| 4.6.4 Project: Population Growth Models | Project | ||
| Precalculus Honors Sem 1 | Name: | ||
| Date: | |||
Population Growth Models
You’ll need a handheld or online graphing calculator for this project. Specifically, you’ll need a way to plot the points in a data set and graph a function on top of those data. Instructions are available at the end of this document.
Background: Exponential Growth
In exponential growth, the amount you have grows by a consistent percentage.
A function for exponential growth that uses the base e is , where
- t is time;
is the value at time t;
is the amount at t = 0, the starting amount; and
- r is the growth rate, expressed as a decimal.
Modeling Population Growth in Madagascar
An exponential growth model is useful when the growth rate stays relatively constant.
The table shows the population of Madagascar between 1960 and 2020. To simplify the numbers, let 1960 be year 0.
| Year | Year (for modeling) | Population (thousands) |
| 1960 | 0 | 5099.4 |
| 1970 | 10 | 6576.3 |
| 1975 | 15 | 7556.0 |
| 1980 | 20 | 8716.6 |
| 1990 | 30 | 11,598.6 |
| 2000 | 40 | 15,776.8 |
| 2005 | 45 | 18,336.7 |
| 2010 | 50 | 21,151.6 |
| 2020 | 60 | 27,691.0 |
1. Use a handheld or online graphing calculator to plot the points in the table. Show the result below. (3 points)
2. Add an exponential curve to the graph. The initial population () is 5099.4, so use
. Try different values for r, the growth rate, until the curve fits the data well. (Hint: r is less than 0.050.)
What is your exponential growth function for the population of Madagascar? (8 points)
3. Compare the population of Madagascar in 2020 with the population your model would predict. Does your model overestimate or underestimate the actual 2020 population? (5 points: 4 for the correct calculation of the estimate; 1 for classifying it as an overestimate or underestimate)
Background: Logistic Growth
A logistic growth curve looks exponential at first, but then the growth rate slows and the curve eventually flattens out. This is often a good population model when resources or space is limited.
The logistic growth function is , where b > 0, and where
- t is time;
- is the value at time t;
- L is the limiting amount, called the carrying capacity;
- b affects the shape of the curve; and
- C affects the position of the curve.
Modeling Population Growth in Cuba
The table shows the population of Cuba between 1887 and 2020. To simplify the numbers, let 1887 be year 0.
| Year | Year (for modeling) | Population (thousands) |
| 1887 | 0 | 1609.1 |
| 1899 | 12 | 1572.8 |
| 1919 | 32 | 2889.0 |
| 1931 | 44 | 3962.3 |
| 1943 | 56 | 4778.6 |
| 1953 | 66 | 5829.0 |
| 1965 | 78 | 7958.2 |
| 1975 | 88 | 9446.4 |
| 1985 | 98 | 10,097.9 |
| 1995 | 108 | 10,888.3 |
| 2005 | 118 | 11,261.6 |
| 2015 | 128 | 11,324.8 |
| 2020 | 133 | 11,326.6 |
4. Use a handheld or online graphing calculator to plot Cuba’s population data. Your graph should include x-values from 0 to 140 and y-values from 0 to 12,000. Describe the pattern you see. (3 points)
5. Explain why a logistic model would be a better choice for Cuba’s population growth than an exponential model. (4 points)
6. Find a logistic function, , that models Cuba’s population growth.
- Assume that 0.039 is a good value for b. Do not change this value.
- Graph
on top of your points. (So the starting value for L = 11,000 and the starting value for C = 15.)
- Experiment with values for L and C until you have a good model for the graphed points. Limit yourself to L-values between 11,000 and 15,000 and C-values between 11 and 15.
What is your logistic model for Cuba’s population? (8 points)
7. Find Cuba on a map. Why does a logistic model make sense for Cuba’s population growth? (3 points)
On Your Own
An exponential function is a good model for population growth when the growth rate stays relatively constant. This type of function can also be used to model the number of people with a contagious disease in the early stages of an epidemic.
However, there are usually limits on growth. Populations are eventually limited by finite space and resources. Diseases cannot infect more than 100% of the population, and they may also be limited by social policies, medical treatments, and growing immunity in the population.
8. It’s your turn to find some real-world data to model with an exponential function or a logistic function.
- Identify the data you want to analyze. Your data should include at least 8 ordered pairs.
- Plot the points on a handheld or online graphing calculator.
- Decide which model to use: exponential or logistic. Justify your choice in writing.
- Using that model, find a function that models your data well.
The work you turn in should
- tell where you found your data,
- list the data in a table,
- justify your choice of model, and
- give the function that is a good fit for your data.
Data search tips:
If you’re interested in logistic functions, look for
- population data (human or animal) for a small island or other area with limited space; or
- data on the number of people in a country/state/city who caught an infectious disease. Choose a disease for which the number of infected people is no longer growing quickly.
If you’re interested in exponential functions, look for
- data for a population’s first years of growth in a large area, such as right after a species was introduced or a city was founded; or
- data on the number of people who caught an infectious disease, starting when the disease was first identified and going through the time it was spreading most rapidly.
(16 points: 4 points for finding suitable data, 4 points for choosing the appropriate model, and up to 8 points for the function modeling the data, depending on how well it fits the data)
Online Graphing Calculator Instructions:
An online graphing calculator may be a better option for this project than a handheld graphing calculator, because you can see in real time how changes to your function affect the graph, instead of having to go back and forth between viewing the function and viewing the graph.
To plot data points and graph a function at the same time on an online grapher:
- Enter the data as a list of points or in a table. You may need to use set notation, or you may be able to simply enter the list of points. Consult the calculator’s Help resource if you need details.
- To enter the function, type it into the second field. You can usually use a caret (^) for exponents and a slash (/) for division. Some sites have a keypad for entering complicated math.
- To change the values of the number(s) you want to experiment with, such as r for an exponential model and C and L for a logistic model, just type in the new value(s). The graph may update as you type, or you may have to press Enter before seeing the new graph. Experiment with different values until your function is a good model for the data.
- On some sites, you can enter letters for the values you want to change and then use sliders to change those values manually.
Handheld Graphing Calculator Instructions:
To plot data points and graph a function at the same time on a TI-83 or TI-84:
- Press [STAT]. The cursor should be on EDIT. Press [ENTER].
- To clear any previous data, arrow to the top of L1 and press [CLEAR] [ENTER]. Then arrow to the top of L2 and press [CLEAR] [ENTER].
- Enter all the x-values into List 1 (L1). Press [ENTER] after each.
- Press the right-arrow button to go to the top of the L2 column.
- Enter all the y-values into List 2 (L2). Press [ENTER] after each.
- Press [2nd] [STAT PLOT]. The cursor should be on 1:. Press [ENTER].
- Select On to turn the plot on. Press [GRAPH] to show the data you entered. (To see your points, you may need to use the [WINDOW] key to adjust the scales on the axes.)
- To enter the function, press the [Y=] key. Clear any existing functions.
- Enter your function as Y1. Use [2nd] [LN] to enter e^.
- Press [GRAPH]. The graph of your function appears along with the plotted points.
- To change your function, press [Y=] and make the changes. Press [GRAPH] again to see if the new graph is a better fit. Repeat until your function is a good model for the data.
Copyright © 2020 Apex Learning Inc. Use of this material is subject to Apex Learning’s Terms of Use. Any unauthorized copying, reuse, or redistribution is prohibited. Apex Learning ® and the Apex Learning Logo are registered trademarks of Apex Learning Inc.
4.6.4 Project: Population Growth Models
