Common Core Algebra I.Unit 4.Lesson 7.More Linear Modeling.by eMathInstruction

Now, we've already spent an entire lesson on doing linear modeling, IE taking real world scenarios, where two variables are related by linear functions. And writing equations predicting outcomes, checking those outcomes, et cetera but linear modeling is one of the cornerstones of common core algebra one. It's one of the things that you should come out of this course being fluent in. Fluent, meaning that you can do it as easily as you do addition subtraction. Okay? So we're going to spend another day on it, another lesson, and let's jump right into exercise one. A warehouse is keeping track of its inventory of cardboard boxes. At the beginning of the month, they had a supply of 1275 boxes left. Okay, so that's what they're starting with. And they use boxes at a rate of 75 per day. But ray asks, how many boxes are left after ten days show the calculation that leads to your answer. Pause the video now, think about this and see if you can come up with an answer. 

All right, let's go through it. Well, for a lot of students, they're going to do the following. They're going to say, well, I'll write. I use a 75 boxes per day. Right? Ten days have gone by. And that means that we've used up 750 boxes. But of course, we started with 1275 boxes. So we subtract off the 7 50, and we find 525 boxes are left. Okay. So after ten days, 525 boxes are left. Now, letter B then asks us to extend this, right? And this is actually almost the biggest thing in algebra. Can you now go from saying how many boxes are left after ten days? To how many boxes left after D days, right? So N is the number of boxes D is the number of days, which of these four choices gives us the number of boxes left after D days. Okay? Pause the video now and see if you can figure it out. All right, let's go through it. Well, think about what we did, right? What we did is that we took 75 and we multiplied it by the number of days, right? And that's how many boxes we had used. But then we took that amount and we subtracted it from 1275. And that gave us the number of boxes left. That's exactly what we did, right? This was 1275 minus 75 times the number of days. So the only difference between these two is in this case, I know the number of days is ten, in this case, the number of days is left as a variable. And therefore, choice three. All right? 

Let's take a look at C if the warehouse needs to order more boxes when their supply reaches a 150. How many days can they wait? Pause the video now, see if you can figure this out. All right, let's go through it. Well, right now we know that this equation, if you will, models, how many boxes we have left, according to how many days we've been using the boxes. Now that one 50 that they gave me is my N it's the number of boxes I have left. So I'm going to put that into my equation and I'm going to let my algebra do its magic, right? I'm going to solve this equation now. I'm going to subtract 1275 from both sides. That's going to be a little bit obnoxious because it's going to leave me with a negative, but I can deal with that. Especially if I'm using my calculator, that will be negative 1125. Sorry, I'm looking over at my sheet sheet every once in a while. IE my answer key equals negative 75 times D, I can now divide both sides by negative 75. And what I'll find is that D equals 15. 15 days before I get down and have to order more boxes. Now, letter D sometimes math can kind of switch around on you.

Right in the middle of a problem. That's a life, right? Letter D if after 5 days, 5 days have gone by. They start using boxes at a rate of 90 per day, right? So we were using them at 75 per day, 9 90 per day. How many days will it be before they run out of boxes? Show the work that leads to your answers. So this is kind of cool, right? Pause the video now and see if you can work through this. This is a real world problem. All right. Let's go through it. Well, what we may want to do is figure out how many boxes we have left after 5 days. So after 5 days, 1275 -75 times 5 leaves us with 900 boxes left. Right? Now we start using them at 90 per day, right? So let me just, let me just set up a real quick. Real quick equation. Let's do 90 times the number of days equals 900. Right? 90 times the number of days equals 900. Divide both sides by 90. And we get ten days. So it's going to take us ten additional days before we run out of boxes or a total. Of 5 plus ten or 15 days. I know, by the way, that that's exactly the same answer we got in C, but that's a complete coincidence. 

Just totally different problem than. Quite frankly, I probably should have designed it so it came out to be different. But it's a total of 15 days. And it's kind of neat. Really what we're dealing with in D and we'll talk about this later on in the course is a piecewise linear function, right? It was linear with the slope of negative 75. For the first 5 days, and then it became linear with a slope of negative 90 for the remaining, I guess, ten days, if you will. All right? So I'm going to clear out the text, write down what you need to. All right, here it goes. Let's do some more with linear models. Exercise two. The cost C in dollars of running a particular factory that produces W widgets can be modeled using this linear function. By the way, if you don't know what a widget is, a widget is an object that a company produces when a, let's say, worksheet designer is too lazy to come up with a real product. So a widget is like an imaginary product that a company produces. Okay? So anyway, the idea though here is that notice, right? Function notation. That's all that is. Function notation. 

All right. Just says that the cost in dollars of running a particular factory is based on the number of widgets it produces, right? That's the number of widgets it produces. It's given by that equation. Now, letter a is just an interpretation. It says, how do you interpret the fact that C parentheses 100 equals 2300? All right? Pause the video. See if you can write down how you interpret that piece of information. Okay, let's do it. Well, here's the input, right? So that's the in. Input, right? So there's 100 widgets. And this is the output. What was the output? That was the cost. So I would interpret it as the following. It costs. $2300 to produce 100 widgets. And a day. At this factory. That's my interpretation. Now that our B says give a physical interpretation for the two parameters, remember the linear parameters in this equation of one 25 or sorry, 1.25. And 2175. So what do those mean in the real world? What are those mean? All right? Pause the video and see if you can write down how you interpret those two numbers. All right, let's do it. I'm going to actually start with the 21 75 first, okay? All right. 21 75 represents the cost to run the factory. 

Even when zero widgets. Are produced. So very often, a company, even if it doesn't make any product in a given day, a bicycle company, a car factory, whatever. It still has costs, all right? These things are what are known as fixed costs and economics. Fixed costs. All right. And you know, I mean, it makes sense, right? I mean, you got to heat the place. There's the rent. There's taxes. There's all sorts of things. So even if you don't build a single widget, it's still cost money to run this factory. Now what is the one 25 represent? Well, this one's a little bit trickier, but remember this is a slope, right? So it's the change in C divided by the change in W let me write it as this 1.25 divided by one. So what this represents, the one 25. Represents that each additional widget. Will raise the cost. By a dollar 25. All right? So each widget that's created, right? An increase in one widget represents an increase in cost of $1 and 25 cents. Okay? And slope always has an interpretation like that. As this thing goes up by one, this thing goes up by whatever. All right? So pause the video now, write down whatever you need to. Okay, let's clear it out. Keep on modeling.

Let's talk about deer in the northeast. We have a lot of deer. I don't know where you're living, but I see them all the time. And unfortunately, my car sees them sometimes. Biologists estimate that the number of deer and Rhode Island in 2003 was 1028, Rhode Island and tiny state. No offense if you're in Rhode Island. And in 2008, it had grown to 1488. Biologists would like to model the deer population P as a function of the year's T since 2000, okay? So in other words, this is my output. And this is my input. Okay. First thing we're asked to do is represent the information that we've been told as two coordinate points. Be careful to know what your values of time are for each year. Why does it say that? Well, it says that because we want T not to be the year, we don't want we don't want our input to be 2003. It's the years since 2000. So in other words, the first piece of information they give us actually occurs at year three. 2003 is three years since the year 2000. And there are 1028 deer. The second piece of information we're given is that year 8, and that's 1488 deer. All right, letter B's has calculate delta P divided by delta T in other words, the average rate of change. Also known as the slope. All right, and it says include units in the answer. 

Why don't you go ahead and do that? You've got the two points in a, calculate that average rate of change. All right, let's do it. So delta P divided by delta T will take our second population 1488. Minus our first population. 1028. Divide it by E minus three. I'm going to do a little cheat sheet here. 460 watch me track the units right now. 460 deer, right? Divided by 5 years gives me 92 deer per year. There's my average rate of change. Letter C asks us to interpret that. So how do you interpret that value? All right, let's go through it. We interpret it by saying, each year. The deer population. Increases on average. By 92. That's it. Right, for every one year, the deer go up by 92. Finally, it asks us to determine a linear relationship between the deer, population, and the years since 2000. Now the nice thing is, here we're doing P equals M T plus B, right? Y equals MX plus B except instead of why we have P and instead of X we have T we already know the M so we can actually start right here. P equals 92 times T plus B the only thing that we don't know is this. But in the last few lessons we've seen this repeatedly. In order to calculate the Y intercept once we know the slope, we simply need to pick one of the two points that we have substituted in. I almost always take the first one because, hey, why not? But it shouldn't matter. All right, we're just going to substitute that in. We'll get 1028 is equal to where am I at here 200 and 76 plus B subtract two 76 from both sides. Running out of space. Way down at the bottom. B is equal to 752.