Project I, MATH 106

Click to download the TeX file for this project.

Guidelines: You should consider your project as a technical paper (like an English paper only different). Part of your grade will be based on the quality of your written presentation. The paper you turn in should have a mix of equations, formulas and prose. Graphs may be copied directly from your calculator, but should be clearly labelled. Use complete sentences, good grammar and correct punctuation. Spelling is also important. You should aim to write your report in such a way that it can be read and understood by anyone who knows the material for this course.

Question 1:

In this question you are to choose a reasonable mathematical model for the population growth of the US, and test it against the data.

Use the data from Table I to find a formula which approximately models the growth of the population over the years 1790-1850. You'll have to consider what {\it sort} of function might describe this growth. Justify your choice of model and comment on how well it fits the data. [You might want to discuss such points as: how accurate the model is over the range 1790-1850; how accurate the model is over the range 1790-1990; the factors which contribute to the discrepancy between the model and the data; and the prediction your model gives for the current US population.]

Year Population Year Population
1790 3.9 1800 5.3
1810 7.2 1820 9.6
1830 12.9 1840 17.1
1850 23.1

TABLE I: US population (in millions), 1790--1850


TABLE II: US population (in millions), 1860--1990

Question 2:

Now try another approach to the problem of modelling this data: Using all the population data in Tables I and II above, plot a graph of the logarithm of the population. Find a straight line which comes as close as possible to passing through these points and use this line to find a new model for the US population. Plot your model against the data and comment on its strengths and weaknesses.

Question 3:

Now let F(t) be the population (in millions) of some country (not necessarily the US) at the time t, measured in years since 1900.

Question 4:

In this question you should use the data in Table III to compare the number of cars in the country with the number of people.

Year 1900 1910 1920 1930 1940
Vehicles 8 458.3 8131.5 23035 27466

Year 1950 1960 1970 1980 1990
Vehicles 40339 61682 89279 121601 143550

TABLE III Automobile registration in the US (in thousands) 1900--1990

Use the data from Table II to estimate, to within 1 year, the first year when there was at least one car for every two Americans. You should include a plot of the data in Table III. Try to find the answer both graphically and analytically. Discuss the strengths and weaknesses of the model for automobile registration you used.

Calculator Tips:

The easiest way to work with a table of data on your TI is to enter it as a list. A list is just what it sounds like: a collection of data like {8, 458.3, 8131.5, 23035\. Notice that it has "curly braces" around it. To enter a list, press LIST to bring up the list menu and use { and }. Store your data lists in some sensibly named variables. (Using STO.) Your data in this project really consists of several pairs of lists; so make sure that you have lists of years saved as well as lists of population and vehicle registrations.

The TI can plot tabular data if you tell it a list of x-coordinates and a list of y-coordinates. Bring up the STAT menu. Press CALC and you'll be asked to give the names of your two lists. Probably you want the years to be your x-coordinates and the other data to be your y-coordinates. Once you enter the names you can EXIT back to the STAT menu. The menu item DRAW allows you to plot your data points.

Warnings: The TI plots not only your data but also any currently selected graphing functions. The viewing window is set using GRAPH: RANGE just as it would be for a graph.) You have two options for plotting your data: SCAT just plots the points, xyLINE plots the points and joins consecutive ones with straight lines.