 So we looked last day We looked last day at half-life at what we practiced some more exponential growth-word problems We said they all fit this template of a equals a zero c to the t over p. They may use different letters For example, I notice in number one Way back here. They're using and and n zero and instead of t over something t over p it looks like whatever that over p was as a fraction as a decimal it was point four three So instead of writing it as a fraction they wrote it as a decimal fine. I can deal with that Here as well instead of using or they are using a but instead of using t they're using n for the number of years fine I can deal with them Were there any of these that you were wondering about I assigned you a bunch. I'm going to be assigning you a few more today go in once Number five Figured someone would ask number five. Did you try number five? Okay, since I've just turned the page. I'm going to do this. I think a Equals a zero c to the t over p. I'm going to write down my little template equation Westlap Carson Okay, I'm pretty sure this 500 is my initial amount. Is that okay? I'm pretty sure this 150 is my final amount. Is that okay? We lose 18 percent, which means I think we keep 72 percent So far so good and It's each time we go through a filter now I've got to translate this a bit. This is not in terms of time But I think my period is every filter every one filter is when we lose 18% wouldn't it be point eight two because I can't do math. Thank you 18 minus how about point eight two? good gosh and They want the number of filters, you know what I'll call that n instead of t because t is time But the it's fitting the same idea instead of years passing its filters passing. Is that okay so far? You got that part? Or not? No, okay now when you solve this Anybody solve this? Did you get this one? What'd you get for your m or your t when you solved it? Okay, so I think you get this You need six point one filters and if you look in the back they did not round it down To six they rounded it up to seven and the reason is it does say less than a hundred and fifty Less than a hundred and fifty grams You know what by round down six filters is not going to be enough just shy of being enough That all right oh See she actually did attach a check to the note Dear mr. Do it. Here's a check That's how you belay That's a bit of a tricky one I probably won't give you one that strangely worded on your test in fact almost certainly It's gonna be a half-life question and some kind of a population growth or population loss question Any others? nine Okay This one is so easy that it's tough and here's what I mean first of all I did in my mind underline the word half-life that tells me that see it's gonna be point five I'm gonna write down my template a equals a zero c to the t over p Who asked me number nine? Amanda what are they asking me to find here? Well, you know what let's use a process of elimination. What's this 30? And we'll give you a hint. It's an amount. It's either the final amount or the initial amount Absolutely. Oh, it's a half-life question. I know see is point five. That's how half-life is defined It's a growth rate of point five. You lose 50 percent every time What's this 12? Well next to it is the word years. I Think that's the total time and we said your growth period is also your half-life Amanda when they tell me The half-life is five point two. They're saying that goes there You know what they're asking me to find in this question. How much we start out with is that okay so far is This an exponent How would I get it by itself then What's happening between the a and the bracket? You know what the initial amount is going to be your final amount divided by? Point five to the power of twelve over five point two and this is straight calculator now I'm gonna go on my calculator 30 divided by Point five to the power of how many terms are there in my exponent Amanda? How many numbers are there in my exponent? Two I better use brackets You know how much you need to start out with a hundred and forty eight point five What does it say to the year's gram hundred and forty you need a hundred and forty nine grams if you want at least thirty grams left over after twelve years Is that okay? Yep, and so I guess my point is I can ask you to solve for this or this as Well as these exponents now if I'm asking you it's all to solve for one of the ones that's an exponent log both sides I'm asking you to solve for one of the ones that's on ground level Just Multipliered of its calculator plug-and-chug and a bit of arithmetic Is that all right? Yep Okay, give me a favor, please Could you turn to page 204 page 204 what we need to talk about today is base e I'm wearing my base e-shirt e it's a scrabble tile and a number It is I made this one up myself actually this was my idea and on the back I have an exponential graph because as Kirsten knows every single exponential graph looks like this or so If you have a base of two it looks like this give a base of e it looks like this remember We said e was this weird strange number e was second function e e was two point seven one eight two eight one eight two It and it looks like your repeats it does not write off your page It goes all haywire in fact if you want E to about 1,000 decimal places here. It is Why is base e? So useful remember this fishing question here Here's what we were doing mathematically when we solve this by having it decreased by five percent each year on January 1st there would be 2,500 fish on December 1st there would still be 2,500 fish on December 31st there would still be 2,500 fish but on December 31st at one minute to midnight boom you lose five percent and Then all throughout the year the fish population stays the same and Katie just before midnight on New Year's Eve Boom you lose another five percent. Is that how populations really work? I think Cassandra populations grow Continuously when you have a big enough population the human population for example There are babies being born every split second not nothing happening until boom right before you move on to the next growth Period and then all the growth occurs Most things grow continuously All the time You know what base is the most useful what base best describes Populations that grow continuously as soon as you the population that's fairly big in the millions Let's say you know what base best describes how they grow base e We call it the continuous growth rate This is why base e is so useful Kirsten as soon as your population is big you can reasonably assume look we're not having a baby every five hours We're having a baby now now in fact even faster than I can snap my fingers as soon as your population is big The growth is occurring instantaneously. It's a much better mathematical model for large populations So the university population an example one Increased by four point five percent per year this increase probably occurred at the beginning of each semester You don't gain university students in February You gain university students in September and you gain them in January so their base e wouldn't make much sense But if you're looking at a bacteria that's spreading base e would make a lot of sense because that beat bacteria is Spreading viral effect. We even use that word in English now. It's we talk about videos going viral. It describes exponential growth It's a base e question Other populations such as the population the world increase at a continuous rate and the formula for continuous growth And I will give you this one on a test. You don't have to memorize this one is that Now you can make an educated guess as to what most of this means Steph, what do you think p zero means? Yep, and you know what p by itself means final your base is e now remember That's not a letter looks like a letter to me. Mr. It's a symbol. It's a number your bases 2.71828182 your bases there Hey Steph, what do you think t stands for? Time and then k stands for your growth constant instead of writing t over p because with base e You don't want to growth period because you want it to be happening all the time You don't want to half-life. You want it to be happening all the time. We don't write it as a fraction We write it as a decimal in front and we call that the growth constant What do we use the letter k because not every single English word math is not all done by English speaking people It was a word for constant in a different letter that began with k. Oh Steph here it is p zero is the initial population P is the final population k is called the growth constant If you're increasing k is positive if you're decreasing k ends up being negative because that's what gives you a negative decay graph Here's an example The intensity i zero of a light source is reduced to i after passing through d meters of a fog According to the formula i equals i zero e to the negative point one two d now first of all How many of you have been in the fog at night before? And if you've driven or been a passenger while driving You'll notice the further away the light goes the dimmer it gets your headlights don't work very far away And you know what it makes sense that that would be a continuous growth because every micro millimeter that you move Your headlights have dimmed by a tiny bit It's not like your headlights are all the same for one meter and then they dim and all the same for one meter And then they give I think every split every split millimeter that you move every split micrometer that you move every spec nanometer That you move the headlights have changed by a tiny bit the strength of the headlights have changed in the fog So the base e this makes sense What's the negative in the exponent telling me? The intensity is getting smaller Okay in what distance oh What do they want me to find here Holly? Which variable do you think stands for distance in the original equation take an educated wild guess from your knowledge of math? You're right D, which means they're probably going to tell me I is your initial intensity and I file final intensity To the nearest hundredth of meter will the intensity be reduced to one quarter of its original value okay? That means they're gonna tell us this and they're gonna tell us this and Remember e is e we're gonna use the letter e on our calculator The key phrase is this last phrase this tells me my final intensity What's my final intensity? What you know what I'm gonna be clever. I'm gonna let it be one quarter point two five if you don't mind And if I do that what's my initial intensity? one By the way, you could also don't write this down. You could also have done final intense initial of four final of one You could also have done Initial of eight final of two That's one quarter. You all wrote Point two five and one there leave that there Don't write this down Supposing I had gone. I'm gonna let my final be One and my initial be four. I would don't write this down. I would plug this into the equation What would I do with that four? I would divide right away and you know what I get over on this side One divided by four, which is what? Point two five, which is why I said to you the other day if they ever give you like a half a population Or one-third a population you can use the original and then divide by three or double or you can just use ones And twos and threes and the ones it's it's much more convenient So I'm gonna do that. Let's see. We said we're gonna let our initial be one our final be point two five And that's gonna give me now you can write this down point two five equals one times e To the negative point two D Ashley will this one make any difference when I divide it over? Okay, I'll leave it there to remind us that we put it there But it's gonna vanish on the next line if that's okay Ashley, where is the D sitting because that's what they want me to find Exponent you know, I'm gonna do up. I'm not gonna take the log of both sides. What's my base? E, you know what I'm gonna take of both sides Ln Yeah, Ln of both sides, which is another base that are this is why your calculator has that base built in actually Ln of both sides. I'm gonna go Ln of point two five and I usually hand write Ln because I'm worried if I print Ln I'll think that's one and later on because I'm stupid that way because that looks an awful lot like that one You don't have to if you're writing looks different from mine, but my printing. Yeah, so I'm gonna hand write Ln every time equals the Ln of e to the negative point one two D So far so good Ryan, what can I do with this exponent? Since it's in no, no, it's inside a law. What can I do with this exponent shruggin? Nathan what can I do with this exponent? This is the whole point of this folks. We have to know this. Oh, I'm a little worried now I Can get it down the whole point of this is to do that because now I can get it down to ground level Yes, yes. Yes. Yes. Yes nod your head. Okay, excellent. I get this Ln of point two five Equals negative point one two D Ln of E That is a case so far Cassandra Remember Ln was actually our way of writing log base. What? When I use Ln, that's the log base. What? You're right. Say it louder Whoever said it say it louder say it means actually say it physically louder not just repeated the same volume say it louder You're right. No, no, sorry. I thought you said the other one. I'm my bad. That's base 10 What was Ln base what? II II and why is that nice? I'm gonna read this but instead of saying Ln of E I'm gonna read that as log base E of E What is the log base E of E? Right Which is why we actually chose to go with Ln instead log because our base was E That was gonna have a log cancel later on. In fact, I have this Ln of point two five equals negative point one two D, how would I get the D by itself now? Well now we're math eight Or math nine. How would I get the D by itself Steph? I think you're right Yeah, D is going to be Whatever the Ln of point two five is All divided by negative point one two D and Not point one two D. Mr. Duke. How about divided by negative point one two. Duh, I need to leave a variable there Hey, you said there's mad night shut up. Oh calculator Ln point two five divided by negative point one two, you know what? eleven point to the nearest hundred eleven point five five meters That's when the light is one quarter as strong as it was From the original location Okay base E How will you know you're supposed to use base E? They'll either use as a trigger phrase continuous growth or you'll see a base equation given to you in the question So part of your homework today If you can turn please To page 208 page 208 you can add number six Now number six is a little bit tricky and deliberately not explaining it because the first thing they're asking you to find is K They're asking you to find K That must mean they're gonna give you X and I zero and I What's X measured in what's right next to the letter X here? What are the units? What's next to this number? I'm pretty sure that's X apparently When X is nine you they'll also tell you I and I zero to find K Then once you found K you could use it for part B. You can't figure that out. I'll talk about it tomorrow or a bit later on in class Really though the kind of question I'm gonna give you is much more similar to number seven How can you glance at number seven and tell that it's a continuous growth base E question? Remember yesterday we opened to a blank page or on a scrap piece of paper last class. We did some examples Same thing except this time. It's going to be changing the base You can either do this right near the same page where we did some more exponential growth equation examples Or on a set of piece of paper What we're gonna look at here. It's ill is how can you change the base if they give you an equation with a certain base Can you rewrite it as a different base and yes you can and There is going to be I think one of these on your test somewhere. Well, I'll ask you to change the base So the heading is changing the base and the first question I gave you is a Population is increasing at a rate of 12% per year a population is increasing at a rate of 12% per year Write an equation describing this so write that far Don't bother putting B and yet because I don't know how much room we're gonna need hey, I don't want to do that Hey, it did it again That's silly. Okay Here do this what you do. What would an equation look like? Well, let's write down our template a Equals a zero c to the t over p the fact that it's 12 percent per year That tells us the growth period. What is the growth period here? It's implied One and we're increasing at 12% I Think the generic equation is going to look like this your final population is going to be your initial population To the power of t over 1 what would your rate your base be right here if we're increasing by 12% Yeah, that's an equation a generic equation. Hello, okay Try that again Down That's an equation a generic equation that describes That particular growth I lost the two there somehow the problem is Doing math with a base of 1.12 is yucky if you're trying to do math in your head I'd like to rewrite this equation as base two I'd like to rewrite that but instead of 1.12 to the t I'd like to to some power. I don't want to change my initial population. I Don't want to change my final population. I want to replace 1.12 to the t with two to the k T Say what hang on bear with me. In fact, I want to replace That 1.12 I want to replace that with a two Now the problem here is this Does 1.12 equal to? No, but you know what? 1.12 equals 2 To some power and traditionally we use a letter k to symbolize that power What number would go there? Oh, I can solve it Where it's gonna be a decimal. I'm sure where is this number sitting an exponent How will I solve for this log both sides? I absolutely agree. So I'm gonna go like this The log of 1.12 Equals the log of two to the k Nathaniel, what can I do with this k? Darn right. That's the whole point of this and in fact I'll get The log of 1.12 equals k log Tyson how would I get the k by itself? I think k is Gonna be the log of 1.12 over the log of 2 and Give it to me to like three. Let's even say four decimal places You get point one six three five What does that mean? What are three dots mean? therefore, I Can replace that one point one two with a two to the point one six three five Which means I can replace that one point one two To the t which is what I had in my original equation with the Two point two point mr. Dude two to the point one six three five To the t Katie is that okay? I'm saying one point one two is the same as two to the power of point one six three five How do I know because I set up the equation to figure that out? And so I can replace that one point one two with a two to the point one six three five and the t and the t Is that okay Katie? Don't write this next bit down Therefore don't write this next bit down My original equation is The same as don't write this down yet because I'm writing it differently. Is this a power to a power? Yeah, what do I do when I have a power to a power? What did we do with the exponents? Multiply them you know what this is how we're gonna write this We're gonna tack the t on there instead of having an extra exponent. I Can rewrite now? This is what you want to write down therefore my original equation Is this base to my original equation? Which was a growth rate of 12% Can be rewritten as a base of two but I have to have that decimal in my exponent because I have to replace the one point one two With two to some power Number two which for some reason has become a number one because I turned the automatic numbering on and I should have not bothered Trying to make a list that's okay Number two Suppose we have a equals a zero one point zero seven to the t By the way, what percent is this population increasing by? Can you see it seven percent? Rewrite this it says as a doubling equation. That means they we want us to rewrite this as base two What's my original base here? What's my original base in this equation? What's my original base in this equation? 1.07 what do I want it to become? So what we're really saying is this I want to replace the 1.07 with a two But that's a nonsense statement right there 1.07 is not equal to two However 1.07 is equal to two to some power it is How can I figure out what exponent I would put there and by the way almost always again if you're changing the base You'll end up with a yucky decimal exponent very rarely will these work out evenly once in a while not very often What am I gonna do now Carson? I'm gonna take a log of both sides Ryan my friend. What can I do with this K? Darn right. Thank you for being back with me. In fact, I'll get this K equals the law mr. Duke you're doing too much work at once Okay, this is frustrating me now often never use automatic numbering. This is what happens when I type in here I'll get the log of 1.07 Equals K log 2. I think K is gonna be the log of 1.07 divided by the log of 2 give me K to four decimal places. What'd you get Andrew? 0.0976 Is that rounded off properly? Okay, so If I want to write this equation as a doubling equation, it's gonna be final amount equals initial amount two to the point 0.0976 That's the same equation if you were to graph that You'd get the same equation as with a base of 1.07 Maybe because we've rounded off a tiny bit after a while it might not quite overlap But certainly for the first million or so it would overlap quite nicely Hey Let's try rewriting this as a tripling equation. That means we want to replace the 1.07 with Not a base of 2 but a base of 3 to the K try that one on your own See if you can figure out what K has to be and what your new equation would look like I'll freeze the screen and I'll do it up here. Yeah We can rewrite any base as any other base the most common ones 2 because 2 is a nice number to do arithmetic with 10 because 10 is a nice number to do arithmetic with oh and If you've been taking a sample of a population and You've been recording the data every month, but you know your population is big enough that it grows continuously base II base II What we're saying here is 1.07 equals e to the K Rewrite this as base e using a zero Sorry a equals a zero e to the kt. I did tell you that whenever I ask you a base e question I will give you this equation. That's the one. We're not asking you to memorize But you know what it's the same as the original equation that we've been looking at except we just write it a bit differently with base e With decimals for the period instead of fractions. We're really saying replace the 1.07 with that Oh now where is the case sitting? It's an exponent. I'm not gonna take the log of both sides Steph Ln of both sides and the K will move to the front. Oh And why would why did we take the Ln of both sides because what is the Ln of e? What is the log base e of e? That K is just going to be the Ln of 1.07 and I get point oh six seven Seven I guess if I round up properly point oh six seven seven Which means my actual equation base e if I take that seven percent growth would look like this What if you get a negative here? What if you get a negative here tells you your population's decreasing and you'd have to double-check your original question to make sure Your growth rate was not seven percent increase but less than one Seven what seven percent decrease good. Gosh is it doing it again? Irritating me now Okay, oh it's gonna stay I guess that's just terrible notes. Oh well last one. I Like this question. I like this question. I like this question. I like this question. I like this question Rewrite p equals p zero times point seven eight to the t as base e By finding the growth constant K and writing your answer in the form p equals p zero e to the k t Change the base from point seven eight to e What we're really seeing here saying here is hey take that Point seven eight and make it Eat of something Try this one on your own. I'll do it slowly up here if you get stuck Okay Back to page 208 so page 208 actually page 211 if you want to be really fussy So now I'm going to assign number 15 and Number 15 in part D They're asking you to rewrite it with a base of one half instead of a base of point nine six They want you to rewrite it as a half-life equation Okay, and then it says use the equation in D to determine the master meaning after one hour 16 is a base e question You know how I can tell just like Lansing at it that 16 is a base e question. Hello 17 18 is good 19 is good This is a long assignment. I've given you a bunch of stuff here But I've given you three classes to work on it And I try to give you time to work on it. Okay So there it is exponential growth on Friday. We're going to look at a specific example of exponential growth called compound interest We're also going to finish off the unit on Friday Take home quizzes next week Test a week from Monday is the plan