Common Core Algebra II.Unit 4.Lesson 6.Exponential Models Based on Percent Growth
Algebra 2
Learning the Common Core Algebra II.Unit 4.Lesson 6.Exponential Models Based on Percent Growth by eMathInstructions
I love I'm Kirk weiler, and this is common core algebra two by E math instruction. Today, we're going to be doing unit four lesson number 6, which has a really long title. Exponential modeling with percent growth and decay. All right. So today, we're going to review quite a bit from algebra one, where we build exponential models based on a constant percent growth or a constant percent decrease, which sometimes we call decay. All right, let's jump right into the growth models and take a look and see what you remember from common core algebra one. All right, exercise number one, very wordy. Suppose that you deposit money into a savings account that receives 5% interest per year on the amount of money that it isn't that's in the count for that year. Assume that you deposit $400 into the account initially. But all right, how much will the savings amount increase? By over the course of the year. I can't read very well right now.
Pause the video right now and try to figure out how much the savings will increase over the course of the year. Well, I hope you didn't have to do any is of or anything like that. We just want to take that $400, we want to multiply it by .05, and it's going to increase by $20. All right. Letter B is real easy. How much money is in the account at the end of the year? That's simple. That's going to be 400 plus 20. Or $420 at the end of one year. Now letter C is the real important part, perhaps the most important part of the entire lesson. By what single number could you have multiplied the 400 by in order to calculate your answer in part B? All right? Pause the video. You might remember this, but let me just explain again, I want to go from the 400 to the 420 in a single calculation as opposed to two calculations. All right. Well, you should multiply by 1.05, but let's make sure we understand why. We've got the 400 and then we're multiplying by 400 times .05. So strangely enough, if we use factoring, we can factor a 400 out of both of these. Being left with a one plus .05 or 400 times 1.05. There it is.
Now what I like to always tell students is that what that really means is that you're ending up with a 105% of what you began with a 100% plus an additional 5%. All right? Letter D says using your answer from part C determine the amount in the account after two years and ten years. Around all answers to the nearest scent. Well, after two years, we will have gotten to multiply by 1.05. Twice, so if we just take our calculator out and punch that in and we will use the calculator soon. So have that TI 84 handy will bring it out in a little bit, but if I just punch that into my calculator, that ends up being an exact answer. $441, but I'll put .00 since I'm rounding to the nearest scent. And then after ten years, we'd have 400 times 1.05 to the tenth. That one's a little bit messier. But it comes out to $651. And 56 cents. All right. Now let's put this into an equation. Letter E, given equation for the amount in the savings account, S of T, as a function of the number of years since the $400 was invested. All right. Well, this is simple enough. We always have that $400 that Y intercept. Times 1.05 raised to the T, right? That's exactly what we had here, where T was two and T was ten. Now, it could be maybe we're interested in doubling our money. Letter F says using a table on your calculator to determine the nearest year.
How long it will take for the initial investment of $400 to double, provide evidence to support your answer. So let's take a look. Let's do a table in our calculator. Let's open up that TI 84 plus. Great. There it is. All right, not a lot of room on the screen, but sorry, we'll cover up the text a little bit here and there. Now what I want to do is I want to go into Y equals. Let's go in there. Obviously, clear out any equations you might have in there, they'll just make things confusing later on. So take a moment to do that. And now let's put in Y equals 400 times might as well put it in just the way it looks, except we'll put an X on a T one O 5, raised to the X all right, make sure that it's in there. Now remember, we started off with 400. So what we're going to be looking at in the table is we're going to be looking for 800. So let's do this. Let's go into our table setup. Let's start our table at zero. Let's make it go by ones. That all set up, taking our time, taking our time. And then let's pop into our table. Let's go. Second graph. And that gets us into our table. Now, of course, what we want to do is we want to just hit the down button and keep looking at these values. And until we see the 800. Now unfortunately, or fortunately, I don't know.
We never really see the 800. What we do see is that at 14, we're at 7 91 97, and at 15, we're at 8 31 57. So, you know, we want to get it to the nearest year. It looks like it's probably closer to 14 than 15, but yeah, I'm not really sure. So let's do this. Let's go back into our table setup. All right. Let's change our table start 14. All right. And let's now change how often our table increments itself or our delta table to .1 to one tenth of a year. All right, and let's hit second graph to put us back into the table. Now, let's scan down. And see, now what I see is I see that my savings at 14.2 years is 799 74, almost double. And my saving after 14.3 years is 8 O three point 6 5, so quite close, so we're going to conclude 14 years to the nearest year it takes 14 years to double. Now, some people would argue, but wait a second at 14 years we haven't doubled. And at 15 years, we've more than doubled. So maybe we should put down 15. That we might put down if the question had asked, what is the minimum amount of time? But this one specifically says to the nearest year and it's closer to 14 years than 15 years when we double. All right. I'm going to put away the TI 84 plus now. It's gone. A little bit of a loud snap. Sorry about that. But hopefully it woke up any sleeping students. And now we can clear out the text.
Pause the video now if you need to. Okay. Got to get rid of it. Don't put that calculator away though. We'll bring it back in a little bit. All right, increasing exponential models. If a quantity Q is known to increase by a fixed percentage P in decimal form. That's key. Thank you can be modeled as Q sub zero times one plus P to the T where Q sub zero represents the amount of Q present at T equals zero and T represents time. Lots of underlining there. So I'm going to get rid of that. All right, let's jump into a problem. Exercise two, which of the following gives the savings S in an account if $250 was invested in an interest rate of 3% per year. Well, think about this a little bit, okay? All right. Well, we could just follow what's in that box, but the plain fact is if we're increasing by 3% per year, that means that we get to multiply by 1.03, right? That's what we multiply by every year. And we're starting off with $250. This just simply tells me how many times I got to multiply, right? So my savings 250 times 1.03 to the T is choice two. So if you got that right, you definitely have the hang of this. Definitely have the hang of this. Now, we're going to immediately segue into the decreasing exponential. But here's the key. At the end of the day, you always multiply the multiplier.
This guy right here always represents the percent you have remaining. That's important. Because when we're increasing, we take a 100% and we add the percent. When we're decreasing, we're going to do quite the opposite. Pause the video now before we get into decreasing exponentials. All right, let's get rid of the text. And move on. Okay. Exercise three says state the multiplier, the base. You would need to multiply by in order to decrease decrease. A quantity by the given percent listed. So let's take an example. Let's say we took, I don't know, $500, and we wanted to decrease it by 10%. What I would do is I'd take 500 and I'd subtract off 500 times 0.1. But that would be 500 times one -.1, which would be 500 times .9. I'm going to put .90, right? So the correct answer is .90. In other words, we would have 90% remaining. You always multiply by the percent you have remaining, right? Whether that's a 105% when we're adding 5% or whether it's 90% when we're subtracting. All right, now if you think you have the idea, if you think you have the idea, pause the video now and do letter B, C and D. All right, let's go through them. Well, for letter B again, we're going to multiply by one minus the percent in decimal form. One minus two, we're going to multiply by .98 because if we decrease by 2%, we have 98% remaining. If we decrease by 25%, we're going to have 75% remaining.
That's pretty easy. All three of those because they're whole number percent. Take a look at this. Often percents that we work with are very small. .5%, not 5%, .5. So that would be one -.005. Or .9 9 5. All right, so that can be a challenge. But this is going to be important because it's going to allow us to build exponential models that decrease instead of increase. So let's do that in the next problem. Pause the video now if you need to, and then we'll move on. All right, let's do it. Decreasing exponential models. If a quantity Q is known to decrease by a fixed percent P in decimal form, then Q can be calculated this way. So this is exactly what we had before. The difference is for increasing we have one plus P to the T and for decreasing we have one minus P to the T besides that they are identical. All right? So let's take a look. If a population of a town is decreasing by 4% per year and started off with 12,500 residents, which of the following is its projected population in ten years, showed the exponential model you used to solve this problem. All right, well, if you think you have the idea of the decreasing, go for it. And then we'll work through the math. All right, let's do it.
Well, if we are decreasing by 4% per year. Then we're going to multiply by .96. How many years? Ten years. So we're going to get to multiply by .96 to the tenth. And we're going to have this. So we just use our calculator 1012 1500 times .96 to the tenth. And what we find if we look at the notes that has the answer on here, it's a messy number. It's like 8310.407, et cetera. But that's closest to choice four. All right. Let's do some more sophisticated modeling. Last problem. The stock price of wind power ink is increasing at a rate of 4% per week. Okay, so I got stock price of wind power ink is in increasing. Increasing at a rate of 4% per week. Its initial value is $20 per share. Okay, great. On the other hand, the stock price of gerbil energy is crashing. It's losing value at a rate of 11% per week, okay? Okay, it's a losing value. If it's price was a $120 per share when wind power was $20, how many weeks will it take for the stock prices to be the same? Model both stock prices using exponential functions then find when the stock prices will be equal graphically. Draw a well labeled graph to justify your solution. All right. Well, let's do wind power.
Let's do Y equals. What did we start off? We started off with 20. It's increasing at a rate of 4% per week. So here's an equation that models its value. On the other hand, gerbil, that's decreasing. I mean, granted, it started off with a $120. But it's decreasing at 11% per week. So there's its function. And what we want to do now, now that we've modeled them, is we'd like to create a graph that shows when these are equal. So let's bring out that TI 84 plus again. And new calculator, plenty of room this time. Let's go into Y equals. And let's clear out the equations from the last problem or equation. All right, now in Y one, I'm going to enter in that function for the wind power. So that's or for wind power ink. So I'm going to put in 20 times 1.04 raised to the X make sure that one looks good. All right, hit enter. Now let's do gerbil energy. 120 parentheses. .89 parentheses. Raised to the X enter. All right. Again, what we want to do is just take a moment, look at it, make sure it looks right, make sure both are correct. All right, what do I want my window to be? Well, I know that the gerbil energy is starting off at one 20. And then it's going down. I know wind power is starting at 20, and it's going up.
So that gives me a little bit of sense for my Y window. What I don't have a great feeling for is my X window. So real quick, let's jump into our table. And just get a sense for what this looks like. So let's go into our table setup. Let's start our table at zero. Let's change that Delta table to one. Now we may even want to make it bigger like 5 or something. But let's stick with one. And then let's pop into our table. Let's do that now. Now what's really cool is remember when powers in Y one, gerbil power, if you will, as in Y two. And let's go down. Now, what we can see right away is that the gerbil power is higher than the wind power. But they're getting closer and closer together. And down here at 11, 11 weeks, that's what X is. You've still got wind power being slightly less than gerbil power. And at 15 weeks, they've traded spots, so that gives us a really good sense for our X window. So what I'm going to do is I'm going to make my X window, I think, go from zero to well, I got to do at least 15, but let's go zero to 20, or this is weeks. And then my Y window, I think I'm going to go zero to I'm going to hit one 20 at least, but why don't we go a 150? All right, so let's do it.
Let's go into our window now. X-Men, let's make zero. X max, let's make 20. Okay, why men? Let's make zero. Y max, let's make a 150. Okay, I'm going to check all those out. Okay. I think we're ready to graph. Remember the first one graph is going to be wind power, the second one graph is going to be gerbil. I just can't say that enough. Sorry, it's just too much fun. And I think I've tried to find a line. All right, let's hit the graph. Okay. Let me see if I can reproduce this. And then we'll find an intersection point. We'll try to reproduce it. Okay, first things first, we got your wind power. Right, this is Y equals 20 times one four to the X, switch colors. This is our durable power. Plus 120 .89 to the X and now let's find that intersection point, specifically let's find out its X coordinate. So let's go into the calculate menu. Remember it's right up here. So second trace, that gets me into calculate. I'm going to go down to the intersect option. So intersect. Let's say curve one, hit enter. Curve two, hit enter. Then it wants to guess, because there's only one intersection point where pi okay. Let's try to get a reasonably near.
Now let's say enter. Thanks a little bit. We have our answer. Right? The coordinate point itself is at about 11.5 comma 31.4. All right? So it's at about 12 weeks. When the stock prices are equal. About 12 weeks. All right, let's pause there. Take a look, and then we'll put away the TI 84. All right, let's get rid of it. There goes the TI 84 plus. Let's also get rid of the text. And let's finish up the lesson. So much of the modeling of exponential functions or modeling with exponential functions, sorry. Based on constant percent growth and decay is material that you did in common core outs were one. But as always, it's been a while, all right? It's important to be able to very quickly look at a multiplying factor and know what percent your increasing an item by and what percent you're decreasing something by. Very, very critical in lessons to come. For now though, I want to thank you for joining me for another common core algebra two lesson by E math instruction. My name is Kirk weiler. And until next time, keep thinking and keep solving problems.