# Project I, MATH 106

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

 1860 31.4 1930 122.8 1870 38.6 1940 131.7 1880 50.2 1950 150.7 1890 62.9 1960 179 1900 76 1970 205 1910 92 1980 226.5 1920 105.7 1990 248.7

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.
• What does it tell you about this country if the function F(t) is not invertible? How about if the graph of F(t) is not concave up?
• Suppose that every month B people are born in this country and D people die. Given that F(0)=A write down a possible formula for F(t). What other factors might affect the vaue of F(t)? Try to modify your model to account for them.

### 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.

Notes:

• When you change the lists you're using the TI doesn't erase the points you've already plotted. So to get several plots on the same screen just go back to STAT: CALC and use different data.
• If you do want to erase what you've plotted there's a menu item; CLRDRW under STAT: DRAW.
• Suppose you have two sets of data plotted and you change the viewing rectangle. If you then go back to STAT: DRAW it will only re-plot the most recent set of data---to plot the other one again start from scratch with the new viewing rectangle.
• The result of {1,3,4} * 10 is {10,30,40\}, and similar things work when you apply any function to a list. This can come in handy for some of the repetitive calculations.
• Once one of you has typed in the data they can transfer it using LINK just as with programs. (The sender should press LINK: SEND: MORE :LIST to choose which lists to send.)