12 - What is Exponential Growth & Decay? (Half Life & Doubling Time) - Part 1 - By Math and Science
Transcript
| 00:00 | Hello . Welcome back . The title of this lesson | |
| 00:02 | is called exponential growth and decay . Also called half | |
| 00:06 | life . Could also titled this lesson . Exponential doubling | |
| 00:11 | time formula and exponential half life decay formula . I'm | |
| 00:14 | excited to teach this because everybody's probably heard of the | |
| 00:17 | concept of the half life of a radioactive element . | |
| 00:20 | The half life of uranium is whatever whatever years . | |
| 00:23 | Uh and so we have kind of in our consciousness | |
| 00:26 | sort of that that is something to do with decay | |
| 00:29 | . But probably you didn't really know until now that | |
| 00:31 | it's an exponential decay . Just like the types of | |
| 00:33 | problems that we've been doing before dealing with money . | |
| 00:36 | Now it turns out that the exact same equations that | |
| 00:40 | govern the growth of money and the decay and the | |
| 00:42 | value . Remember the depreciation of assets ? We talked | |
| 00:46 | about that in the last couple of lessons those were | |
| 00:48 | exponential equations . It was exponential growth compounding interest formula | |
| 00:53 | . It turns out that the exact same equation governs | |
| 00:56 | other things in nature . Other things in nature , | |
| 00:58 | notably for instance , population growth of bacteria . For | |
| 01:02 | instance when you put bacteria in addition let them grow | |
| 01:05 | , they are going to exponentially the population's gonna exponentially | |
| 01:09 | grow . It does not grow in a line , | |
| 01:10 | the population of bacteria or virus or something grows exponentially | |
| 01:15 | right . Also populations in the world do not grow | |
| 01:19 | like populations of people , they do not grow linearly | |
| 01:21 | , they grow exponentially also the decay of elements into | |
| 01:25 | other elements does not decay in a line down the | |
| 01:28 | decay in an exponential fashion down . So what I | |
| 01:31 | want to do is at the beginning of the lesson | |
| 01:33 | , tell you and show you what these exponential growth | |
| 01:36 | and decay uh equations are in terms of population growth | |
| 01:41 | and also decay radioactive decay . And then what we | |
| 01:44 | want to do after I show you what the equations | |
| 01:46 | are , we're going to back up the truck a | |
| 01:47 | little bit and we're gonna start from that compounding interest | |
| 01:50 | formula which you already understand and know and I'm gonna | |
| 01:53 | show you how these half life decay formulas and how | |
| 01:56 | this doubling time formula comes about from it . And | |
| 01:59 | then we're gonna solve some problems that deal with population | |
| 02:02 | growth and radioactive decay . That's all just in the | |
| 02:05 | back of your mind . Just keep in mind that | |
| 02:07 | it's all exponential growth and decay . Okay . So | |
| 02:11 | I don't expect you to understand this yet . I | |
| 02:13 | am not gonna go into great detail but I want | |
| 02:15 | to show you what these equations are and we're gonna | |
| 02:17 | derive these equations so you'll know exactly where they do | |
| 02:19 | come from . So we have something called the doubling | |
| 02:21 | time exponential growth formula . And here's what it is | |
| 02:25 | . This equation is the exponential equation that governs the | |
| 02:28 | population growth of bacteria in a Petri dish or even | |
| 02:32 | population growth of people in a city or in a | |
| 02:35 | nation or something like that . And you can see | |
| 02:37 | that the number two is in here is the base | |
| 02:39 | of the exponent . That's why it's called the doubling | |
| 02:41 | time exponential growth formula . It's an exponential formula because | |
| 02:45 | the uh the exponent the time variable is up in | |
| 02:49 | the exponent there . And what you have here is | |
| 02:51 | you have an initial population , you have a final | |
| 02:54 | population and then you have these variables here which I | |
| 02:58 | want to talk about a little bit later . But | |
| 02:59 | basically D . Is the doubling time for instance you | |
| 03:02 | might know that in a dish of bacteria you might | |
| 03:05 | know instead of talking about it growing at 3% per | |
| 03:08 | year . Like we talk about money we generally in | |
| 03:11 | population growth we don't talk about how many percent per | |
| 03:13 | year . We generally say the population doubles in four | |
| 03:17 | hours or the population of this bacteria doubles in six | |
| 03:20 | hours . Okay , so this D . Is what | |
| 03:23 | we call the population doubling time . So if we | |
| 03:25 | know that the bacteria is doubling every six hours then | |
| 03:28 | D . Is going to be six for six hours | |
| 03:30 | . And then we can project in the future however | |
| 03:32 | many hours we want in the future what the population | |
| 03:35 | will be . If we know the initial population , | |
| 03:37 | we know what the doubling time is and we know | |
| 03:40 | how far in the future we want to look . | |
| 03:42 | Okay now I'm gonna show you that this equation comes | |
| 03:44 | directly from the exponential equations that we've already learned . | |
| 03:48 | That comes from compounding interest . It's the same exact | |
| 03:50 | equation is just written in a different way here . | |
| 03:52 | But when we talk about population growth we talk about | |
| 03:54 | doubling time . Right ? Same thing with half life | |
| 03:57 | radioactive decay . You've probably heard of uranium decays and | |
| 04:01 | so many thousands of years or so many millions of | |
| 04:03 | years . The half life is so many thousands of | |
| 04:05 | years or whatever it is . Notice that this equation | |
| 04:07 | in this equation are exactly the same equation . We | |
| 04:10 | have an initial population . In this case for radioactive | |
| 04:13 | decay , we talk about the final amount of atoms | |
| 04:16 | in the initial amount of atoms , whereas in populations | |
| 04:19 | we talk about the final population and the initial population | |
| 04:21 | . But the math is exactly the same . Instead | |
| 04:24 | of a doubling , we have a half . Because | |
| 04:27 | this is not a growth of the population , it's | |
| 04:29 | the decay of the amount of atoms into some other | |
| 04:32 | atoms . So what we have is in one half | |
| 04:34 | here and here instead of A . D . We | |
| 04:37 | have an H . The H . Is not the | |
| 04:39 | doubling time . Because the uranium is not getting more | |
| 04:42 | and more , it's having itself , it's going less | |
| 04:44 | less . So if we know that the uranium , | |
| 04:47 | I'm just making this up that half of it decays | |
| 04:49 | in 1000 years , we would say that has a | |
| 04:52 | half 1000 year half life . So this would be | |
| 04:54 | the number that you put in here , 1000 years | |
| 04:57 | will be the half life . So if we know | |
| 04:59 | how much uranium we start with and we know how | |
| 05:02 | long it takes for half of a sample to decay | |
| 05:05 | . And we know how far in the future . | |
| 05:06 | We want to take a look like how many thousands | |
| 05:08 | of years down the road . Then this is an | |
| 05:10 | exponential equation that will tell us how many uranium atoms | |
| 05:13 | we have left or how many grams of uranium will | |
| 05:15 | have left . So for the population growth up here | |
| 05:17 | it's an exponential growth curve . And for the population | |
| 05:21 | decay here it's an exponential decay . So growth and | |
| 05:24 | decay , it's exactly the same equation really . It's | |
| 05:27 | just that in here we have a one half , | |
| 05:29 | that's what's making it go down down down every year | |
| 05:32 | . And here we have a number , which is | |
| 05:33 | the number two , which is making it go up | |
| 05:35 | up up . So what I want to do is | |
| 05:37 | just keep these in the back of your mind , | |
| 05:39 | keep in mind that they are exactly the same equation | |
| 05:41 | . Just one makes it go up and one makes | |
| 05:43 | it go down and now we want to talk about | |
| 05:45 | where do they come from ? We could solve problems | |
| 05:46 | with these right now , you know , you wouldn't | |
| 05:48 | know really where they come from , You certainly wouldn't | |
| 05:50 | know that they come directly from what you already know | |
| 05:53 | . So I want to back up the truck a | |
| 05:54 | little bit , cover these guys up and let's start | |
| 05:58 | from what we know and let's figure out that these | |
| 06:01 | equations for half life and doubling time . Formula exponential | |
| 06:04 | doubling time for populations come from exactly what we know | |
| 06:08 | . So we know that the growth of money is | |
| 06:16 | something that we'll look at in the last three lessons | |
| 06:19 | we have extensively covered what this is and this is | |
| 06:22 | the following equation . The amount of money I have | |
| 06:25 | in the future is equal to the principal times one | |
| 06:28 | plus some annual rate divided by the number of compounding | |
| 06:31 | periods raised to the power of MT . This equation | |
| 06:35 | should be familiar to you if it's not familiar to | |
| 06:37 | you , it just means that you haven't looked at | |
| 06:39 | any of my lessons and exponential growth of money . | |
| 06:42 | So you need to go back to that part . | |
| 06:43 | All this stuff builds on each other . So this | |
| 06:46 | is , it governs how much money I have in | |
| 06:48 | the future . When I start with a certain amount | |
| 06:50 | of money and I have a certain interest rate and | |
| 06:51 | I'm compounding so many times per year in times per | |
| 06:54 | year . It turns out that this exact same equation | |
| 06:58 | that governs how how money grows in a bank account | |
| 07:01 | in terms of an interest rate . When you get | |
| 07:03 | so many percent interest per year . This equation also | |
| 07:07 | governs population growth . This exact equation that governs how | |
| 07:10 | money grows in a bank also grows . How many | |
| 07:13 | bacteria is going to be in a Petri dish ? | |
| 07:15 | Because it's all exponential growth , it's all exponential growth | |
| 07:20 | . So uh in order to kind of make some | |
| 07:23 | progress here and connect the dots from here to the | |
| 07:25 | equations , I just showed you over there . Um | |
| 07:28 | I'm gonna ask you to just take a simple example | |
| 07:31 | with me . Let's take an example . Okay , | |
| 07:33 | let's say the population growth Rate instead of money , | |
| 07:40 | we're gonna talk about population growth rate , annual growth | |
| 07:43 | rate of the population . Let's say it's 1 4 | |
| 07:47 | per year . This means that if I have a | |
| 07:52 | Petri dish of some kind of bacteria or virus and | |
| 07:55 | I look on year number one and I know how | |
| 07:57 | many how many uh bacteria I have in their account | |
| 08:01 | them let's say . And then I look one year | |
| 08:03 | later I'm gonna have 1.4% additional bacteria in that dish | |
| 08:09 | the next year . And then the third year after | |
| 08:11 | that we'll have 1.4% of Of your number two . | |
| 08:15 | And so that's why it grows exactly the same way | |
| 08:17 | that money grows . But let's say that the growth | |
| 08:19 | rate is 1.4% per year . Okay so if we | |
| 08:23 | put it into this equation the one that we know | |
| 08:26 | we're gonna change the variables a little bit instead of | |
| 08:28 | the amount of money . And the principal we're gonna | |
| 08:30 | change these letters a little bit . We're going to | |
| 08:32 | have the final amount of bacteria that we have is | |
| 08:35 | going to be equal to some initial amount . These | |
| 08:37 | are exactly replacements for these variables right here . And | |
| 08:41 | then what we're gonna have is one plus this over | |
| 08:43 | this and all that stuff . But what's gonna end | |
| 08:45 | up happening is it's gonna be 1014 to the power | |
| 08:49 | of teeth . Why does it equal this ? Well | |
| 08:52 | because it's an annual growth rate per year that means | |
| 08:54 | it's it's uh you can think of it as compounding | |
| 08:57 | once per year . So n is one . So | |
| 08:59 | you have one plus this is the rate 11.4% means | |
| 09:03 | remove the decimal two spots . So 20.14 here we | |
| 09:08 | add to one we get 1.14 one time per year | |
| 09:11 | and is one . So this is what the equation | |
| 09:13 | comes out to be . So this is the initial | |
| 09:18 | number I'm going to talk about instead of bacteria . | |
| 09:20 | I'm going to talk about the number of people population | |
| 09:24 | , initial number of people . This is the final | |
| 09:29 | number people . This is the time in years . | |
| 09:36 | So so far you should be totally with me . | |
| 09:38 | As long as you know what this formula is from | |
| 09:40 | previous lessons . This is the equation that governs the | |
| 09:43 | growth of money with an initial amount of money , | |
| 09:45 | a final amount of amount of money and interest rate | |
| 09:47 | and compounding so many times per year . In this | |
| 09:50 | case we're just growing uh 1.4% per year . So | |
| 09:54 | this equation comes out to this and we just relabel | |
| 09:56 | these variables in terms of populations . So when you | |
| 09:59 | see in as a variable , it means the number | |
| 10:01 | of something , number of bacteria , number of people | |
| 10:05 | later on when we do half life , it will | |
| 10:06 | be a number of atoms . That's the same thing | |
| 10:08 | . It's the amount of something instead of the amount | |
| 10:10 | of money we use in to be the number of | |
| 10:14 | people or whatever it is . Alright now , I'm | |
| 10:17 | gonna do a big note here . Note this is | |
| 10:20 | something that's not obvious , but it's important for us | |
| 10:22 | to do this , to connect the dots from this | |
| 10:24 | to those equations over there . A 1.4% growth rate | |
| 10:30 | . Annual growth rate means that this population doubles approximately | |
| 10:38 | doubles every 50 years . This is not something you | |
| 10:45 | would know by looking at this , this is something | |
| 10:47 | I'm telling you , and I'm gonna show you why | |
| 10:49 | it's the case . You can think of population growth | |
| 10:51 | in two ways and that's what we're basically gonna do | |
| 10:53 | . You can think of it as growing 1.4% per | |
| 10:56 | year in this example , or you can think of | |
| 10:59 | it as the population just doubles every 50 years . | |
| 11:01 | When we talk about money , we usually talk more | |
| 11:04 | about how much growth you're getting every year in terms | |
| 11:06 | of percent per year . But when we talk about | |
| 11:08 | populations , either it's bacteria or people populations or if | |
| 11:12 | it's the number of atoms , we don't talk usually | |
| 11:14 | about the number of percentage per year . We talk | |
| 11:17 | about the doubling time . Or how long does it | |
| 11:21 | take for the population to double ? It turns out | |
| 11:23 | that these are just two different ways of expressing the | |
| 11:25 | exact same thing . And now I need to to | |
| 11:28 | connect those dots to show you that that's the case | |
| 11:30 | . 1.4% growth rate is approximately doubling every 50 years | |
| 11:34 | . How do I know that ? Because put right | |
| 11:38 | here , because of the following thing . If I | |
| 11:42 | put this into this formula right here , this is | |
| 11:44 | the growth rate formula for a population , right ? | |
| 11:47 | If I put it in here , then uh over | |
| 11:50 | here in not times 1.014 to the power of 50 | |
| 11:57 | is approximately equal to two And not . So if | |
| 12:01 | I start , if I wanted to figure out how | |
| 12:03 | many people I ended up with 50 years in the | |
| 12:06 | road , what would I do if I wanted to | |
| 12:08 | know how many people I have in the population 50 | |
| 12:10 | years later ? What would I do ? I would | |
| 12:11 | put how many people I started with this is the | |
| 12:13 | growth rate for this example and I would put 50 | |
| 12:15 | years in here . That's all I've done here . | |
| 12:17 | How many people I start with what the growth rate | |
| 12:19 | is to the power of 50 . That's what I'm | |
| 12:21 | going to have 50 years from now . But what | |
| 12:23 | I'm telling you is if you put this exponent in | |
| 12:25 | your calculator , what you're going to find out that | |
| 12:28 | 1.014 , go ahead and grab a calculator to the | |
| 12:32 | power of 50 is approximately equal to two . Okay | |
| 12:37 | , that's why these are equivalent because in not is | |
| 12:40 | the same on both sides . The population doubles because | |
| 12:43 | 1.01 for this particular interest rate or growth rate , | |
| 12:47 | When you put an expanded into 50 years , it | |
| 12:49 | just so happens that it comes out to be about | |
| 12:51 | two . That is why for this particular it's not | |
| 12:54 | something you would just no , it's just I'm telling | |
| 12:56 | you that for a annual growth rate of 1.4% , | |
| 12:59 | it turns out that that's about 50 years . Because | |
| 13:02 | when you take that rate and raise it to the | |
| 13:04 | power of 50 you get about two . Okay , | |
| 13:07 | now why am I doing all this stuff ? So | |
| 13:09 | you got to work with me a little bit here | |
| 13:10 | ? So let's go just a little bit farther . | |
| 13:12 | Let's change this around and say this is basically an | |
| 13:15 | equation here . So Even though these this is not | |
| 13:19 | exact , I'm gonna go and put an equal sign | |
| 13:20 | here and make it an equation , even though it's | |
| 13:21 | not quite exact , it's 2.00 something and I'm going | |
| 13:25 | to take and Saul I'm gonna move this exponent to | |
| 13:27 | the other side by doing the following thing . 1.014 | |
| 13:33 | is equal to the two to the power of 1/50 | |
| 13:36 | . You'll see why I'm doing this in just a | |
| 13:38 | second , make sure you understand this . If I | |
| 13:39 | take this and raise it to the power of 1/50 | |
| 13:42 | then the exponents cancel . So that's why they're gone | |
| 13:45 | and if I do it to the left I can | |
| 13:46 | do it to the right and raise this to the | |
| 13:48 | one over 50th power . So now you should agree | |
| 13:50 | with me that the number one point oh 14 is | |
| 13:53 | equal to the number two to the power of one | |
| 13:56 | over 50th . It comes directly from from all of | |
| 13:59 | the exponent here , grab a calculator , take , | |
| 14:01 | take this fraction , raise two to the power of | |
| 14:02 | that and you're gonna find out that it equals one | |
| 14:04 | point Close , very close to equal 1.14 . Okay | |
| 14:10 | , now finally I'm ready to show you a little | |
| 14:12 | bit of the punch line here . Okay , So | |
| 14:15 | there is two equivalent ways to think about population growth | |
| 14:26 | . There's probably more than two , but there's two | |
| 14:28 | that we're gonna talk about in this lesson . The | |
| 14:30 | first way is the way that we talked about a | |
| 14:32 | long time ago . How much does the population grow | |
| 14:34 | in percentage every year ? You could say that this | |
| 14:37 | population is in is equal to some initial population Times | |
| 14:43 | 1.014 to the power of thi this equation , in | |
| 14:46 | words you would say that the population grows 1.4% per | |
| 14:54 | year . That's one way to think about it . | |
| 14:55 | But like I said we don't usually talk about this | |
| 14:58 | is how we look at it if it was money | |
| 15:00 | right ? But it's not money and so it's something | |
| 15:03 | else or we can think of it in terms of | |
| 15:06 | doubling time we can say that n . Is equal | |
| 15:10 | to and not notice it's 1.1 point oh one forward | |
| 15:16 | to the power T . What would I put in | |
| 15:17 | here ? 1.14 we just said was equal to two | |
| 15:21 | to the power of 1/50 raised to the power of | |
| 15:24 | T . All I did was I said this number | |
| 15:26 | I just saw for what it is . I'm going | |
| 15:27 | to stick it in there . So this means that | |
| 15:31 | in is equal to in not and then multiplied by | |
| 15:38 | two to the power of multiply the exponents . Because | |
| 15:40 | it's an experiment raised to an exponent . It's T | |
| 15:43 | to the T . Over to I'm sorry to to | |
| 15:46 | the power of T over 50 . Right , so | |
| 15:49 | this is the other way to look at it . | |
| 15:51 | So this is one way to look at in terms | |
| 15:53 | of a population growth rate . This is another way | |
| 15:55 | to look at it and this way to look at | |
| 15:57 | it in words is that the population doubles every 50 | |
| 16:07 | days . I'm showing you buy a concrete example with | |
| 16:12 | numbers that the equation that governs money , which is | |
| 16:15 | exactly where this comes from and has exactly the same | |
| 16:18 | form as a percent growth of money every year . | |
| 16:21 | You can think of a population growing X percent per | |
| 16:23 | year or you can convert that into a doubling . | |
| 16:27 | What we call a doubling time formula . This is | |
| 16:29 | the formula that we're learning in this lesson . This | |
| 16:31 | is the equation that I showed you at the beginning | |
| 16:33 | , I said it was the doubling time exponential growth | |
| 16:36 | formula . The final amount of people that you have | |
| 16:39 | is equal to the initial amount of people times two | |
| 16:41 | to the power of this exponent I'm gonna talk to | |
| 16:43 | you about in a second . What this really means | |
| 16:48 | is that what you're really doing here is the exponent | |
| 16:51 | is basically telling you The exponent here in the bottom | |
| 16:56 | , this is the doubling time , 50 years is | |
| 16:58 | when the population doubles when I put a value of | |
| 17:02 | time in here . This whole exponent when you put | |
| 17:05 | a value of time in there , is telling you | |
| 17:07 | how many doubling periods you have . And it's better | |
| 17:09 | to really show you what a chart . So let | |
| 17:12 | me go over here . We have a problem here | |
| 17:13 | that we're gonna work in just a second , but | |
| 17:14 | I'm gonna go and use the bottom half of this | |
| 17:16 | board to continue my thought process over here . So | |
| 17:19 | we're gonna have something like n . Is equal to | |
| 17:21 | end , not Multiply by 2 to the T over | |
| 17:24 | 50 . This experiment is telling you how many doubling | |
| 17:27 | periods I have . Okay , let's say I look | |
| 17:30 | 50 years in the future , I'm gonna put 50 | |
| 17:32 | and for T50 over 50 is one , that's only | |
| 17:35 | one doubling period . So I'll take the population started | |
| 17:38 | with and multiplied by two because then the exponent will | |
| 17:40 | be 11 doubling period . If I then instead looked | |
| 17:44 | 100 years in the future then t would be 101 | |
| 17:47 | 100 divided by 50 is two . Which means I | |
| 17:49 | have to doubling periods . I would multiply by 22 | |
| 17:53 | times for two doubling periods . So this exponent with | |
| 17:55 | the fraction gets a lot very confusing for a lot | |
| 17:58 | of students . All it's doing is it's it's forcing | |
| 18:00 | you to tell me are forcing me to tell you | |
| 18:03 | how many doubling periods I have Here every doubling periods | |
| 18:07 | 50 years when I put the time in and do | |
| 18:09 | the division it's how many doubling periods . And I | |
| 18:10 | actually calculating down the road , that's what I'm going | |
| 18:13 | to use for my xbox . So for instance if | |
| 18:16 | I'm going to do a little chart which I am | |
| 18:18 | I would say this is the time in years and | |
| 18:23 | then this is the population . Okay , so in | |
| 18:27 | zero years , Let's go way on out like here | |
| 18:30 | at zero years , what's going to happen ? I | |
| 18:32 | put zero in for T2 to the zero is one | |
| 18:36 | , and then one times and not , I'm going | |
| 18:37 | to have an initial population and they're not . That's | |
| 18:39 | what it means to start at zero . Okay , | |
| 18:42 | But what if I look 50 years in the down | |
| 18:45 | the road ? 50 years , Okay , then what's | |
| 18:47 | gonna happen is gonna be 50/50 , which is one | |
| 18:50 | . So the population then is going to be in | |
| 18:53 | not times two to the first power . So I've | |
| 18:57 | doubled it . And that makes sense because we said | |
| 18:59 | the doubling time was 50 years . So in 50 | |
| 19:01 | years I'm doubling my initial population . Now , let's | |
| 19:04 | go down there and say we're looking 100 years on | |
| 19:06 | the road down the road . So that's gonna be | |
| 19:08 | 100 over 50 . That's too . And so your | |
| 19:10 | population is going to be in not times two squared | |
| 19:13 | , which means uh In Not Times two . And | |
| 19:16 | then again times too , so it doubled again . | |
| 19:18 | Okay , and we can play this game again , | |
| 19:20 | will go down one more 150 years . Then it | |
| 19:24 | would be in not times two to the three power | |
| 19:28 | , because 1 50 divided by 50 gives me three | |
| 19:30 | , it's three doubling period . So this is one | |
| 19:32 | doubling period to doubling period . Three doubling period . | |
| 19:35 | I hope you can see that this is an exponential | |
| 19:37 | formula , exactly in the same way that this is | |
| 19:40 | an exponential formula . The exponents appear in the in | |
| 19:42 | the the variable is up here in the exponent same | |
| 19:45 | thing here . The variable is up here in the | |
| 19:47 | experiment . It's just 22 different ways of saying exactly | |
| 19:50 | the same thing . If you want to talk about | |
| 19:52 | the population growing X percent per year , you would | |
| 19:54 | use something like this . But if you want to | |
| 19:56 | talk about the population doubling every however many years you | |
| 20:00 | would use something like this . It yields the exact | |
| 20:03 | same curve . The equation is like a little bit | |
| 20:05 | different , but they yield exactly the same curve . | |
| 20:08 | So we can generalize this instead of talking about 1.4% | |
| 20:11 | per year and talking about 50 year doubling time , | |
| 20:14 | we can generalize it to what we call the doubling | |
| 20:17 | time . Exponential growth formula . We have an initial | |
| 20:20 | population , we have a final population , it's doubling | |
| 20:23 | every D . Years . This is the doubling time | |
| 20:25 | . Now I'm using years but the doubling time might | |
| 20:28 | be given to you in hours . If the doubling | |
| 20:30 | time was 10 hours then you would put a 10 | |
| 20:33 | here . But then the time that you put up | |
| 20:35 | here also should be expressed in hours . So when | |
| 20:37 | you do like let's say the doubling time was 10 | |
| 20:40 | hours . But you were gonna look 20 hours in | |
| 20:43 | the future then how many doubling periods would you have | |
| 20:45 | ? 20 divided by 10 ? You'd have to doubling | |
| 20:48 | periods . So this division in the expo is just | |
| 20:51 | telling you how many doubling periods I have . And | |
| 20:53 | then of course you do the multiplication here And double | |
| 20:57 | that many number of times . And that's how the | |
| 21:00 | growth is going to happen . So in population growth | |
| 21:02 | of bacteria or population growth of people or population growth | |
| 21:05 | of of things like that , we don't usually talk | |
| 21:08 | about 5% per year growth . We say the doubling | |
| 21:11 | time of this colony is 15 hours . So then | |
| 21:14 | you will put 15 hours here . The two is | |
| 21:16 | there . You have an initial population . And then | |
| 21:20 | if you want to know how much you have down | |
| 21:21 | the road , you have to stick how many hours | |
| 21:23 | or how many days or whatever it is down the | |
| 21:25 | road that you're looking and you're going to get the | |
| 21:27 | exact same exponential growth curve that you get . When | |
| 21:31 | we talked about money , it's exactly the same thing | |
| 21:33 | . And I tried to show you that it's exactly | |
| 21:35 | the same thing by starting from The growth of money | |
| 21:39 | equation . This is what the population growth is in | |
| 21:42 | terms of percent per year coming from this equation . | |
| 21:44 | And then we just take an example , 1.4% is | |
| 21:47 | 50 years , showing you that when you put it | |
| 21:49 | in there , the percentage can be expressed in terms | |
| 21:52 | of a doubling time , essentially a doubling with a | |
| 21:55 | base of two like this . And when you stick | |
| 21:57 | it back in there , you can you can get | |
| 21:59 | from this equation directly to this equation which means they're | |
| 22:01 | the same thing which equation you use Just depends on | |
| 22:05 | the problem that you have . Okay now the next | |
| 22:08 | thing I want to talk about is half life . | |
| 22:10 | Now we've discussed it a little bit already , we'll | |
| 22:13 | solve our problems in just a second . Notice the | |
| 22:16 | half life equation is exactly the same equation as this | |
| 22:19 | one . The only difference is you have a one | |
| 22:21 | half here instead of a two here and instead of | |
| 22:24 | tea over D . We call it T over H | |
| 22:25 | . H . Means half life . So when you | |
| 22:28 | if you know that a sample cuts in half the | |
| 22:31 | number of atoms or its population , if it cuts | |
| 22:34 | in half every 10 days let's say then H . | |
| 22:38 | Is equal to 10 . And this is the same | |
| 22:40 | same sort of thing . So if you have a | |
| 22:42 | 10 day half life and you look 10 days in | |
| 22:45 | the future , that's one half life period , then | |
| 22:48 | that means this is exported as one . And then | |
| 22:50 | you're just multiplying the population by one half because after | |
| 22:53 | one half life period you should be at half the | |
| 22:56 | population and and so on and so forth . The | |
| 22:58 | exponent that you put here . This division is telling | |
| 23:01 | you how many half life periods you're looking in the | |
| 23:03 | future and then you're just multiplying by one half that | |
| 23:06 | many times . Okay , so I think it's pretty | |
| 23:10 | good idea to take a look at the half life | |
| 23:13 | in a tiny , tiny bit more detail . Just | |
| 23:15 | like we did a chart here for this , let's | |
| 23:17 | say um let me actually get myself just a tiny | |
| 23:21 | bit more room . Let's close this off . Like | |
| 23:24 | this . Let's just say that for an example that | |
| 23:28 | uh half life , yeah , this Of some radioactive | |
| 23:34 | isotope or something like that is 1600 years . That | |
| 23:39 | means that in 1600 years , half of that sample | |
| 23:43 | uranium or whatever it is , it's going to transmute | |
| 23:46 | and turn into some other atom . It's gonna decay | |
| 23:48 | and on average half of it will decay in 1600 | |
| 23:51 | years . That's what that means . Okay , so | |
| 23:53 | then if you were going to look at this in | |
| 23:55 | equation form , looking at the radioactive decay , what | |
| 23:58 | that would mean is that N . Is equal to | |
| 24:00 | end ? Not times one half T over 1600 . | |
| 24:05 | That's what happens . You put the half life into | |
| 24:08 | the H location and then this is the equation that | |
| 24:10 | would govern . Okay , what would this look like | |
| 24:13 | if we then took a look out over time ? | |
| 24:16 | Just like we did for this one . When we | |
| 24:18 | had one doubling period , we doubled uh The population | |
| 24:22 | . We had to doubling periods 100 years out . | |
| 24:24 | We w we multiplied by essentially by four because we | |
| 24:27 | doubled it twice . How is it going to look | |
| 24:29 | here ? Yeah . So let's do the same kind | |
| 24:32 | of thing . So we'll do time in years . | |
| 24:37 | Okay ? And then we'll have amount of atoms left | |
| 24:42 | . Okay ? And we'll go ahead and draw a | |
| 24:44 | line all the way across like this . All right | |
| 24:46 | . So what happens at time zero ? This means | |
| 24:48 | no time has elapsed at all . If we put | |
| 24:50 | a time of zero in here , it's zero over | |
| 24:52 | 1600 so it's 01 half to the zero power is | |
| 24:55 | one . And then times this means we still have | |
| 24:58 | all of our sample left . We start out within | |
| 25:00 | not and in zero years we have exactly what we | |
| 25:03 | expect to have . Okay , Now let's take this | |
| 25:06 | thing out . Since we have a half life of | |
| 25:08 | 1600 years . Let's see what happens after 1600 years | |
| 25:11 | . Elapse If we put 1600 and the exponent and | |
| 25:14 | divide by this , we're gonna get one , which | |
| 25:16 | means one half to the one power . So that | |
| 25:19 | means we're taking in multiplying in not just times one | |
| 25:22 | half , which means we're going to cut the whatever | |
| 25:24 | we started within half , which is exactly what a | |
| 25:26 | half life is . We said half of it should | |
| 25:29 | be around after 1600 years . And that's exactly what | |
| 25:31 | we get from our equation . Now let's go out | |
| 25:35 | to to half life periods 3200 years . If we | |
| 25:38 | put 3200 years divide by the 16 we're gonna get | |
| 25:41 | to . And then what we're going to get is | |
| 25:44 | the exponent will be two . So to be in | |
| 25:46 | not times one half square . So what that means | |
| 25:50 | is we'll take the initial after 3200 years , we | |
| 25:52 | will have the initial amount of atoms times one half | |
| 25:56 | , but then times one half again . So it | |
| 25:58 | cut itself in half , two times the first time | |
| 26:01 | it cut himself in half was after 1600 years . | |
| 26:03 | But then after another 16 years , that cut himself | |
| 26:06 | in half again , that means That it's really 1/4 | |
| 26:10 | of the initial population , one half times one half | |
| 26:13 | is 1/4 . And that's what ends up happening here | |
| 26:15 | and we'll go out one more time here . What | |
| 26:18 | about 4800 ? If I put 4800 here , you're | |
| 26:21 | gonna get three . And so it's gonna be in | |
| 26:23 | not times one half cube . So you take the | |
| 26:28 | initial population times one half times one half times one | |
| 26:31 | half . That's how much you're gonna have after three | |
| 26:33 | half dive periods or left . So if you want | |
| 26:35 | to look at a concrete example , let's say That | |
| 26:40 | the initial amount of the stuff we had was 10 | |
| 26:43 | g That was and not then after 1600 years we | |
| 26:47 | cut this in half and we're gonna have five g | |
| 26:50 | . And then after 3200 years , if you take | |
| 26:53 | 10 times one half times one half again , you're | |
| 26:56 | going to get 2.5 g . Notice you have it | |
| 26:59 | here and then we have it again here . And | |
| 27:02 | then over here you're going to get 1.25 g because | |
| 27:06 | we have it again here . So one half times | |
| 27:09 | one half times one half times the initial gives us | |
| 27:11 | to 1.25 g here . And then of course we | |
| 27:14 | could take this out and say what about two years | |
| 27:17 | later it's going to be exactly what the equation is | |
| 27:20 | here . It's going to be in not times one | |
| 27:22 | half T over 1600 whenever the time is . So | |
| 27:28 | here the numbers I picked where nice multiples here . | |
| 27:31 | So I could calculate easy , but you can put | |
| 27:33 | any time you want into there , Anywhere in between | |
| 27:37 | 1600 and zero , you can put any number you | |
| 27:39 | want and it's going to predict exactly how much of | |
| 27:40 | the sample that you have for exponential growth of money | |
| 27:43 | . The curve goes up like that for exponential decay | |
| 27:46 | , it goes down like that . So I think | |
| 27:49 | at this point we need to solve a couple of | |
| 27:51 | problems . Ultimately , this is the equation that you're | |
| 27:55 | going to use for half life decay . And this | |
| 27:57 | is the equation you're going to use when the population | |
| 28:00 | growth is given to you in terms of doubling time | |
| 28:03 | . If you're ever given a problem where you're given | |
| 28:06 | that the population growth is however many percent per year | |
| 28:09 | then you're going to use the regular compounding growth formula | |
| 28:13 | that we have . All right . So let's go | |
| 28:17 | ahead and do our first problem . A bacteria population | |
| 28:22 | size and not That's the size the initial size of | |
| 28:25 | the population . It doubles every 12 hours by How | |
| 28:28 | much does it grow in two days . It's kind | |
| 28:32 | of a vague problem because it doesn't tell you how | |
| 28:34 | much you started with or how much you're trying to | |
| 28:37 | end up with . It's just telling you that however | |
| 28:39 | much you start with doubles after this much time . | |
| 28:43 | So you're given not a percentage , you're given the | |
| 28:46 | doubling time . So you know , you're gonna have | |
| 28:47 | to use the doubling time formula . So the very | |
| 28:49 | first thing you want to do is write that equation | |
| 28:51 | . Now the number of bacteria that we have in | |
| 28:56 | the future is going to be equal to the initial | |
| 28:58 | amount of bacteria that we have times two to the | |
| 29:01 | power of T over the doubling time by now you | |
| 29:04 | should be reading this exponent to be telling you how | |
| 29:07 | many doubling periods do we have ? That's what it's | |
| 29:10 | telling you . And that's the exponent we're gonna use | |
| 29:12 | on the two . Okay so I have first . | |
| 29:16 | The very first thing you have to do is you | |
| 29:18 | have to realize that you're given days in the problem | |
| 29:21 | for one of the times and the doubling time was | |
| 29:23 | given in a different unit called hours . You have | |
| 29:26 | to make sure that the times that you're using in | |
| 29:28 | these equations have the same units always . So we're | |
| 29:31 | gonna take two days and just say that's equal to | |
| 29:33 | 48 hours . So I have the same unit here | |
| 29:35 | . This is the doubling time um equation here . | |
| 29:38 | And so now I have to start putting things in | |
| 29:40 | place . I'm trying to say that the number of | |
| 29:44 | bacteria in the future is going to be equal to | |
| 29:47 | whatever it is . I start with Multiplied by two | |
| 29:51 | and then I'm looking how many hours down the road | |
| 29:54 | , 48 hours And the doubling time is 12 hours | |
| 29:59 | . So what does this mean ? It means the | |
| 30:00 | doubling time is 12 hours . But I'm actually looking | |
| 30:02 | four times that down the road . So what happens | |
| 30:06 | is this is two to the fourth power . This | |
| 30:08 | exponent , the only purpose of it is to tell | |
| 30:10 | me how many doubling periods I have . The problem | |
| 30:13 | tells me the population doubles every 12 hours . And | |
| 30:16 | when you do the exponent math , the division , | |
| 30:18 | it tells me how many doubling periods do I have | |
| 30:20 | ? Four doubling periods . So what happens is What | |
| 30:25 | this means is that the final population here in is | |
| 30:29 | going to be equal to two to the fourth power | |
| 30:31 | , which is 16 times the initial population . Okay | |
| 30:35 | , so you can write this in words however you | |
| 30:38 | want . The problem says the bacteria population doubles every | |
| 30:41 | 12 hours by how much does it grow in two | |
| 30:43 | days . All you can say the population is 16 | |
| 30:48 | times larger In two days or after two days . | |
| 30:59 | This is where a lot of students get confused because | |
| 31:01 | I never gave you what the initial population was . | |
| 31:03 | And so they don't really know what to do . | |
| 31:04 | I'm just trying to figure out how much bigger it | |
| 31:06 | is . So if you don't know what the initial | |
| 31:08 | population is , leave it in there until and just | |
| 31:11 | tell me how much bigger it is . You see | |
| 31:13 | how little math is here . It's just telling me | |
| 31:16 | How uh long it takes the population to double the | |
| 31:20 | exponent calculates how many doubling periods I have in this | |
| 31:22 | case . four . So that tells me the population | |
| 31:25 | is now 16 times larger . It's gonna be the | |
| 31:29 | similar sort of deal for the half life uh types | |
| 31:32 | of problems . And this problem , it says the | |
| 31:35 | half life of carbon 14 is 5730 years . How | |
| 31:39 | much of a 10 mg sample will we have after | |
| 31:43 | 4500 years elapse . Okay , now , first of | |
| 31:48 | all , before you do any math , what do | |
| 31:49 | you think is gonna happen ? It's telling me that | |
| 31:51 | the half life of this carbon 14 is 5730 years | |
| 31:55 | . That means that however much carbon I start with | |
| 31:57 | , I expect to have about half of it after | |
| 32:00 | 5730 years . But in this problem , I'm only | |
| 32:03 | going out 4500 years in the future . I'm not | |
| 32:06 | even going one half life period down the road . | |
| 32:09 | So I expect to not quite have decayed , I'm | |
| 32:12 | going to decay . I'm just not going to get | |
| 32:13 | all the way to the halfway point . I'm not | |
| 32:15 | gonna have half of my sample left , I'm gonna | |
| 32:17 | have a little more than half left because I didn't | |
| 32:19 | even decay one half life period . Now , that's | |
| 32:22 | in words , what happens ? How did we do | |
| 32:25 | it in terms of math ? What we say , | |
| 32:27 | is that the final amount of material that we have | |
| 32:30 | is equal to the initial amount of material that we | |
| 32:31 | have times one half raised to the power of T | |
| 32:35 | over H . Where H . Is whatever the half | |
| 32:37 | life is . The having time is another way to | |
| 32:40 | write that . So then you just fill in the | |
| 32:43 | material . You know that the initial amount of the | |
| 32:45 | sample is 10 right ? It is 10 if you | |
| 32:48 | want to And then one half its cutting itself in | |
| 32:51 | half every how often ? Every 5730 years . However | |
| 32:55 | , I'm only going 4500 years in the future . | |
| 32:58 | That's T and it's 5730 years in the future . | |
| 33:03 | Years is the same unit as years . So I'm | |
| 33:05 | okay using these numbers . But notice what this fraction | |
| 33:08 | is gonna be whenever you do this division , it's | |
| 33:10 | going to be 10 times one half when you do | |
| 33:13 | 4500 divided by 57 30 you're gonna get 0.78 53 | |
| 33:20 | . Which means the exponent here means it's not even | |
| 33:22 | one , which means I didn't even get to one | |
| 33:25 | having period , I didn't even get to one half | |
| 33:27 | life . So what you do is in your calculator | |
| 33:30 | , take one half to the power of this and | |
| 33:32 | then multiply by 10 , you're gonna get 10 times | |
| 33:36 | 0.58 oh two . Notice that this is now not | |
| 33:40 | cut in half , it's a little bit more than | |
| 33:42 | half . And so what you're gonna get is 5.8 | |
| 33:45 | oh two mg , 5.8 oh two mg . Which | |
| 33:48 | is a little bit more than half of my 10 | |
| 33:51 | . The reason it's a little bit more than half | |
| 33:52 | is because I didn't quite get to 5700 years . | |
| 33:55 | If I would have taken exactly the 50 730 years | |
| 33:58 | , the exponent would be a one and then it | |
| 34:00 | would be multiplying by half and then I would have | |
| 34:01 | exactly five . Okay , so this exponential decay of | |
| 34:08 | half life period , It's interesting to look at a | |
| 34:10 | little chart , we talked about exponential growth a lot | |
| 34:14 | . Let's just do a little graph here . It's | |
| 34:16 | not gonna be for this particular problem , but in | |
| 34:18 | general for the half life decay , you have time | |
| 34:21 | here , Then the initial sample has some in not | |
| 34:25 | some initial in this case it was 10 mg , | |
| 34:27 | but we're going to leave it and not , that's | |
| 34:29 | the initial amount of atoms that I have . What's | |
| 34:31 | gonna happen is also which colour ? This thing is | |
| 34:35 | gonna exponentially decay down . Like this never gonna quite | |
| 34:39 | get to zero , but it's gonna exponentially decay down | |
| 34:41 | . And then when it gets to about half this | |
| 34:44 | is in not over to right here , this is | |
| 34:49 | one half life period . Remember h is the amount | |
| 34:53 | of years it takes for half of the sample to | |
| 34:55 | decay . So eight years in the future is when | |
| 34:57 | this thing has decayed to about half of what its | |
| 35:00 | value is here . If I were to go another | |
| 35:02 | having period down here , I'm not gonna drop but | |
| 35:05 | if I go another H period here then it will | |
| 35:08 | cut it in half again and again and again and | |
| 35:10 | again . That's what's going on . Every time you | |
| 35:11 | go h years in the future it cuts the previous | |
| 35:15 | value in half and half and half . Ok so | |
| 35:18 | it's a lot of material . We've learned not one | |
| 35:21 | equation , we've learned two equations . We've learned that | |
| 35:24 | we can express exponential growth in terms of doubling time | |
| 35:27 | . And we've also learned that we can express it | |
| 35:29 | in terms of half life decay . The equations are | |
| 35:33 | exactly the same thing really . You have the initial | |
| 35:35 | population in the final population , you have the time | |
| 35:38 | in years or hours or whatever . And then you | |
| 35:40 | have the the the the having time which we call | |
| 35:43 | half life , you have the doubling time which we | |
| 35:45 | call doubling time . And I hope that you understand | |
| 35:48 | by taking a look at these charts that these equations | |
| 35:51 | , even though they look different , they are exactly | |
| 35:53 | the same as the original compounding interest equation that we | |
| 35:56 | have . This is one way to look at the | |
| 35:58 | situation . This is another way to look at exactly | |
| 36:01 | the same growth . Notice everything on the left here | |
| 36:03 | with the end , in the end , it is | |
| 36:04 | exactly the same , this and this are exactly the | |
| 36:07 | same curve . It's just you have to have the | |
| 36:09 | doubling time given to you in the problem . If | |
| 36:11 | you're going to express it like that , so if | |
| 36:13 | you're given doubling time , use the doubling time formula | |
| 36:15 | . If you're given half life , use the half | |
| 36:17 | life formula . Otherwise , just know that they are | |
| 36:19 | exactly the same curves of exponential growth and decay that | |
| 36:22 | we've studied in the past . |
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