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.
YearPopulation196094.092196598.8831970104.3451975111.5731980116.8071985120.7541990123.5371995125.3272000126.7762005127.7152010127.579
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.]

You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

cheekabooy
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
x1y10940929400059888394000101043459400015111573940002011680794000251207549400030123537940003512532794000401267769400045127715940005012757994000
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
k=0.165732
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(e0.07655t)+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.
image

Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Relevant Questions

asked 2020-11-03

The exponential models describe the population of the indicated country, A, in millions, t years after 2010.Which country has the greatest growth rate? By what percentage is the population of that country increasing each year?
India, A=1173.1e0.008t
Iraq, A=31.5e0.019t
Japan, A=127.3e0.006t
Russia, A=141.9e0.005t

asked 2021-02-25
The population of California was 29.76 million in 1990 and 33.87 million in 2000. Assume that the population grows exponentially.
(a) Find a function that models the population t years after 1990.
(b) Find the time required for the population to double.
(c) Use the function from part (a) to predict the population of California in the year 2010. Look up California’s actual population in 2010, and compare.
asked 2021-09-17
Determine whether the function given by the table is linear, exponential, or neither. If the function is linear, find a linear function that models the data. If it is exponential, find an exponential function that models the data.
x f(x)
-1 8/7
0 8
1 56
2 392
asked 2021-03-11
The exponential function f(x)=42.2(1.56)x models the average amount spent, f(x), in dollars, at a shopping mall after x hours. What is the average amount spent, to the nearest dollar, after four hours?
asked 2021-08-16
A report tells us that in 2009, there were 870 gray wolves in Idaho, but that the population declined by 19% annual rate of decrease continues.
a) Find an exponential model that gives the wolf population W as a function of the time t in years since 2009.
W=870(1.19)t
b) It is expected that the wolf population cannot recover if there are fewer than 25 individuals. How long must this rate of decline continue for the wolf population to reach 25?
asked 2021-06-11
Which set of ordered pairs could be generated by an exponential function?
A. (1,1)(2,12)(3,13)(4,14)
B. (1,1)(2,14)(3,19)(4,116)
C. (1,12)(2,14)(3,18)(4,116)
D. (1,12)(2,14)(3,16)(4,18)
asked 2021-05-14
Find a recursive rule that models the exponential decay of y=32,000(0.95)t