WEBVTT
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all right, We are given a table with various
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population values over time, and we have a lot
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of things to do. So I hope you packed
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a snack because this is going to take us a
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while. So in part A, we want to
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come up with an exponential model based on these two
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points from the table, and then we're going to
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use that model to come up with predictions for 1919
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50. So to find the model, we first
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have to find the value of K we're going to
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use R P equals p 92 the Katie population growth
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equation. And we're going to use these two points
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. So we'll use for a final population 18 a
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980 and we'll use for our initial population 790.
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And for time, what has elapsed between the year
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17 50 the year 1800 is 50 years. So
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we use a 54 t, and we're going to
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solve this for K. So first we divide both
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sides by 7 90 it's going to reduce to 98
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79th and then we're going to take the natural log
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of both sides and then we're going to divide by
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50. So we get K is a natural log
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of 98 79th over 50 so we can substitute that
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into our model along with the initial population of 7
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90 And for our model, we have p of
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tea equals 7 90 e to the power natural log
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98 79th over 50 times T. Okay, so
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we're going to use that to make our prediction for
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the year 1900. So if it's 1900 then it
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has been 150 years since the year 17. 50
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. So this model assumes that t equals zero.
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Is the year 17 50? Okay, so for
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our prediction for the year 1900 we're going to use
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T equals 1 50 substitute that end to our population
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model, and we get approximately 1508. Now,
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that's 1508 million people. The actual value reading off
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the table was 16 50 16 50 million people.
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Okay, we're going to do the same thing for
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the year 1950. So if it's 1950 that it
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has been 200 years since the year 17 50 so
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we put 200 into our population model and we get
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approximately 18 71 18 71 million people, the actual
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model of the actual value from the table WAAS 2560
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. So notice that both of the values we got
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from this model were lower than the actual values.
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All right, that entire process, we're going to
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repeat for two different ordered pairs and make a different
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prediction. So for part B, this time we're
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using 18. 50 and its population of 12.
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60 million and we're using 1900 and its population of
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16. 50 1,000,000. We're going to go through
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the same process again. So we need to find
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the k value for this one using p of t
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equals P, not B to the K T.
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And then we'll have our model and then we can
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make our predictions. So we're substituting 16 50 for
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p of tea and we're substituting 12 64 p not
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. And this time again, 50 years has elapsed
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from 18 50 to 1900. So we have each
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of the 50 k. We're solving this for K
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. So we're going to start by dividing both sides
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by 12 60 that fraction will reduce to 55/42.
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55/42 equals each of the 50 K. Then we
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take the natural log of both sides, and then
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we divide both sides by 50. So we have
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our K natural log of 55/42 divided by 50.
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Now, we substitute that into our model, and
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we have p of t equals 12 60. The
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population in the year 18 50 times e raised to
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the power natural log of 55/42 over 50 times.
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T. That's the model we're going to use for
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part B. Where t equals zero. Is the
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year 18 50. Okay, so the prediction we're
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making this time two of them, I believe,
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um or is it just one? Just one.
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We're making a prediction for the year 1950. So
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the year 1950 would be 100 years since the year
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18 50. So we're going to use T equals
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100. We substitute 100 into our model, and
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we get approximately 2161 million now, the actual value
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in that year based on our table was 25.
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60 million. So another case of this model that
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we came up with giving us a value that's lower
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than the actual value. Now we have one more
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time to go. We're going to do Part C
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same idea. But this time the points were using
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are from 1900 with a population of 16 50 and
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from 1950 with a population of 25 60. Okay
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, let's use those values into our population growth equation
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and find the value of K. So the final
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value is 25 60 the initial value of 16 50
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and the time that has elapsed is 50 years.
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Let's divide both sides by 16 50 on the fraction
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will reduce 2 to 56/1 65. Take the natural
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log on both sides and divide by 50. And
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that gives us our K value natural log of 2
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56/1 65 all divided by K by 50. So
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we substitute that into our model P of t equals
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initial value 16 50 times. He raised to the
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power natural log of 2 56 over 1 65 over
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50 times teeth and in this case, t equals
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zero is the year 1900 and the prediction were interested
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in making for part C is for the year 2000
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. So the year 2000 is going to be 100
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years since the year 1900. So we're going to
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use T equals 100 substituted into our model. Put
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it in the calculator on would get approximately 3972 million
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for the population, while the actual population was significantly
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higher. 6080 million and that's over six billion people
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. So why the discrepancy? Well, if you
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think about what happened between 1919 50 and 2000 significant
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advances were made in medical technology in the last 50
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years or so, and so life span would be
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incredibly altered. We couldn't use a model from the
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early 19 hundreds because of medical advances.