Semester Projects

Vegetable meets Mathematics Photo courtesy of flickr user anroir. Some rights reserved.

Project Goals

The goal of these projects is to give you the opportunity to make your own connections between mathematics and modern society by considering a wide variety of problems ranging from economic and environmental issues to social and political situations that can be modeled and solved by mathematical means. They will help you establish connections between Math 10250 and your other courses, and they will allow you to make contributions in areas of your interest and expertise. In addition, they will provide you with an opportunity to interact and collaborate with your classmates.

Choosing a Topic

Topics can be chosen from the following:

1. Chapter Projects from Chapters 2 - 5.
2. The projects described below under "Other Project Topics."
3. Other courses you are able to establish a connection with math 10250.
4. Anything that you find interesting and is approved by your teacher.

Rules for the Project

Please follow the guidelines below when producing your project.

1. You can work in groups of size 1-4 students (from any section of Math 10250).
2. Each group submits one (typed) paper (and an electronic copy).
3. Each member of the group receives the same score - a number between 0 and 10 - which will count toward your 20 participation points.
4. The first draft is due by November 14 and final version due by December 10. Your project must state your project title, the names of your team members and which class sections each team member is from.

Alternative Project Topics

1. The Social Security. Some experts project that the Social Security shortfall over the next 75 years will be about four trillion dollars. Is that true? How do they know? Make your contribution in the national debate about saving Social Security using ideas and techniques you learned in Math 10250.
2. The Deficit. Visit the Webpage of the Congressional Budget Office (CBO) at http://www.cbo.gov/ and try to make sense of the numbers you find in "Current Budget Projections". Note that income streams are useful in making projections.
3. GDP. The Gross Domestic Product (GDP) of the U.S. economy during 2005 grew at the annual rate of about 3.5% while China's GDP grew at the annual rate of about 9%. Assuming that both countries will sustain that rate of growth forever, find the time at which China's GDP will be equal to the GDP of the U.S.
4. The Dollar. What are the fundamental causes for the fall of dollar's value?
5. Sub-prime Loans. What are sub-prime loans and what they have to do with the current housing and banking crisis?
6. Arctic National Wildlife Refuge: To drill or not to drill? A question for public debate these days is whether the Arctic National Wildlife Refuge (ANWR) contains enough oil to make its extraction worth both the economic cost and the environmental risk. Make your contribution by doing the numbers.
7. Oil Price. Is the current oil price the result of world demand & supply or/and market manipulation? Draw your own conclusions by collecting data from reliable sources and analyzing them using the mathematics you learned in Math 10250.
8. Wind Energy. Collect data about wind energy production in the U.S. since 2000 and draw a curve that fits these data. Also, draw the oil-price curve using data from reliable sources. Furthermore, compare the shape of these curves and make sense of the current projections of wind energy production for the next 10-20 years. Finally, find out for which country in the world the percentage of the energy it uses from wind is maximum.
9. Solar Energy. Collect data about solar energy production in the U.S. since 2000 and draw a curve that fits these data. Also, draw the oil-price curve using data from reliable sources. Furthermore, compare the shape of these curves and make sense of the current projections of solar energy production for the next 10-20 years. Finally, find out for which country in the world the percentage of the energy it uses from the Sun is maximum.
10. Mountains Beyond Mountains (preview). In this inspiring book Tracy Kidder describes the quest of Dr. Paul Farmer, a man who would cure the world. Curing infectious diseases and bringing the lifesaving tools of modern medicine to those who need them most is his life's calling. Read this book and use the mathematics you have learned in Math 10250 to try to understand, analyze and propose possible solutions to the global health problem.
11. Universal Health Care. What are the benefits and problems of a universal health care system? Examine and compare the health care system of the U.S. and one or two from other developed counties like the U.K., Germany, France Japan, etc.
12. The End of Poverty (preview). In the preface of this book its author Dr. Jerey Sachs (Quetelet Professor of Sustainable Development at Columbia University, Director of the Earth Institute, and Director of the United Nations Millennium Project) writes: "When the end of poverty arrives, as it can and should in our generation, it will be citizens in a million communities in rich and poor countries alike, rather than a handful of political leaders, who will have turned the tide. The fight for the end of poverty is a fight that all of us must join in our own way." Read this very interesting book and use the mathematics you have learned in Math 10250 try to understand (quantify, analyze) poverty as a world problem, and propose possible solutions that our generation can realize.
13. Top Ten. What are the top 10 major challenges for your generation? Provide some number to justify your choices.
14. A. Income distribution and Lorentz curves. The way that income is distributed throughout a given society is an important object of study for economists. The U.S. Census Bureau collects and analyzes income data, which it makes available at its website, www.census.gov. In 2001, for instance, the poorest 20% of the U.S. population received 3.5% of the money income, while the richest 20% received 50.1%. The cumulative proportions of population and income are shown in the following table:

Proportion of population	Proportion of income
0	0
0.20	0.035
0.40	0.123
0.60	0.268
0.80	0.499
1.00	1.00

For instance, the table shows that the lowest 40% of the population received 12.3% of the total income. We can think of the data in this table as being given by a functional equation y = f(x), where x is the cumulative proportion of the population and y is the cumulative proportion of income. For instance, f(0.60) = 0.268 and f(0.80) = 0.499. Such a function (or, more properly speaking, its graph) is called a Lorentz curve.

1. Show that f(x) = 0.1x + 0.9x² is a possible Lorentz curve. Also, compute the income received by the lowest 0 %, 50%, and 100% of the population.
2. Show that f(x) = 0.3x + 0.9x² is not a Lorentz curve.
3. For the Lorentz curve in (i) show the following properties:
   1. f(0) = 0, f(1) = 1, and 0 ≤ f(x) ≤ 1 for all 0 ≤ x ≤ 1,
   2. f(x) is an increasing function,
   3. f(x) ≤ x for all x, 0 ≤ x ≤ 1.
4. Explain why properties (a)-(c) hold for every Lorentz curve.
5. Write many other different formulas for Lorentz curves.
6. Using real data produce Lorentz curves for the U.S. and Canada in 2005.
7. Sketch the graph of a Lorentz curve and compare it with the line y = x.
B. Coefficient of Inequality. If the Lorentz curve of a country is given by f(x) = x then its total income is distributed equally. Otherwise there are inequalities present in the distribution of income, which are measured by the following number:  
coefficient of inequality = 2 ∫₀¹ ( x - f(x) ) dx 
which is also called the Gini Index.
1. Compute the coefficient of inequality when f(x) = 0.1x + 0.9x².
2. Show that the Gini Index is the ratio of the area of the region between y = f(x) and y = x to the area of the region under y = x, and provide an economic interpretation of this ratio.
3. Using real data estimate the Gini Index of the U.S. and Canada in 2005.
15. The Coca-Cola can. In this project, we will investigate whether a Coca-Cola can is designed to minimize the amount of aluminum used for the volume of soda it contains.
    1. For a cylindrical can, closed at the top and bottom, with given volume V , find the ratio h/d of height to diameter that minimizes the total surface area A.
    2. Second, by measuring the height and the diameter of the base of a Coca-Cola can, determine whether it minimizes the surface for the volume it contains.
16. What does calculus have to do with change? The two central concepts in calculus are the derivative (instantaneous rate of change) and the integral (total change). Write in your own words the way you understand these concepts. Give examples from mathematics and its applications to demonstrate them.