Hello wonderful mathematics people. This is Anna Cox from Kellogg Community College. Exponential growth in DK. The formula A equal a sub naughty of E to the KTA. Sub naughty is the initial amount, K is a constant, T is time, and just A is the ending amount. Now, if our constant is greater than 0, then we're talking about growth, and if our constant's less than 0, we're talking about decay. So for exponential growth, it's just going to be a graph that's an exponential graph that's increasing. Now in story problems, our time is never going to be 0. So we're really talking about this portion of the graph here where our time is positive in exponential decay. It's going to be an exponential graph that's decreasing. And once again, due to the time factor, we're going to be having it be only from the time being positive and exponential growth, our amounts are going to be increasing and exponential decay our amounts are going to be decreasing. Let's look at some examples. If we had $25,000 invested at 4% continuously, we're going to use a equal A, not E to the RT. So we want to find out how much we have. If we invested 25,000 compounded continuously is the E portion, the rate is going to be .04 and we're going to do it for one year. So if we have E equal 25,000 E to the .04 T, we're going to get, Dang it, If we look at a $25,000 investment at 4% compounded continuously, our final amount is going to equal the initial investment times E to the rate, which in this case is .04. And let's say we want to know what happens after one year. So a time is going to be 1. If we then pick up our calculator and plug this in, we're going to get a equal $26,020.20 thirty cents, sorry, at the and at the end of approximately 1 year, we've made $1000 by not doing anything other than investing our money at 4%. Let's look at doubling time. To do doubling time, we want to figure out how long it will take that 25,000 to turn into 50,000. So we started with 25,000 and we wanted to end with 50,000 E .04 T. So in this case, we're going to divide each side by 25,000, IE there's a double, there's a 2 right there. So twenty 50,000 / 25,000 is 2. So you could think of every first dollar invested, you get out $2.00. So it really doesn't matter how much the 25,000 and the 50,000 is, if we knew that we started with P amount and we're going to end with 2P amount, the P and the 2P will cancel and it will always be one and two for a doubling time. Now to solve this, we're going to put this into logarithmic form. We're going to get .04 T equal the natural log OF2SO2T is going to equal the natural log of 2 / .04. We plugged that into our calculator and I don't know, I think it's something like 17.7 years. Looking at some more examples, what if we had population growth? If we started with a population that was 203.3 million in 1970 and it grows to 300.9 million in 2007. So in 2007 we have 300.9, that's our ending amount. In 1970 we have 203.3. That's going to be our starting amount, E the K, we don't know in this in this problem, we're going to find it. But the T, how long was it? It was 37 years. So now we're going to take the 300.9, divide it by 203.3. That's going to equal E to the 37 K taking the natural logarithm to get rid of that E So natural log of 300.9 divided by 203.3 would equal 37 K or K is going to equal 137th of that natural log of 300.9 / 203.3. Now I personally never round until the very, very last step. And the reason is that we don't want to have any rounding errors as we're going. Errors will be compounded if we round as we go. So we're going to get K is approximately .011. Now when will it be 315,000,000? So we're going to have 315. We start with the 203.3 E. We now know that K, that K was that 137th natural log 300.9 / 203.3, but we don't know our time this time. So we're going to do the same steps. We're going to divide each side by the 203.3 and we're going to get E to the 137th natural log 300 and .9 / 203.3 times T Then if we take the natural log to get rid of the base E and then we're going to divide each side by that K value to get T by itself. So T is going to be approximately 40 years. Now it asks for when, so we need to take the original 1970 and add 40 years to it. So in approximately 2010 we will have 315 million. If we look at a carbon dating problem, we want to know after 5715 years, amount of carbon present is half what was there originally. So if we had originally P amount after 5715 years, we are having half that amount present, E or K we're going to find and our T is 5715. Now some sources state that that number could be as 5730 ± 40. So it could have been 5690 up to 5770 depending on which source you're using. So the PS here are going to cancel and we're going to take the natural log to get rid of the base E. So the natural log of 1/2 is going to equal K * 5715 or K is going to be 1 / 5715 times the natural log of 1/2. Well, that's going to be approximately negative. Why is it negative? Because it's a decay .00012. If this 505,715 change by a little bit, that K is not going to change by very much. It's still very small. What happens if we found, say, something that had 76% of its original carbon 14 and we want to know what year it is? Well, if it's got 76% left, we would have .76 of the original. So if we called, the original PP would equal P * E to that K, which is approximately -.00012 * t The PS on each side are going to cancel. We could have used other variables, we could have used A's. So we're going to take the natural log of .76 and that's going to equal -.00012 T So to get T by itself, lane of .76 divided by that K and we're going to get T. Being approximately 2268 years old, a logistic growth model is similar to an exponential with the fact that it's going to increase exponentially to a certain location and then all of a sudden it's going to start decreasing and it's going to get closer and closer and closer to some limit for the growth. The limit may be the population size or how big of an area the population has. There are all sorts of different possibilities for the limiting of the growth, and So what happens is as it increases exponentially, at some point the resources decline or the quantity of population declines, and so it decreases at some rate of growth until we get to the limiting factor of that growth. The equation is a equal constant C / 1 plus AE to the negative BT. We're going to look at a problem that's going to have A being 30,000 inhabitants. This is going to be about a flu epidemic. So we're given the equation A equal 30,000 / 1 + 20 E to the -1.5 T So we want to know how many people became I'll with the flu when the epidemic began. Well, if the epidemic began, the time was 0. If the time is 0, negative 1.5 * 0 zero E to the 0 power is one. Anything to the 0 power is 1. So we end up with 30,000 / 1 + 20, which is approximately 1429. So at the time of the beginning of the flu epidemic, there were 1429 cases out of the 30,000. How many people were ill by the end of the fourth week? Well, now instead of the time being zeroed, the time is going to be 4 for the 4th week. So 30,000 / 1 + 20 E to the -1.5 * 4. If we grab our handy dandy little calculators, that's going to come out with 28,583 approximately. So in week four we went from 1429 week zero to 28,583 and week 4. Now the next question is what's the limiting size? Well, as T gets larger and larger and larger, E to something really, really big is going to be something really, really big. So this bottom is basically going to end up going to one if we go out far enough. So our limiting factor here is really just going to be the original population, because we can't ever have more than the original quantity of people coming down with the epidemic. So that 30,000 is our limiting factor. There's also Newton's law of cooling, which is a DK equation, and it says the temperature. The final temperature equals the room temperature plus the original temperature of the object minus the room temperature times E to the KT. So C is the constant temperature of the surroundings, T naughty the original temperature, T is the final temperature, little T is the time, and K is some constant. Let's look at what happens to a cake. If a cake is 210° and it's left to cool in a room that's 70°. After 30 minutes, the cake is 140 degrees. So we're going to say the final temperature of 140 is equaling to the room temperature of 70 plus the original temperature of 210 minus the room temperature of 70 all times E to the K constant, which we don't know times 30. If we compute this out, we're going to subtract 70 from each side. 210 -, 70 is 140 E to the 30K. Divide each side by 140. So we get 1/2 equal E to the 30K. Take the natural log of 1/2 to equaling 30K. So K is really just 130th, the natural log of 1/2. So that's going to be approximately -.0231 say we want to figure out what's going to happen 40 minutes later. So 40 minutes later, we're going to say, what's the final temperature if we started with 70 plus that 210 -, 70, which was the 140 E to that constant which we just found. And because it's a constant, it's never going to change negative .0231 * 40. Well, if we plug all this into our calculator, we get 70 + 140 E to the negative .0231 * 40 and our handy dandy calculator tells us it's approximately 126. So 126° for our cake. 40 minutes later, what time would it give us? 90°? So if we put in 90°, we'd have 90 equaling 70. Plus are 210 - 70 E to the negative .0231 * T Subtract the 70. We get 2140 E to the negative .0231 T. If we divide each side by 1:40, we're going to get one seventh equaling E to the negative .0231 T Take in the natural log to get rid of that base E and we get one seventh natural log of 1/7 equaling -.0231 * t So to get T by itself, we take the natural log of one seventh. Divide by -.0231 and we're going to come up with approximately 84. So 84 minutes after the cake was taken out of the oven, it'll be a 90°. Thank you and have a wonderful day.