The table gives the midyear population of Japan, in thousands, from 1960 to 2010.

Albarellak 2021-01-31 Answered

The table gives the midyear population of Japan, in thousands, from 1960 to 2010.
Use a calculator to fit both an exponential function and a logistic function to these data. Graph the data points and both functions, and comment on the accuracy of the models. [Hint: Subtract 94,000 from each of the population figures. Then, after obtaining a model from your calculator, add 94,000 to get your final model. It might be helpful to choose t=0 to correspond to 1960 or 1980.]

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Expert Answer

Answered 2021-02-01 Author has 83 answers

Step 1
Following the suggestions given by the problem, let
t= the number of years after 1960 (so t=0 is 1960)
enter the population numbers with 94000 subtracted from each.
Using Desmos, first add a table. (clic on "+" at the upper left) and enter the numbers. It should look something like in this image:
Step 2
y1  abx1
R2=0.7692 e1
a=10665.3 b=1.02682
Step 3
Next, you can get an exponential form y=abx by typing in the next box below the one with the table:
y1   a bx1
The values for a and b appear below. It should look similar to what is the lower right in the image above. Desmos gives a different answer than the book's. (Compare it in the graph at the bottom). The model with the proper variable names and shifted back up by 94000 would be:
P(t)=10665.3(1.02682)t + 94000
Step 4
Next in another box below, type this to get a logistic model:
y1  /M  1 + A ekx1
y1  abx1
PR2=0.7692 e1
a=10665.3 b=1.02682
y1  M1 + Aekx1
R2=0.9927 e2
M=33086.4 A=12.3428
Step 5
This time it does given an answer similar to the book's. Using the right variables and shifting back up by 94000, we have:
P(t)=33086.41 + 12.3428e0.165732t + 94000
Step 6
Using GeoGebra 5 and entering the same numbers in as a list of points, the command FitExp does give the same answer as the book. (shifted it back up by 94000 here)
P(t)=1909.7761e0.07655t + 94000
=1909.7761(1.0796)t + 94000
Step 7
The logistic model is better than the exponential. The Desmos one seems to stay closer to the points overall than the other exponential one.

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