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#14 Hot-lanta"In 1990, the population of Atlanta was about 3,000,000 and at that time was increasing at a rate of about 70,000 per year Write an equation in sl...

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#14 Hot-lanta"In 1990, the population of Atlanta was about 3,000,000 and at that time was increasing at a rate of about 70,000 per year Write an equation in slope-intercept form to model this situation where X represents the years since 1980 and y represents the population in millions b) What is the y-intercept and what does it represent? What is the slope and what does it represent? d) Use the equation to predict the population in 2035. Use the equation to predict when the population hits

#14 Hot-lanta" In 1990, the population of Atlanta was about 3,000,000 and at that time was increasing at a rate of about 70,000 per year Write an equation in slope-intercept form to model this situation where X represents the years since 1980 and y represents the population in millions b) What is the y-intercept and what does it represent? What is the slope and what does it represent? d) Use the equation to predict the population in 2035. Use the equation to predict when the population hits 3,500,000.



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since $1997,$ the population of North Dakota has been decreasing by about 3290 people per year. The population was about $650,000$ in 1997 .
a) Write a linear equation to model this data. Let $x$ represent the number of years after $1997,$ and let $y$ represent the population of North Dakota. b) Explain the meaning of the slope in the context of the problem. c) According to the equation, how many people lived in North Dakota in $1999 ?$ in $2002 ?$ d) If the current trend holds, in what year would the population be $600,650 ?$

We have a situation describing the prison population over various years. We need to model it with an equation. We're told that in 1990 the prison population was 740,000. And in 2000 this population was one million, 320,000. Now, these can be rewritten as two points. They don't really look like points. Right now, However, the problem recommends that we set t equals zero. That is, time equals zero at the year 1990 and T is just replacing our X variable. So this 0.1990 will be represented by a zero in the X spot. And now, to avoid having to write all the zeros every single time, I'm going to be writing this in thousands. So we'll just write zero comma, 740 showing that there were 740,000 prisoners when this experiment started. Is in the year were calling zero similarly weaken safer. This other point, Tom, in 2000 that was 10 years after 1990. So the X value is 10 and our why value. Since we're putting this into thousands, we remove three zeros, cutting it down to 1320 So we now have two points. It's time to calculate the slope of this line that we're going to be a because we need the slope in or to make an equation. Slope is given by the formula. M equals y Tu minus y one over x two minus x one Where x one y one and x two y two are just some points. I'm going to be using the 1990 data for X one y one and the 2000 data for X two y two. Let's plug in. We have Why two or 13 20 minus y 1 740 Divided by X two, which is 10. My specs one, which is zero. This comes down to 580 divided by 10 or cancelling a zero from both 58/1. Thus, that's the slope of our line. All that remains now is to create an equation of this. We're going to be using Point Slope form, which has the generic format. Why minus y one is equal toe m times x minus X one where m is the slope and x one y one is a point. We're going to be using the 0.0 740 for X one y one. So we have y minus 740 equals 58 times X minus zero or Y minus 740 equals 58 x, which could be a rearranged to give us why equals 58 x plus 740 Before we do anything else, Let's take a look at the slope here. This is a problem with riel world applications. So the slope isn't just some number on a page. This has really meaning here. So 58 that's our slope can also be written as 58/1. You may remember from the reading that slope is equal to change. And why over change in X, where this triangle delta symbol means change. So the Y variable in this case is number of prisoners. I'm going to write just prisoners for that, and the X variable is years. So this is modeling prisoners over years. So if we increase years by one, we'll increase prisoners by 58 or more accurately, 58,000. Because, remember, we are dealing in thousands. Oh, our. If we go backwards one year, we'll decrease the population by 58,000. In essence, the slope is just telling us the number of prisoners per year. That is what the slope tells us. It's just how many prisoners are going to be added or subtracted per year, and this comes down to being 58,000 prisoners per year. Now that we know what the slope means, let's try one application of this. We're going to plug in a number, um, into T and determine what how many prisoners are going to be in the prison population at that time. So let's say we're dealing with the year 2007. That is 17 years after 1990. So, uh, that would be t equals 17 or X equal 17. Let's play the into occasion. Why equals 58 times 17 plus 740 Now I have already done this in my calculator 50 year time. 17 plus 740 gives us 1726 prisoners. So that's the number of prisoners projected to be in prisons in 2007. Of course, this isn't thousands, so it's really 1,726,000 prisoners who will be in in prisons in the year 2007. So we have created an equation modeling this situation. We have interpreted the slope as prisoners per year and we have determined that there will be 1,726,000 prisoners in prisons in 2007.

All right, let's first take a moment to digest the information given to us in this problem. First of all, we can identify our variables. X is a number of years after 1998. In other words, 1998 is represented by the X value zero. They give us that there's 1,284,000 people in year 2001. So that means for an X value of three are why value is 1,284,000. The Y value. Thus is the population. They give us one more other in Pete important piece of information, which is that we're increasing population by 95,700 each year. So this is the quantity m or the slope. In this case, in order to write the linear equation, we want to find B, which is our y intercept or value when X equals zero notice from the quantities that were given for the year 2001. We need to go back three years. So in order to find B, we can subtract three years worth of growth. So 86 3 groups of 8700 from the quantity given at your three. One million, 284,000. So 1,284,000 minus three times 8700 gives us RB value 1,257,900 So this is the number of people that there were in 1998. Now for part A, we can write our linear model 8700 are slope X plus are starting value of people one million, 257,000 equals Why so now we have written in equation for this situation. Part B is asking us to explain the meaning of the slope. Well, we discuss that are Slope Slope is equal Teoh 8700 which is the increase in population each year. The next part of the question wants us to figure out the number of people living in Maine in 1998 in 2000. Tooth Well, fortunately for us, we already found 1998. That is our be value. So 1998. There's 1,257,000 people living in Maine in 2002. We need to go one year beyond 2000 won, which is what they gave us. So looking at our table here, we want to go one year beyond 2001 to get 2002. So that means we can add 8700 to our quantity from 2001. When we do this will need to dio 1,000,284 1000 plus 8700. This will give us the quantity of one million 292 1000 700. So there's our answers for part B. Remember to always include your labels people finally in part d we want to solve for the number of years after there would be a quantity of people of 1000 or one million Excuse me one million 431,900. So we have this number of people that is our why value and we want to know solve for X. So if we replace why with this quantity in our equation, let me just shrink my writing down a little bit here so I can fit it all in one line for you to see, we can now simply solve for X subtracting the initial quantity. 1,257,000 from both sides. We find 8700 x is equal to 100 174,000 900. Our final step here who? I have one too many zeros. There we go. Our final step here now is to divide by 8700 giving us X equal to 20. So remember, the X is the number of years after 1998. So in order to solve for the year, we need to do 1998 plus 20 which gives us the year 2018 as our final solution. Let's go back and review all of the work that we have done. First, we solved for the B value 1001 million, 257,000 allowing us to write our equation for the given situation. The B value represents the initial number of people in the year 1998. The slope then represents the increase in population each year, increasing 8700 part C. We solved for the number of people in these two given years and part d, we solved for the year 2018 which has 1,431,900 people

So this problem tells us that since 1995 the population of Russia has decreased by about 505,000 people per year. That it also tells us that in the year 2000 the population was about 146 million. So we're going to use the variables p for population and T for time. But very specifically, it's in years since 1995. That's gonna be important to write a linear equation in slope intercept form to predict the population of Russia two years after 1995. So first it mentioned Slope intercept. So I just want to write the slope intercept equation Y equals MX plus B. But we're not using the variables X and Y. We're using the variables p for population and T for time. So the first thing I want to do is figure out. Should people for why or for X or vice versa and t for wire X? And so to do that, I use this little trick, and I know that because why is the dependent variable and X is the independent variable that why always depends on X and in this case, problem, the population, which is P is going to depend on the time. And so what that would mean is, if I just match these things up, that instead of using why I'm gonna use the variable p And instead of using X, I'm gonna use the variable T. And so then I could rewrite my soap intercept equation to say P equals M t plus B. So all I've done is I've swapped out why for P and expertise. They also tell me essentially, my slope, they tell me that it's decreasing, so it's important. That means myself is negative by about 505,000 people for years. So that's gonna be my slope. So I can go ahead and put that in right away. So negative. 505000 Now what they don't tell me is what the population was in 1995. Now think about this. If t represents the number of years since 1995 imagine you were standing in 1995. How many years has it been since 1995? It's been zero. So in the year 1995 it's been zero years since 1995. Remember that any time we're finding a why value or in this case, population, when X or this Case t zero. What we're really finding is the y intercept, and so that's what I don't know. But what I do know is that in the year 2000 the population was about 146 million, and so what I could do with that information is I can create an ordered pair, remember nor ordered pairs normally go X comma y. But in this case, they would go t com api. And so what I need to figure out from here is what are my T and P values now? P is easy. That's population. So in the place of P, I could put delectable too much face there. I could put 146 1,000,000 now for my TVL. You might think that you want to put the Year 2000 here, but remember, T represents the years since 1995. So if I put 2000 that would mean it's been 2000 years since 1995. If you're in the year 2000 how many years has it been since 1995? It's been five. So that's gonna be my T value. So now what I can do back in my slope intercept equation over here is I can replace tea with five and p with 146 million. I'm gonna go ahead and do that. So 146 1,000,000. I've already got my slope, which is negative. 505,000 multiplied by five plus my y intercept, which I don't know. So I'm gonna go ahead and multiply Negative five. Heard that 5000 times five. And that's gonna get me negative. Two million 525,000 plus B. So I just gave myself a little more space here. So now to get beat by itself, I'm going toe Add this 2.5 million onto the other side. Now think about this. The reason this is gonna make sense is because, remember, 146 million is the population in the year 2000 but it's been going down for five years. So when I add this, this number is gonna get bigger, meaning that my population in the year 1995 is going to be higher than it was in the year 2000 which is going to make total sense. So let's go ahead and add that on. So, plus 2,525,000 and whatever I do to one side I have through the other plus 2,525,000. And so now maybe is isolated. And I would get for my why intercept 148 million 525,000. And so my soap intercept equation Sorry, I was using the wrong variables again. Would be p equals negative 505,000 T plus 148 million 500 25,000. And so what this shows me is that if t war zero, this would go away and remember t were zero then we're in the year 1995. This is my population in the year 1995 and every year that passes, it's decreasing by that much. So now my next step is to use this equation to predict the population of Russia in the year 2015. So what they're giving me is a time. This is a time. But remember, it has to be in years. Since 1995. So if you were in the year 2015 that's been 20 years since the year 1995. So that's gonna be the T value that I'm gonna use. I'm gonna find Pete, so I'm gonna have p equals negative 505,000 times 20 plus 148,000 148,525,000 when I multiply. Negatives 505,000 times 20. Time to get negative. 10 million. 100,000. Add that to my 148,525,000. And that means that the population of Russia, 20 years after 1995 which is the year 2015 would be 138 million 425,000. Now, just to double check myself, I want to make sure that remember, in the year 2000 it was 146 million and then in the year 20 1538 million. So it's still going down, which is really good news

All right. This problem Were given the time population 2003, which is 14, And in 2007 2000 one of them is a far we are told that the population is changing when you need and then I asked how much it's changed Christianity and telling, well we have the population here into the family population to Coventry. Uh huh. The difference between the two, Well, that would have been pouring money fortunately money cheaply 93. That's funny. The second part. Yes. How much how long did it take to girl? Unfortunately anyone before 2100 people people and that is just the difference in the years party S for the average population growth per year. So for that we're in to grant part a Population grew by 7°. They were divided by years from part B. I think four years and you get really some class points 75. Next question I have What is the population in the year 2000? All right, well, I've been after we just take You know, the population is changing 105 points back to the year. I would know the difference between 3 2000 which is typically used really multiplying them together to get Yeah, the whole thing just right. I'm gonna take a politician into 11 3 just fortunately long this effect and from that position 93 35 yet you got I would make it Roundup can have two points official, pardon me asked us to find equation Starting in the year 2000 world. You could say operation because you a question here. That much money or no correct no other Children Liberator 35 when I went back to the creator and just when you multiply that by number of years we have to mhm Yeah, a number of years. Clearly. What is the problem? Mm The last part. Uh huh. Using our information. Find position 7 13. Well heartbreaking is my up close look. Seriously. 537 sign. Uh huh. To 14. 14 years after another. Uh huh. 3300. What? Good five. And that's Obama.


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