The table gives estimates of the world population, in...

The table gives estimates of the world population, in millions, from 1750 to 2000. (a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures. (b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population. (c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.

The table gives estimates of the world population, in millions, from 1750 to 2000.

(a) Use the exponential model and the population figures for 1750 and 1800 to predict the world population in 1900 and 1950. Compare with the actual figures.
(b) Use the exponential model and the population figures for 1850 and 1900 to predict the world population in 1950. Compare with the actual population.
(c) Use the exponential model and the population figures for 1900 and 1950 to predict the world population in 2000. Compare with the actual population and try to explain the discrepancy.

Key Concepts

Exponential Growth Model

The exponential growth model is a mathematical representation where the rate of change of a quantity is proportional to its current value. This model is commonly used to describe populations, investments, and other processes that grow or decay at a consistent percentage rate over time.

Parameter Estimation

Parameter estimation involves using known data points to determine the constant growth rate in the model. By using values from different times, one can solve for the growth rate parameter, often by employing logarithmic transformations.

Forecasting Using the Model

Once the growth rate is estimated, the exponential model can be applied to predict future values. Forecasting involves substituting future time values into the model to project the population or quantity forward, assuming the continued applicability of the growth rate.

Model Calibration and Data Fitting

Calibrating the model requires selecting appropriate historical data points to ensure the growth rate accurately reflects the process being modeled. This alignment is critical to making reliable future predictions.

Comparison with Actual Data and Model Limitations

Comparing the model’s predictions with actual data helps to assess its accuracy and highlight discrepancies. Such comparisons reveal the limitations of assuming a constant exponential growth rate, as real-world processes may experience changes due to external factors, resource constraints, or other dynamic influences.

Transcript

00:03 All right, we are given a table with various population values over time, and we have a lot of things to do, so i hope you packed a snack because this is going to take us a while.

00:15 So in part a, we want to come up with an exponential model based on these two points from the table, and then we're going to use that model to come up with predictions for 1900 and 1950.

00:27 So to find the model, we first have to find the value of k.

00:30 We're going to use our p equals p.

00:33 P0e to the kt population growth equation.

00:36 And we're going to use these two points.

00:38 So we'll use for our final population, 980.

00:45 And we'll use for our initial population, 790.

00:49 And for time, what has elapsed between the year 1750 and the year 1800 is 50 years.

00:55 So we'll use a 50 for t.

00:57 And we're going to solve this for k.

00:59 So first we divide both sides by 790, and it's going to reduce to 98 -79.

01:05 And then we're going to take the natural log of both sides, and then we're going to divide by 50.

01:14 So we get k is the natural log of 98, 79th, over 50.

01:20 So we can substitute that into our model along with the initial population of 790, and for our model we have p of t equals 790 e to the power, natural log 98 .79th over 50.

01:40 Times t.

01:42 Okay, so we're going to use that to make our prediction for the year 1900.

01:50 So if it's 1900, then it has been 150 years since the year 1750.

01:57 So this model assumes that t equals zero is the year 1750.

02:04 Okay, so for our prediction for the year 1900, we're going to use t equals 150, substitute that in to our population model, and we get approximately 1 ,508 million people.

02:22 The actual value reading off the table was 1650, 1650 million people.

02:35 Okay, we're going to do the same thing for the year 1950.

02:38 So if it's 1950, then it has been 200 years since the year 1750.

02:43 So we put 200 into our population model, and we get approximately 1871, 1871 million people.

02:55 The actual value from the table was 2 ,560.

03:03 So notice that both of the values we got from this model were lower than the actual values.

03:08 All right, that entire process we're going to repeat for two different ordered pairs and make a different prediction.

03:15 So for part b, this time we're using 1850 and its population of 1260 million, and we're using 1900 and its population of 1650 million.

03:31 And we're going to go through the same process again.

03:33 So we need to find the k value for this one using p of t equals p -not, e to the k -t.

03:40 And then we'll have our model, and then we can make our predictions.

03:43 So we're substituting 1650 in for p of t, and we're substituting 1260 in for p -not, and this time, again, 50 years has elapsed from 1850 to 1900...

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