can we come up with a formula to model the growth of this bacteria right here. So let's notice some observations we made. So the initial bacteria was 500 so our initial amount is going to be we can call it the initial population that turns out to be 500. Okay, because our goal for a formula is going to look something like the population is going to look like an initial population times e to the kt that's what we're trying to figure out here. So we have the initial population is 500. The bacteria is growing at a rate of 40% per hour so this tells us two things. One T is going to be measured in hours right so we're going to talk about hours, hours since, whoops for a letter there, since the start, whatever that is. So the 500, the 500 mark, and then our growth rate k is going to be 40% so 0.4 per hour. When you talk about your growth rate rates are always ratios right it should be something per whatever right so it's going to be per hour in this situation here. So when we put that together we see that our our model is going to be the population with respect to time is going to equal 500 e to the point 40 and we're going to measure their growth in hours so that's our formula. We're going to be using to model this and so then we can ask questions like okay what is the estimated amount after 10 hours. So after after 10 hours how many bacteria should we expect to have well essentially what we're trying to do is we're trying to compute p of 10 right so we're trying to figure out 10 hours here would mean that T equals 10. So that's where we need to compute what's p of 10 so we would take 500 times e to the point four times 10, which point four times 10 of course just before, but we're going to get e to the fourth right, you know, given e is some irrational number 2.7 etc etc right. I'm not going to bother doing this without a calculator so use a calculator to help you out here. A consoled your calculator is going to tell you that e to the fourth is approximately 54.5981 try to have a lot of decimal places here. It kind of depends on what you're trying to model here if in this case we're talking about bacteria bacteria are organisms you can't have a fraction of an organism it's either alive or it's dead right you can't just take a fraction of it. It's like oh we're going to count the kidney of this person or we're going to count the. I don't know the nucleus of this bacteria know you got to count the whole thing so you got around to the nearest integer, but add a couple extra decimal places to keep track of things right. Because of course if you multiply this by a big number that's going to cause the decimal to move over a couple we need to have a lot of decimals here in a scientific setting we might be paying attention to significant digits. We're not going to worry about that in this mathematical setting right here 500 times this approximation is going to give us 27,300. Right if we round to the nearest decimal and again I should probably mention that these of course are approximations that we were exact up to e to the fourth. But after that one started console console dinner calculator these are approximations so we should expect about 27,000 bacteria. After 10 hours of growth assuming their growth rate remains constant throughout that duration and so again this is just the typical example of how one does calculations with exponential growth.