 What we said last day was an exponential growth word problem used the same equation over and over a template. Now Ryan they may use different letters in fact in your homework sometimes they'll say oh use an n instead of a t or whatever I really don't care what letters they want to use what I said Matias was this so it's always going to be final amount equals the initial amount times whatever your growth constant is if it's a percentage it's one plus or one minus to the power of the total time divided by the growth period. I said you want to memorize that Nathaniel for your test if you don't have that memorized for your test you only be able to start these questions there's going to be at least two maybe three of these on your test. What I need you to do though is actually find a blank page you can either go to the back of your book where there's a blank page or you can do this on a separate piece of paper and as a heading all I've got is some practice exponential growth word problems and here's the first one. So as a heading practice exponential growth questions either on a separate piece of paper if you prefer your notes that way or you can go in the back of your book as a bunch of blank pages you can put it back there if you want to as well I don't care whatever works for you we're in and doing four of these and you can copy out question number one. Question number one says the population currently has 230 million people if the population is increasing by 7% every two years how long will the population take to double the population currently has 230 million people the population is increasing by 7% every two years how long will the population take to double the first thing you have to be able to do here is to recognize that this is not a logarithmic scale question this is not a pH decibel or Richter scale question because there is going to be one of those on your test words like increases decreases doubles triple words like that are your hints this is an exponential growth word problem and our first approach we said cursed and when we get this we're going to write that template an equation and we're going to do it on every question today and hopefully by the end of that you have it memorized Katie we're going to say a equals a zero c to the power of t over p and then we're going to list our data this 230 million now that's the initial population I could write 230 000 000 but since they gave me the population in millions I'll just put a 230 and my final answer will be in millions or whatever like if I if they need me to find a population here hey what are they asking me to find in this question final population initial population growth rate total time or growth period what are they asking me to find here do you think what does how long mean now how long can either be used for time it can also be used for growth period it's one of the exponent things let's pause and just remember that that must mean they told me the other stuff what's my final populate oh I forgot to put a zero there what's my final population some of you're going why didn't you put in a zero there a national sorry my bad what's my final population how'd you get 460 because it says to double what if it said to triple I'd go 200 690 okay what's my growth rate my growth constant well increasing by 7% means what am I gonna put here do you remember Nicole one my original population plus the 7% increase sorry isn't it what okay is that right what if it was decreasing by 7% it'd be one minus 0.07 or 93 0.93 and there's some other possibilities we'll be looking at some of those today as well what's this to well this is your growth period because it says it's growing every two years what if it was every three years my growth period be three what was every month my growth period be one twelfth whatever this is a decimal or fraction or whatever and they're asking me to find total time so when I plug in numbers Carson I'm going to get final amount equals initial amount 1.07 to the power of t over two now the mistake that drives me crazy that I have to now stop and give you zero for the rest of the question is the kids that go like this oh I'm going to go 230 times 1.07 don't write this down oh this is really 460 equals 246.1 to the t open why is that such garbage is there an exponent on the 1.01 then don't you dare be multiplying before the e in bed mass complete garbage instead because you will almost always have a coefficient here Katie you always have initial population sometimes your initial population Katie will be one which is really nice a little easier we're always going to on this line divide by the coefficient 460 divided by 232 equals 1.07 by the way this is going to give us a little shortcut next time if they say how long to double I could use 230 and 416 but can you also see I could use initial of one and final of two what if they said how long to triple I could use 230 and 690 or if I'm really clever I can use an initial of one and a final of three because when you divide that's what you'll end up with anyways a little shortcut Amanda where's my exponent sitting oh where's my exponent where's my variable sitting so how am I going to solve it darn right once you've written that down I need to look up because I need to beat a sloppy notation out of you to prevent you from making dumb mistakes every year every year Holly I see some kid and it's clear that there I had a great eight teacher that lovingly beat this out of me they do this don't write this down this is yucky they write a fraction as two half lines instead of one top and one bottom clearly and this confuses the heck out of them because they don't know what to do with that too oh please if you ever write a fraction like that I'll freak on you the t is on the top the two is on the bottom and just extend the bar write that down because now you've got clearly on the top step you've got clearly on the bottom and I'm even going to suggest that you all know what to do now how would I get the t by itself it's math eight yeah in fact I think I'm going to times by the two because it's dividing the t yes I'm going to divide by the log of 1.07 because it's timesing the t and cross multiply I think t ends up being two log two divided by the log of 1.07 which is what now go to your calculators try having fun and wandering again usually that means I'm getting tired and need to move around what do you get let's see two log two divided by the log of 1.07 now let's decide unless they say different we'll go to two decimal places 20.49 almost 20 and a half years bang on is that all right example two almost identical to example one except all I've done is I've changed how often it increases a population has 230 million people if the population is increasing by 7% every half year how long will the population take to double in fact you probably use ditto marks for most of the question and just replace the phrase every half year kirsten you know what I'm going to do first I'm going to write that because I got to memorize the stupid thing anyways I'm going to list my data off in the margin here somewhere a equals you know what let's get really really clever let's try that shortcut because we know things are going to double I'm going to let my initial population be one and my final population be two because if I put a 230 and a 460 there Katie I get the same thing is that okay didn't lose you there you don't have to do that but you know I'm in there enough to like shortcuts my growth rate kirsten is still going to be 1.07 because we're still increasing by 7% they're still asking me to find t but what's my growth period this time where do you get the 0.5 you're right every half year let's plug in chug uh final amount two equals initial amount one 1.07 to the power of t over 0.5 is that okay holly again do you have to do this stupid little shortcut now you could have put the 230 and the 460 but it's kind of nice if they ask you double or triple use a population initial of one because now technically I would divide both sides by one holly is that going to make any difference at all when I divide both sides by one then holly where is my variable sitting I'm going to take the log of both sides and then I'm going to move the exponent to the front ah but I'm not going to write it as a stupid small slanty fraction I'm going to write it as a big two-line fraction extend the fraction bar t by itself oh I'm dividing by 0.5 I think I'll multiply on the other side I'm timesing by 1.07 I think I'll divide on the other side I think t is going to be 0.05 here 0.5 times the log of two divided by the log of 1.07 I got 5.12 5.12 years big difference 20 and a half years and the other question just our five years in this question kind of a counter-intuitive answer is why I said a lot of politicians a lot of city planners they don't understand exponential growth and how just changing a tiny thing makes a huge difference in your answer let's rank let's ramp up the nerd component example three radioactivity example three a radioactive substance weighs 631.7 grams what's the half-life of this substance if there is 219.2 grams left after 19 days radioactive substance weighs 631 grams that's decaying that's how radioactivity works energy is being converted sorry mass is being converted into energy and so the mass goes down when it gives off radioactive energy there's a little physics lesson for you some of you that are in physics may have heard the conservation of energy law rephrase slightly differently what is the half-life of the substance if there are 219.2 grams after 19 days what is the half-life if there are 219.2 grams after 19 days hey Tyson you know I'm gonna do first heck why not I'd like you to all underline the word half-life this is how half-life is defined if it's a half-life question your growth rate is 0.5 it's a half it's 50 c is 0.5 I'm not going to tell you that I'm going to expect you to remember that but is there an easy trigger rate way to remember that your growth rate is 0.5 what words do we underline and half as a decimal is okay that means all the rest of it should be in there what's the 631.7 I think that's what you're starting out with what's this 219.2 final amount and you'll notice your final amount is smaller because it's radioactive decay most common mistake kids put the initial and the final in the opposite places they got them mixed up and then when they solve for time they get a negative time nature is saying actually no in the past you had that much that's why it's negative in other words you'll know you've messed that up again negative time now what's this question asking me to find what is the okay half-life is the period this question Katie is asking me to find p that means they must have told me t the total time that's elapsed 19 days tricky part is p is measured in days half-life is measured in days and t is measured in days and so it takes a bit of practice of just reading the questions Amanda so you can tell what they want you to find the half-life of the total time and all I can say is I'm gonna give you a bunch for practice and you'll figure it out but I can say if they ask you to find the half-life that's a period let's plug in our numbers 219.2 equals 631.7 a half to the power of 19 over p is what we're trying to find now what yeah now I made these numbers up and so it probably won't work out evenly don't round off I'm gonna use the answer feature on my calculator to carry extra decimals I would never round this off to two or three decimal places because it's going to introduce a lot of error I would carry five or six 219.2 divided by 631.7 now this is a fluke that we have all those zeros there holly so I can round off probably the .347 and get a legitimate answer but I'm still going to use the answer button on my calculator I'm going to write .347 but in real life I should carry at least six decimal places because exponential errors if you round off the rounding off errors increase exponentially so they make a big wave of good difference that makes sense now my little rant so I'm going to write .347 equals .5 to the 19 over p actually where's the variable sitting you know I'm going to have to do what am I going to have to do with my variable sitting in exponent folks log both sides see how clever I was I left a space after the equal signs put the log in there did I yell at you about writing proper fractions big fractions not little small ones Katie this is where it really shines I marked this question two years in a row on the provincial exams of the summer and I saw a whole bunch of kids don't write this down do this and they froze did not know what to do for rag jean I'm going to suggest if you don't write that as a smooshy splinty fraction but if you clue in that's on the top and that's on the bottom okay I'll put the over one there just this one feminist it's great eight on cross multiplying right physics 12 you can use stuff moves diagonally you're going to get p log point three four seven equals 19 log point five you're going to get p equals 19 log point five divided by the log of point three four seven what's the half length of this substance and you notice again I'm using the answer button do you get 12.44 so this substance loses half of its mass every 12.44 days after 24.88 days it'd be one quarter as big I almost guarantee in fact I will tell you there's going to be a half-life question on your test it's going to be probably two word problems rock sand one of them going to be a half-life the other one probably gonna be population one more example four a radioactive substance has a half-life of seven point four days how long until only five percent of the substance remains a radioactive substance has a half-life of seven point four days how long until only five percent of the substance remains I would do two things right away actually while I stand in your way I would underline the word half-life that's going to tell me something important tells me see but even before that Ryan you know what I'm going to do oh I'm still writing my bad Ryan what am I going to do first yeah a equals a zero c to the t over p ready oh half-life mr. duix says I know this what's that seven point four uh karson what it's the half-life and I said the half-life is the period this time they've given us p p is seven point four okay what's that five percent final amount you see I'm kind of going process of elimination here right I'll look at the numbers figure out what those are and then I'll fill in the blanks final amount is as a decimal point zero five because that's what five percent is okay take a guess what percent did I start out with do you think as a percentage a hundred percent if they ever give me a question and they're talking about percentages now I gotta distinguish between there are percents in this question step but there was also percents in number one and number two can you see there number one and number two though the percents were clearly a growth rate here the percent is clearly an amount and I can't teach you how to recognize the difference between the two all I can say is read carefully anyhow I think my initial amount is 100 percent which is what is the decimal Andrew one which means I think when they say how long this time it's t not how long is the half-life period how long is the total time t how long until all right let's plug in what we got here here final amount point zero five initial amount one point five to the t over seven point four try solving this on your own I'll do it slowly up here you get just shy of 32 days 31.98 days yep half-life question with percentages involved can you turn now go back to your workbooks page 206 page 206 page 206 example seven page 206 example seven now we're talking about radioactive iodine from Chernobyl in page 206 example seven it's talking about radioactive iodine 131 that 131 is not a mass or anything like that it's the atomic number like uranium 236 but you're not going to use that in the actual question step says this in april 1986 nuclear accident of Chernobyl contaminated the atmosphere with quantities of radioactive iodine 131 the half-life is 8.1 days determine the number of days to the nearest day it took to reduce to two percent of the original level very similar to number four that we just did so try this one on your own I'll freeze the screen and I'll do it up here see if you get what I got yes no yes final amount point zero two initial amount one hundred percent c is point five p is 8.1 ask me to find t is that okay rucksack sorry still working yep back ronati yep okay how else can they mess around with this what else can they ask take a look at example five at the top of the page here patient feeling ill had a sample of bacteria taken from her throat say ah the sample contains 387 bacteria cells 24 hours later the sample was carefully recounted and it was found to contain 812 cells find the doubling period to the nearest tenth of an hour katie you know i'm gonna do first ah sure why not i feel better it's no longer blank it gives me a systematic approach i'm gonna go a equals a zero c to the t over p now katie this is a bit of a tricky one some things are obvious i hope okay can you tell me what my initial amount is yep and what's that 812 katie i think you're right i couldn't hear yep do me a favor folks underline the word doubling period and what they're telling you in this question by saying doubling period they want c to be two what if they said tripling period they'd want your base to be what are they asking me to find in this question by the way find the what manda okay so i'm gonna write down p equals question mark that means amanda they must have told me time this one's a bit tricky because there is one more number that appears in this question however it was written out in english not written out like nums what's the total time elapsed in this question kirsten 24 so the equation that i'm gonna end up solving kiddo solving kiddo is 812 equals 387 times 2 to the 24 over p what do i do first before i do anything else divide by okay you get 20.7028 blah blah blah blah i'll write 20.7 but i'll use my answer button on my calculator be aware of the letter p sitting exponent so how am i going to solve it take the log of both sides yep uh hey i can fit a shmusha log in there and i can fit a log in right there and i'll get this the log of 20.7 equals 24 over p log 2 is that okay how do i get the p by itself i think p is going to end up being 24 log 2 divided by log of 20.7 oh hang on don't use 20.7 if you do it use your answer button do you get just shy five and a half days oh what does it say to the nearest tenth of an hour 5.5 hours so every five and a half hours this population doubles at 11 hours you have twice as much 16 and a half hours you have twice four times as much now we also need to look at a couple of more weird things in example six it talks about changing the base we're going to look at that next class we're also going to look at why base e is much nicer to use as our base this is where base e really shines that's going to be next day for today giving you lots of time i ready to say try numbers one and two you can try number three number four but can you all look up at number four please there is a typo in number four instead of two thousand can you make it 20 000 put an extra zero on it please otherwise you can't get the answer in the back of the book you'll be a decimal point out and you'll be freaking out and i wouldn't want you to ever have to freak out at least not in five is good five you have to think about the answer a tiny bit because you'll round the answer a certain way they won't and you'll say why but then you'll go oh because you're clever six and seven are for later because their base e as is eight base e i like number nine nine like number 10 11 is good 12 is good 13 is good 14 is good i'm going to temporarily pause there now that's not all for homework because next class i'm going to be continuing with today's lesson it'll be about half an hour or so then you'll have class i'll start a couple more questions from here i'll also be giving you a take home quiz on half life and victor scale and work problems but boys and girls you have the remainder of class to work on your own you don't often get this much time so enjoy it