math 102 mathematics of sustainability projects
TRANSCRIPT
Math 102 Mathematics of Sustainability
Projects
Contents
Project 1 Trends in US and MN Food Systems‐‐‐‐‐‐‐‐‐‐‐‐Page 2 Project 2 Ecological Footprints‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐Page 17 Project 3 Population Growth & Resource Capacity‐‐‐‐‐Page 24 Project 4 Student Selected Problem‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐Page 33
Updated January, 2010
Written by Rikki Wagstrom, Department of Mathematics
Metropolitan State University
With assistance from the following:
Ken Meter, Crossroad Resource Center
Brad Ewing, Global Footprint Network
This project was made possible through a Center for Teaching and Learning grant with funding from the Office of the Chancellor, Minnesota State Colleges and
Universities
Project 1 Trends in US and MN Food Systems Part 1 Trends in US Food System
1. Think of a food item that you purchased when you were a child (for example, a candy bar). If you bought that same food item now, it probably costs more. This is largely the effect of inflation—the general rise in prices of goods and services. Suppose you spent $0.30 on a candy bar 30 years ago. Today, with inflation, that same candy bar would sell for approximately $0.99. It appears that you have spent more for today’s purchase, but the reality is that the value of the U.S. dollar has decreased. With inflation, both costs and incomes grow. So, even though costs grow, our ability to purchase also grows. Spending $0.99 today is equivalent to making a $0.30 purchase 30 years ago. Figure 1 below shows two graphs displaying trends in U.S. per capita food expenditure between 1953 and 2008. (Source: Food CPI and Expenditures: Table 7. USDA Economic Research Service. www.ers.usda.gov. Accessed August 1, 2009.)
Graph A (labeled “In US Dollars”) shows expenditures in each year. Graph B (labeled “In 2008 US Dollars”) shows the same expenditures, but given in terms of their 2008 US Dollar equivalent. According to the data shown in these two graphs, how much did the average person living in the U.S. spend on food in 1953, and what is the equivalent expenditure in 2008 dollars? By what percentage has per capita food expenditure in the U.S. grown between 1953 and 2008? Use graph A to answer this question and then use graph B to answer the question. Why is the percentage growth for graph A considerably larger than the percentage growth for graph B?
Out[38]=
1960 1970 1980 1990 2000Year
1000
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3000
Figure 1: Per Capita Food Expenditure
In US Dollars
A
B
In 2008 US Dollars
3
Suppose that you heard a statement on the radio that Americans are spending a lot more on food now than they were fifty years ago. Use graphs A and B, as well as your work above, to provide a counter argument.
2. The graph in Figure 2 below displays the trend in U.S. disposable personal income per capita
between 1953 and 2008. (Source: Food CPI and Expenditures: Table 7. USDA Economic Research Service. www.ers.usda.gov. Accessed August 1, 2009.)
By what percentage has per capita disposable personal income grown between 1953 and 2008?
In 1953, what percentage of an average person’s disposable income was spent on food? What percentage is being spent on food currently (in 2008)?
In order to cover the social and environmental costs of food production, some individuals advocate for an increase in food prices and argue that Americans spend less on food today than they used to. Other individuals, who support competitive markets that keep costs low for consumers, argue that Americans are spending more on food now than at any time in the past. Using your work in exercises 1 and 2, explain how it is possible for both sides to make these arguments.
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Figure 2: US Disposable Personal Income Per Capita
In 2008 Dollars
4 3. The money that you spend on food in a grocery store or restaurant pays for the manufacturing,
packaging, transporting, and marketing of the food as well as payment to the farmer who produces the food. The table on the following page shows the percentages of retail prices received by farmers for different food groups. Source: Retail price, farm value, farm‐to‐retail price spreads, and farm value share for selected foods, USDA Economic Research Service. www.ers.usda.gov . Accessed August 1, 2009). Compare the percentages received by farmers for “prepared foods” with the percentages received for other food categories. What do you notice? The graphs in Figure 3 below display the increasing trend in the percentage of U.S. food expenditures that are devoted to manufacturing, packaging, transporting and advertising, and the decreasing trend in the percentage received by the farmers. (Source: Marketing bill and farm value components of consumer expenditures for domestically produced farm foods. USDA Economic Research Service. www.ers.usda.gov. Accessed August 1, 2009.)
Why do the two graphs appear to be mirror images of each other? Over the past 50 years, growth in food science and technology has made it possible to produce many new and convenient foods for consumers. These foods largely fall under the category of “prepared foods” or “processed foods”. What potential explanation does the table on the next page provide for why the percentage of food expenditures received by farmers has been decreasing over the last 50 years? Explain.
Out[40]=
1960 1970 1980 1990 2000Year
0.2
0.4
0.6
0.8
Figure 3: Percentage of US Food Expenditure
Manufactoring, Transportation, Marketing
Farm Share
5
Farm share of retail price Food 1998 1999 2000 2001 2002 Percent Animal products: Eggs, Grade A large, 1 doz. 52 47 53 60 62 Beef, choice, 1 lb.** 47 49 49 46 44 Chicken, broiler, 1 lb. 54 49 48 44 43 Milk, ½ gal. 41 39 34 29 26 Pork, 1 lb.** 25 25 31 30 23 Cheese, natural cheddar, 1 lb. 39 32 29 26 23 Fruit and vegetables: Fresh-- Lemons, 1 lb. 25 23 22 22 22 Apples, red delicious, 1 lb. 19 19 21 22 24 Potatoes, 10 lbs. 15 19 17 16 16 Oranges, California, 1 lb. 21 28 15 17 19 Grapefruit, 1 lb. 18 18 16 16 16 Lettuce, 1 lb. 21 24 24 27 28 Frozen-- Orange juice conc., 12 fl. oz. 38 37 33 32 30 Broccoli, cut, 1 lb. 11 10 12 14 15 Corn, 1 lb. 8 8 7 6 6 Green beans, cut, 1 lb. 7 6 5 5 6 Canned and bottled-- Peas, 303 can (17 oz.) 22 22 22 24 23 Corn, 303 can (17 oz.) 23 22 22 18 17 Applesauce, 25-oz. jar 14 14 16 17 18 Pears, 2-1/2 can 16 13 13 13 13 Peaches, cling, 2-1/2 can 15 14 14 15 15 Apple juice, 64-oz. bottle 32 19 18 18 17 Green beans, cut, 303 can 14 13 13 13 13 Tomatoes, whole, 303 can 7 7 7 8 8 Dried-- Beans, 1 lb. 20 20 19 18 17 Raisins, 15-oz. box 29 36 16 45 30 Crop products: Sugar, 1 lb. 32 31 27 23 20 Flour, wheat, 5 lbs. 20 18 19 19 18 Shortening, 3 lbs. 26 18 15 13 10 Margarine, 1 lb. 26 17 15 13 10 Rice, long grain, 1 lb. 22 19 14 11 9 Prepared foods: Peanut butter, 1 lb. 26 23 22 21 20 Pork and beans, 303 can (16 oz.) 11 11 11 9 9 Potato chips, regular, 1-lb. bag 8 9 8 7 7 Chicken dinner, fried, frozen, 11 oz. 13 13 14 15 15 Potatoes, french fried, frozen, 1 lb. 11 11 10 8 7 Bread, 1 lb. 5 5 5 4 4 Corn flakes, 18-oz. box 4 4 4 4 4 Oatmeal regular, 42-oz. box 6 5 5 5 5 Corn syrup, 16-oz. bottle 3 3 3 3 3
6 Part 2 Agricultural Economic Trends within Minnesota In Part 1, you explored the decreasing farm value share of U.S. food expenditures. In Part 2, you will explore the Minnesota food system from the perspective of a farmer whose income depends on this system.
1. (Trends in commodity prices) A commodity is a product that is mass‐produced for sale in a market. The selling price of a commodity is determined by both the demand for the commodity and the available supply. Common commodities produced by farmers include
• crops such as corn, wheat and soybeans, • livestock (animals) and animal products such as cattle and hogs, milk and eggs
Figures 4, 5, and 6 on the following page display graphs showing trends in the prices received by farmers for their corn, soybeans, wheat, cattle, hogs and milk during the period from 1960 to 2008. (Source: Marketing & Outlook. Farmdoc. www.farmdoc.illinois.edu/manage/uspricehistory/us_price_history.html . Accessed August 1, 2009 and Quick Stats, USDA National Agricultural Statistics Service. www.nass.usda.gov. Accessed January 2, 2010.) Crop prices shown are Minnesota annual averages. Livestock and milk prices are U.S. annual averages. Compare the trends in each of the commodities. In what decade do the prices peak? What do you notice about the pre‐peak prices compared to the post‐peak prices?
Suppose a farmer produces some or all of the commodities shown in these graphs. If this farmer wants to earn the same income in the post‐1970’s as he or she was earning in the pre‐1970’s, would the farmer need to produce more or less than he or she produced during the pre‐1970’s period?
Sections 5 and 6 in Chapter 2 of Michael Pollan’s Omnivore’s Dilemma provide historical context for the trends observed in Figures 4, 5 and 6. According to Pollan, what factors lead to the peak prices, and what factors lead to the drop in prices afterwards?
7
Out[82]=
1970 1980 1990 2000Year
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Figure 4: Crop Price Received Per Bushel
Corn
Wheat
SoybeansIn 2008 Dollars
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1970 1980 1990 2000Year
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100
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200
Figure 5: Livestock Price Received Per 100 Pounds
Cattle
HogsIn 2008 Dollars
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Figure 6: Milk Price Received Per 100 Pounds
In 2008 Dollars
8 Consider the changes that occurred in the corn price received by farmers between 1960 and 2007. What were average corn prices in the years 1960, 1974 (peak year), and 2007? Make sure that you give units! Between 1960 and 1974, by what percentage did the price of corn increase?
Between 1974 and 2007, by what percentage did the price of corn decrease?
Between 1960 and 2007, by what percentage did the price of corn decrease? What was the average rate of growth in corn prices between 1960 and 1974? Give a one‐sentence non‐technical interpretation of your answer. What was the average rate of decrease in corn prices between 1974 and 2007? As before, give a one‐sentence non‐technical interpretation of your answer.
2. (Trends in production) The graphs in Figures 7, 8, and 9 show milk production trends for Minnesota dairy farmers during the period from 1924 to 2008. (Source: Quick Stats. USDA National Agricultural Statistics Service. www.nass.usda.gov. Accessed August 1, 2009.) Figure 7 displays average milk productivity per cow. Figure 8 displays the number of milk cows in the state. Figure 9 displays the total milk production for the state. In the year 2000,
• How many milk cows were there in MN? • How much milk did each cow produce, on average? • What was the total milk production for MN? Answer this question in two different ways:
First, by reading the value on the graph in Figure 9, and second, by calculating it using your answers to the previous two questions.
9
Out[111]=
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Figure 7: Average Annual Milk Productivity
In Pounds Per Cow
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Figure 8: Average Number of Milk Cows
In Thousands
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Figure 9: Total Milk Production
In Millions of Pounds
10 (Fill in the blank. Show your work in the space below.) Between the years 1924 and 2008, milk productivity per cow in Minnesota increased at an average rate of _____________________________________ or by _________________ percent. This is equivalent to growing by a factor of __________________. How do you think it is possible for farmers to increase the milk productivity of a cow this much? (Fill in the blank. Show your work below.) Also, between the years 1940 and 2008, the number of dairy cows raised in Minnesota decreased at an average rate of _____________________________________ or by _________________ percent. Notice the dramatic spikes in Minnesota’s total milk production during the 1940’s, 1960’s and 1980’s. What explanation do Figures 7 and 8 provide for these spikes?
3. (Trends in Farm Income & Expenses) The graphs in Figure 10 on the following page show the trend in total cash receipts received by Minnesota farmers as well as the trend in total production expenses incurred by Minnesota farmers. Cash receipts represent the income received from the sale of crops and livestock. Production expenses represent the expenditures incurred by farmers in the production of agricultural commodities, specifically livestock and crops. Production expenses include agricultural inputs (feed, livestock and poultry, seed, fertilizer, pesticides, etc.), labor expenses, interest payments, rent paid to landlords, property taxes, etc. (Source: Farm Income Data Files. USDA Economic Research Service. www.ers.usda.gov/data/farmincome/FinfidmuXls.htm. Accessed August 1, 2009.)
11
Why are cash receipts and production expenses so much higher in the 1970’s? You can refer back to your earlier work in this project and/or your reading in Chapter 2 of Omnivore’s Dilemma if you need assistance. During which years are the cash receipts equal to the production expenses? During which time periods are the cash receipt totals larger than the expense totals? During which time periods are the cash receipt totals smaller than the expense totals? What does it mean for Minnesota farmers when the cash receipts are less or equal to the production expenses?
Out[31]=
1940 1960 1980 2000Year
5.!109
1.!1010
1.5!1010
Figure 10: Cash Receipts and Production Expenses
Cash Receipts
ExpensesIn 2008 Dollars
12 During the time period from 1950 to 1970 what was the approximate maximum value for agricultural production expenses? During the time period from 1980 to 2008, what was the approximate minimum value? Give your answer in billions of dollars. In comparing production expenses before and after the 1970’s decade, what do these two numbers tell you?
Use your answers to the five questions above to summarize some basic differences in Minnesota farming profitability before and after the 1970’s decade.
By what percentage did cash receipts grow between 1950 and 1980? What was the average annual rate of growth? Give your answers in complete sentences. Remember to give units. By what percentage did cash receipts decrease between 1980 and 2000? What was the average annual rate of decrease? Give your answers in complete sentences.
13 4. Since the 1930s, the federal government has offered financial incentives to farmers for the
production of certain commodities (corn, soybeans, barley, wheat, dairy, livestock, etc.). These incentive payments are called subsidies. They are intended to offset financial losses incurred by the farmers producing these commodities. In addition to federal subsidies for agricultural commodities, the government also provides disaster payments and payments to farmers who are willing to put aside some land for conservation. The graphs in Figure 11 below show the total government payment to Minnesota farmers during the period from 1995 to 2006 and a breakdown showing the largest components of the payment. (Source: Farm Subsidy Database (2006). Environmental Workgroup. http://farm.ewg.org/farm/ . Accessed August 1, 2009.)
Determine the total government payments in the years 2000 and 2004. Give your answers in billions of dollars. Complete the following table.
Percentage of Total Government Payment Corn Soybeans Wheat Dairy Conservation Reserve Program 2000 2004
1996 1998 2000 2002 2004
5.!108
1.!109
1.5!109
Figure 11: Top Govt Subsidies in Minnesota
Corn
Soybean
Wheat
Dairy
Conservation Reserve Program
In 2007 Dollars
Total Gov't Payment
14 What general observations do you have from comparing these government expenditures?
5. The graphs in Figure 12 below show how government payments have affected the total farm income for Minnesota’s farmers. The boldfaced graph represents the combined sum of the cash receipts and the government payments.
In what year were government payments largest? Determine the approximate total government payment in that year. Please show your work. Give your answer in billions of dollars.
In exercise 3, you identified periods of time when farming was not profitable in Minnesota. If you now include government payments into the calculation of farm income, is farming profitable during these time periods? Explain.
1940 1960 1980 2000Year
5.!109
1.!1010
1.5!1010
2.!1010Figure 12: Cash Receipts and Government Payments Combined
Cash Receipts
Expenses
Cash Receipts Plus Gov't Payments
15 Around the year 2000, there were approximately 82,000 farm owners in Minnesota. Determine each of the following quantities for the year 2000: Total farm income (cash receipts + gov’t payments) = ______________________ Total production expenses = ______________________ Total farm profit for Minnesota farmers = ______________________ Profit per farm owner = ______________________ The profit that a farmer makes needs to cover annual living expenses for his or her household. This includes food, medical, dental, auto, and life insurance, car payments, utilities, home mortgages, etc. Do you think the profit that you estimated above is adequate to cover these expenses? Explain your answer.
6. The table below shows the total number of Minnesota farmers and the number reporting losses on
the U.S. Agricultural Census. (Source: Table 5. U.S. Agricultural Census. USDA National Agricultural Statistics Service.)
1987 1992 1997 2002 Total Number of Minnesota Farmers 85,079 75,079 78,755 80,839
Number of Minnesota Farmers Reporting Losses 31,086 29,303 31,623 34,883
For each of the four years above, determine the percentage of farmers reporting losses. How has the percentage of farmers reporting losses changed over time?
16 The graph in Figure 13 below shows the number of self‐employed farmers in Minnesota between the years 1969 and 2008. (Source: State Economic Profiles. Bureau of Economic Analysis. www.bea.gov/regional/spi/action.cfm. Accessed August 1, 2009.)
(Fill in the blank. Show your work below.) Between the years 1985 and 2007, the number of self‐employed farmers in MN decreased at an average rate of _____________________________________ or by _________________ percent.
7. Assemble the information that you have learned from analyzing the data and graphs in this activity. Summarize the key findings from this activity.
8. Write down 3 questions that this activity has raised for you regarding food and agricultural systems both nationally and in Minnesota.
Out[55]=
1975 1980 1985 1990 1995 2000 2005Year
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Figure 13: Number of Self!Employed Farmers
Project 2 Ecological Footprints
Introduction In this project, the first of a two-part series on sustainability, you will study Ecological Footprints and their underlying calculations. First of all, what is an Ecological Footprint? An Ecological Footprint measures humanity’s demand on the planet’s biological resources. The Global Footprint Network (www.footprintnetwork.org ) defines an Ecological Footprint as follows: “the amount of biologically productive land and water area an individual, a city, a country, a region, or all of humanity uses to produce the resources it consumes and to absorb the waste it generates with today’s technology and resource management practices.” Simply put, your Ecological Footprint measures the land and water area on the planet that is needed to sustain your lifestyle. Included in your footprint would be items such as the following:
• Crop and pasture land for food production • Forested land for paper and wood products • Crop land for production of fibers • Built-up (developed) land for housing, roads, schools, industry, etc.. • Forested Land for absorbing atmospheric carbon dioxide emitted from burning fossil fuels
The list is actually much larger than this, if you consider all of the biological resources needed to manufacture everything that you own and consume and to absorb the wastes you generate. If you could add up the Ecological Footprints for every person on the planet, you would have an estimate for how much land and water is demanded to sustain all of the humans currently living on the planet. In your third project, you will explore the finite supply of biologically productive land and water on the planet. Comparing the demand on our biological resources with the supply available will give you a way of measuring how sustainably we are currently living. Determining a person’s total Ecological Footprint is VERY difficult because there is so much to include and requires data that may not currently exist. Quoted estimates probably underestimate the real value since some aspect of land use is typically left out of the calculation due to inadequate data sources. In this project, you will only explore carbon dioxide footprints for automobiles and footprints for some common food commodities. Consequently, the footprints that you will calculate will be low estimates of a person’s total Ecological Footprint. Part 1 Carbon Footprints from Cars When fossil fuels (like gasoline, natural gas, diesel, coal, etc..) are burned, carbon dioxide is emitted into the atmosphere. The amount emitted depends on the type of fossil fuel. For gasoline, approximately 19.37 pounds of carbon dioxide are emitted into the atmosphere with each gallon consumed. (This amount varies slightly depending on the octane level. Source: U.S. Environmental Protection Agency, www.epa.gov/appdstar/pdf/brochure.pdf.) Some of the carbon dioxide emitted is sequestered (i.e. absorbed) by the oceans and vegetation, especially forests, on the planet. The excess carbon dioxide remains in the atmosphere and warms our planet. Because of current concerns about global warming, people are interested in knowing how much forested land is required to absorb the carbon dioxide released into the atmosphere from various sources,
18 including gasoline-powered cars. This area of land is called the “carbon footprint”. In a geographic region dominated by northern temperate middle-aged forests (common to northern most regions of the U.S. and parts of Canada), carbon dioxide is absorbed at an average annual rate of 0.8 tons per acre. The global average sequestration rate is closer to 0.04 tons per acre.
(a) The function
€
A =3.535DCER
estimates the carbon footprint from one person’s annual auto emissions. Each of the variables represented in this function is given in the table below. Use unit analysis to derive this function. Recall that there are 2000 pounds in 1 ton.
Independent Variables Dependent Variable Carbon dioxide sequestration rate = C
(tons per acre) **C represents how many tons of carbon dioxide are absorbed by the vegetation on an acre of land. This quantity is variable because the amount of
vegetation on an acre of land varies. Fuel efficiency of car = E
(miles per gallon) Daily driving amount = D
(miles per day per car) Average ridership = R
(people per car)
Carbon footprint = A (acres per person per year)
19
(b) Varying Fuel Efficiency: Consider a car driven 40 miles per day with an average ridership of 1.57 riders (which is the U.S. average). Suppose the sequestration rate is 0.80 tons of carbon dioxide per acre of land. Substitute these values into the function from part (a) and simplify. You should see that A is now a function of E. Write down this function and use it to complete the table.
Varying Daily Miles Driven: Now, consider a car that has an average fuel efficiency of 28 miles per gallon which carries an average of 1.57 riders (U.S. average). Suppose the sequestration rate is again 0.80 tons of carbon dioxide per acre of land. Substitute these values into the function from part (a) and simplify. You should see that A is now a function of D. Write down this function and use it to complete the table.
Varying Ridership: Now, consider a car that has an average fuel efficiency of 28 miles per gallon which is driven 40 miles a day, on average. Suppose the sequestration rate is again 0.80 tons of carbon dioxide per acre of land. Substitute these values into the function from part (a) and simplify. You should see that A is now a function of R. Write down this function and use it to complete the table.
Ridership Carbon Footprint 1 rider 2 riders 4 riders
Varying Sequestration Rate: Now, consider a car that has an average fuel efficiency of 28 miles per gallon which is driven 40 miles a day and carries 1.57 people, on average. Substitute these values into your function from part (a) and simplify. You should see that A is now a function of C. Write down this function and use it to complete the table.
Fuel Efficiency Carbon Footprint 15 mpg
(large SUV)
28 mpg (average sedan)
48 mpg (hybrid sedan)
Daily Miles Carbon Footprint 20 miles per
day
40 miles per day
80 miles per day
Sequestration Rate Carbon Footprint 0.80 tons per acre
(forested area)
0.04 tons per acre (global average)
20 (c) Describe how the carbon footprint changes as you change each of the input variables in part (b). Based
on your results, what are some ways for a person to reduce their carbon footprint?
(d) Now, it’s your turn. Use the function from part (a) to determine the carbon footprint for either yourself or someone you know.
Part 2 Food Footprints The amount of land required to produce the foods that we eat is called our “food footprint”. Estimating the entire food footprint for a person is very difficult. For example, if you wanted to determine the food footprint of a box of macaroni and cheese or a can of pop, you would need to account for the crop land required to produce all of the raw ingredients, plus the built-up land used for manufacturing, plus the land required to absorb the carbon dioxide emissions generated in the manufacturing and distribution processes, etc. For this reason, food footprints of processed foods are quite complicated. On the other hand, food that is grown on the land and consumed directly (for example, vegetables and fruits) have somewhat simpler footprints. For example, you estimated the food footprints of potatoes and bananas earlier in this course: • About 915,479 acres of potato crop land are required annually to meet the U.S. demand for potatoes. • About 708,571 acres of foreign land are required annually to meet the U.S. demand for bananas.
Recall that these estimates did not include the land area required to absorb carbon emissions generated in the production and transport of these commodities. So, these estimates are probably low. In this part of the project, you will use unit analysis to estimate the footprints for apples, beef and coffee. Together with bananas and potatoes, these five foods comprise a substantial part of the diet of many people in the U.S. and can give a low estimate for many peoples’ food footprints.
21 (e) (U.S. Apple Consumption) In 2004, fruit bearing apple trees in the U.S had an average yield of 13.5
tons of apples per acre. (Source: USDA www.nass.usda.gov). Per capita consumption of apples in that same year was 50.4 pounds. (Source: U.S. Apple Association, www.usapple.org/media/publications/applenews/2005/nov2005.pdf .) Use unit analysis to estimate the acres per capita of apple groves required to meet the current U.S. demand for apples.
(f) (U.S. Coffee Consumption) The annual U.S. demand for coffee is currently about 140 billion cups. The average yield among the largest coffee-producing countries (Brazil, Ethiopia, Columbia, Peru, Guatemala, Honduras, India, Indonesia, Mexico and Viet Nam) is approximately 964 pounds of beans per acre. Coffee grows on trees. An average tree produces about 4.8 pounds of coffee beans, which produces approximately 16 gallons of coffee. (Sources: United Nations Food & Agriculture Organization: FAO Statistics Division www.fao.org , and Frank Catano, Society and Environmental Sustainability, presentation at 2008 SENCER Institute.) Use unit analysis to estimate the acres per capita required to meet the U.S. demand for coffee each year. Recall that the population of the U.S. is approximately 301 million people.
22 (g) (U.S. Beef Consumption) Let’s consider beef consumption in the U.S. First, a few pieces of
information:
• In January 2006, 2.06 billion pounds of beef were produced in the U.S. from slaughtering 2.67 million head of cattle raised in the U.S.
• In 2004, per capita beef consumption in the U.S. was 42.3 kilograms per person. (Sources: www.nass.usda.gov, www.fas.usda.gov/dlp/circular/2006/06-03LP/bpppcc.pdf, and Major Uses of Land in the United States, 2002/EIB-14, Economic Research Service, USDA) Let C represent the acres of pasture a single cow grazes over the course of a year. Let A represent the acres of pasture per capita required to meet the U.S. demand for beef over a year. Use unit analysis to create a function relating the variables A and C. Choose A to be the output variable. Recall that there are approximately 2.2 pounds in one kilogram. Use your function to complete the following table:
Grazing Acres Required Per Cow Over a
Year Grazing Acres
Required Per Person 2 acres per head (highly vegetated pasture)
5 acres per head (less vegetated pasture)
23 For cattle that are finished on corn (e.g. on feedlots), land is also required to grow the corn feed. A typical corn yield in the Midwest is about 160 bushels per acre. Each bushel of corn weighs about 56 pounds. The feed-to-meat ratio is about 7 pounds of corn to about 1 pound of meat. You can also use information from your work estimating grazing land. Use unit analysis to estimate the acres of feed corn per capita required to support U.S. corn-fed beef consumption.
Now, combine the per capita footprints for grazing land and corn crop land together to obtain estimates for the U.S. per capita beef footprint:
Grazing Acres Required Per Cow Over a
Year Total Acres Required
Per Person 2 acres per head (highly vegetated pasture)
5 acres per head (less vegetated pasture)
(h) Compare the per capita apple, banana, potato, beef and coffee footprints that you have now estimated. Which commodity requires the greatest land use? Which requires the least land use? Are any of these values substantially more or substantially less than the others?
Part 3 Comparing the Footprints
(i) How do the food footprints in Part 2 compare to the carbon footprints in Part 1?
Project 3 Population Growth & Resource Capacity
Part 1 Population Projections Between 1950 and 2005, population growth in the U.S. has been nearly linear, as shown in the graph below. (Source: Population Division of the Department of Economic and Social Affairs of the United Nations Secretariat. 2007. World Population Prospects: The 2006 Revision.)
If you looked at population growth over a longer period of time, you would see that it is not actually linear. However, over the relatively short period of time above, the growth looks nearly linear. A statistical technique called linear regression can create a linear function that approximates the actual population growth over this period very well. It turns out that this function is
€
P = 0.00247t + 0.159
where t represents the time variable measured in years since 1950 and P represents the (approximate) population of the U.S. measured in billions of people. (If you take statistics, you’ll probably learn how to obtain this function.) The graph of this linear function is shown in the figure above.
(1) Just to make sure that you understand how to work with this function, use it to complete the following
table. The actual population values are given. If you are working with the function correctly, the values you obtain should be close to the actual population values!
Year Actual Population (billions of people)
t (years)
P (billions of people)
1960 .186158 1990 .256098 2005 .299846
(2) Use the linear function to determine the approximate year when the population of the U.S. first reached 200 million people. Set up an equation and determine the answer algebraically.
25 (3) Explain what each of the values 0.00247 and 0.159 means in practical terms, using everyday language.
(i.e. What are these values telling you about the population of the U.S.?)
Shown on the graph below is the actual population growth during the 1950-2005 period, and two different population projections out to the year 2050. The first projection was generated by the United Nations. (See reference on previous page.) The second projection was obtained by extending the linear approximation above out past the year 2005. From the graph, you can see that the linear projection is relatively close to the U.N. projection between 2005 and 2050. For the remainder of this project, we’ll use the linear approximation to make population projections because the equation for this function is easier to work with.
(4) Use the linear approximation to make population projections for the following years. (Use the equation of the function, not the graph, to obtain your projections.)
Year t (years)
P (billions of people)
2008 2025 2050
(You should check that your values match the values shown on the graph.)
(5) Use the linear function to give the approximate year when the population of the U.S. first reaches 350 million people. Set up an equation and determine the answer algebraically.
1960 1980 2000 2020 2040Year
0.1
0.2
0.3
0.4
0.5
U.S. Population !billions of people"UN Projection
1950!2005 Population DataLinear Projection
26 Part 2 Population Growth, Ecological Footprint and Biocapacity Recall from your reading in the Living Planet Report 2008 that the Ecological Footprint represents humanity’s demand on the Earth’s biological resources whereas the Biocapacity represents the supply of biological resources available for consumption. Specifically, the Ecological Footprint measures the area of biologically productive land required to support human demands for food, fiber, timber, energy and space for infrastructure, and also to absorb waste products. The Biocapacity measures the area of biologically productive land (i.e. forest land, cropland, and grazing land) available. You have already estimated a few components of the current U.S. per capita Ecological Footprint, namely carbon footprints from automobiles, and footprints for the production of apples, potatoes, bananas, coffee and beef. Estimating a person’s total Ecological Footprint would require much more work. For example, you would need to consider the total land and water area required to produce all of a person’s food, to absorb carbon dioxide emissions not only from driving a car, but also from heating a home and using electricity. Add to this the amount of land needed to produce all of the wood/paper/fiber products a person uses. Add to this the amount of land needed for a person’s housing unit, and their share of all the developed land and infrastructure within the country. As the list goes on and on, the data requirements and the number of calculations grow. The Global Footprint Network (www.footprintnetwork.org ) has done extensive work in estimating both Ecological Footprints and Biocapacities for cities and countries all over the world. For the remainder of this project, we will use the results from their calculations because they are much more complete. Note: In your last project, you measured footprints in units of ‘acres’. The Global Footprint Network measures Ecological Footprints and Biocapacities in ‘global acres’ rather than ‘acres’. A global acre is a unit of land area that also takes into consideration how biologically productive the area is. One global acre is equivalent to one acre of land with world average productivity. Since the world’s productivity varies each year, so will a global acre. To keep matters simple, we will assume that a global acre remains the same over time, and just treat it as a unit of land area. (If you want to learn more about global acres, visit the Global Footprint Network’s website www.footprintnetwork.org .) The graph below shows the trends in U.S. per capita Ecological Footprints and Biocapacities between 1961 and 2005. (Source: Global Footprint Network (2008). All rights reserved.)
1970 1980 1990 2000Years
5
10
15
20
US Per Capita
Ecological Footprint
Biocapacity
Average Ecological Footprint 1971!2005
27 (6) The total Biocapacity of the US has remained fairly constant over time, at roughly 3.38 billion global
acres. Despite this fact, the per capita Biocapacity is declining. Why is this?
(7) Notice also that the per capita Ecological Footprint has fluctuated since the early 1970s. The horizontal line shows the average per capita Ecological Footprint between the years 1971 and 2005—approximately 21.063 global acres. If the per capita Ecological Footprint continues to linger abound 21 global acres, what will happen to the total Ecological Footprint of the U.S.? Will it increase, stay the same or decrease? Explain your answer.
(8) In what year do the two curves intersect each other? What is the U.S. population during this year?
(9) In the years preceding the intersection point, how do the per capita Ecological Footprint and Biocapacity compare? How do they compare in the years after the intersection point?
(10) In the years following the intersection point, the U.S. entered a state referred to as overshoot. In terms of sustainability, what does overshoot indicate? How do you think it is possible for a country experiencing overshoot to meet the needs/demands of its residents?
(11) In 2005, the per capita Ecological Footprint was approximately 23.01 global acres while the per capita Biocapacity was approximately 10.80 global acres. How many times larger is the Ecological Footprint than the Biocapacity? (In other words, what number would you have to multiply the Biocapacity by in order to obtain the Ecological Footprint?) This number will tell you how many copies of the U.S. land area would be required to meet the current resource demands of people living in the U.S.
28 (12) You saw from the graph above that the per capita Biocapacity of the U.S. is decreasing over time.
Determine a function that models this decrease. In other words, determine a function equation in which t represents an input time variable (in years since 1950) and B represents the per capita Biocapacity output variable (in global acres per person). Recall that the total Biocapacity of the U.S. has been roughly constant at 3.38 billion global acres. (Hint: if you divide ‘billions of global acres’ by ‘billions of people’, the ‘billions’ cancel and you are left with global acres per person.)
(13) Use your per capita biocapacity function to complete the following table.
(14) Suppose that the U.S. per capita Ecological Footprint continues to linger about 21.063 global acres for the foreseeable future, and that the per capita Biocapacity continues to decrease. Use your per capita Biocapacity function to determine the approximate year in which the per capita Ecological Footprint is three times the per capita Biocapacity. Write down the equation that you would have to solve and then solve this equation. (Hint for solving equation: What can you do to eliminate fractions from an equation?)
Year t B (global acres per person per year)
1950 2008 2020 2050
29 Part 3 Ecological Footprint Reduction Recall that the U.S. per capita Ecological Footprint in 2005 was approximately 23.01 global acres. The table below shows estimates for the different components of the U.S. per capita Ecological Footprint in the year 2005. (Source: Global Footprint Network (2008). All rights reserved.)
Cropland Grazing Land
Forest Land (for wood,
fiber & fuel)
Carbon Sequestration
Land
Built-up Land Total
U.S. Ecological Footprint per capita
(global acres per person per year) 3.41 0.74 2.52 16.09 0.25 23.01
The graph below shows the trends in these components over the period from 1961 to 2005.
(15) Recall that the per capita Biocapacity in 2005 was 10.80 global acres. The U.S. is experiencing overshoot. According to the table, which one component of the per capita Ecological Footprint is solely responsible for the overshoot?
In next few exercises, you will investigate hypothetical measures to reduce the U.S. carbon footprint over time (and therefore the overall Ecological Footprint) with the objective of ending overshoot by the year 2050.
(16) (Option 1) Suppose that the U.S. had adopted energy policies in 2005 that would require the country’s per capita carbon footprint to be cut by 5% every 10 years up to 2050. With these policies, would the per capita carbon footprint decrease linearly or exponentially over time? Let s represent the number of years since 2005 and let C represent the per capita carbon footprint. Determine the equation for this function.
Out[12]=
1970 1980 1990 2000Years
5
10
15
Per Capita Ecological Footprints
Cropland
Grazing Land
Forest Land
Carbon Land
Built Up Land
30 (17) (Option 2) Now suppose that the U.S. had also adopted policies in 2005 that would require the
country’s total carbon footprint (not per capita) to be cut by 5% every 10 years up to 2050. With these policies, would the total carbon footprint decrease linearly or exponentially over time? Let s represent the number of years since 2005 and let E represent the total carbon footprint. Determine the equation for this function.
(18) Now, lets see what affect these proposed carbon footprint reduction policies will have on the overall Ecological Footprint. Use the linear population growth function, the per capita Biocapacity function and the two carbon footprint functions to make projections for the 2020 and 2050 per capita Ecological Footprints. You can organize all of your calculations on the table provided. Space has been provided for you to write in your function equations.
Year 2020 2050 t (years since 1950) s (years since 2005)
Population
€
P = 0.00247t + 0.159 (billion of people)
Per capita Biocapacity (global acres per person)
(1) Per capita carbon footprint—Option 1 (global acres per person)
Total carbon footprint—Option 2 (billions of global acres)
(2) Per capita carbon footprint—Option 2 (global acres per person)
(3) Sum of per capita footprints for forest, cropland, grazing, and built-up land—assume unchanged from 2005 (global acres per person)
Per capita Ecological Footprint — (1) + (3) **If assuming Option 1. (global acres per person)
Proj
ectio
ns
Per capita Ecological Footprint — (2) + (3) **If assuming Option 2. (global acres per person)
31 (19) When you compare the two Ecological Footprint projections to the Biocapacity projections in the
year 2050 you should see that overshoot still exists. Back in exercise 11 you determined that the 2005 per capita Ecological Footprint was _______ times the per capita Biocapacity. With the carbon footprint reduction policies implemented, we would hope that this factor is at least smaller in the year 2050. Calculate this factor assuming Option 1 is used. Then calculate the factor assuming Option 2 is used.
(20) Recall the two carbon footprint reduction policies that you have been considering: • (Option 1) Reducing the per capita carbon footprint by 5% every 10 years. • (Option 2) Reducing the total carbon footprint by 5% every 10 years.
In exercise 19, you should see two things:
• The 2050 factor for Option 1 is larger than the 2005 factor. • The 2050 factor for Option 2 is smaller than the 2005 factor.
So, why is Option 2 more effective than Option 1 at reducing overshoot, and how is it possible that the
factor for Option 1 is larger than the 2005 factor, given that Option 1 still reduces the Ecological Footprint?
(21) Propose a policy or a series of policies (similar to the kinds of policies described in (16) and (17) which, if implemented in 2005, would eliminate overshoot by the year 2050. There are many possibilities here. You should make an effort to create policies that you think might be plausible. (For example, eliminating overshoot by reducing the carbon footprint by 10% annually is probably not plausible.) In order to do this, you are welcome to consider reducing not only the carbon footprint but other components of the Ecological Footprint as well. For example, you could consider reductions in forest land use, agricultural land, or built-up (i.e. developed) land footprints. If you wish, you can also propose policies that slow population growth in the U.S., but you may not consider policies that reverse population growth. (In other words, you can not require the U.S. population to decrease.)
Defend your policies by creating mathematical functions that would describe the footprint reductions over time. Use these functions to make projections in the year 2050 and show that overshoot is eliminated. You will want to use a separate sheet of paper for this exercise.
32 Concluding Comments
In this project, you developed mathematical models (functions) that enabled you to study a VERY complicated problem. When you were MUCH younger, did you play with models? For example, did you play with dolls, model cars, model rockets, or build structures with LEGOs, etc.? Models are simplified versions of something that, in real life, is more complex. For example, model toys don’t usually have all of the inner anatomy of the real-life object. Well, the same is true with mathematical models. In order to create mathematical models, we usually have to make some assumptions which simplify the problem. These assumptions make it possible to mathematically study the more complex real-life problem. However, aspects of realism are lost when you do this. In the end, the predictions made by the models have to be considered in light of the assumptions made, and you have to keep these assumptions in mind when using your predictions to answer real-world questions. This project was intended to give you a taste for how mathematics is used to study complex problems of current interest, but also to give you a sense for the magnitude of the reductions in human consumption required for the U.S. to live within its means.
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Project 4 Student Selected Problem
For this project you will learn how to ask a refined quantitative question, choose appropriate mathematical ideas and calculations to explore or answer your question, and construct a technical report detailing what you have learned through your mathematical work. The purpose of this project is for you to use mathematical reasoning to learn something that you didn’t know before, not to persuade. You will begin by choosing a sustainability topic of interest to you. Once you choose a topic, you will pose a question that you can explore using the mathematical concepts that you have learned in this course. The mathematics might include analyzing growth and decay trends over time using graphs or tables and calculations of average rates of change or percentage change. Or, you might try to estimate the value of some quantity using unit analysis. Or, you might try to study the relationship between some variables of interest by creating and analyzing a function. To assist you in organizing your work on this project, you will begin by completing the worksheet shown on the next page of this packet. An example of a completed worksheet is also included in the packet. Successful completion of this worksheet is crucial to the overall success of your project report. The worksheet is factored into your project grade and your instructor will use your worksheet responses to provide you with important feedback while you are working on your project. You will be completing this project in three stages:
Stage 1: Identifying a topic and question. You will do this as you complete questions (1) and (2) on the worksheet.
Stage 2: Obtaining necessary quantitative information/data and planning the
mathematics that you are going to use to explore your question. You will do this as you complete questions (3) and (4) on the worksheet.
Stage 3: Performing the needed mathematical work and interpreting your results. At
this stage, you will write a 4‐5 page, single‐spaced, typed final report. Your report must be organized as follows:
1. Introduction—Includes topic of interest and the question that you are exploring. (You can
just copy your responses to questions (1) and (2) from the worksheet.)
2. Mathematical analysis of the question—In this part of the report you carefully lead the reader through the mathematical ideas and calculations that you briefly summarized in question 4 from the worksheet using the quantitative information that you listed in question 3 from the worksheet. You should explain how the mathematics you are using helps you examine your proposed question. If you make any assumptions that impact your work, you should clearly state them. In general, mathematical work/calculations need to be shown. If several identical calculations need to be made, show only one of the
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calculations as an illustration. (For example, if you are calculating average rates of change over 10 different time periods, only show the calculation for one of the time periods. For the other time periods, just give the calculated values.)
3. Interpretation of mathematical results and conclusion—In this part of the report you discuss in detail the results of your mathematical work and interpret the results in the context of your proposed question using non‐technical language. In other words, you will be discussing what you have learned about your question through your mathematical work. In this section, you should also address any weaknesses in your work that limit the ability of your project to fully answer your question, and briefly discuss any modifications that would need to be made for a more complete exploration of your question, if necessary.
4. Bibliography—You must cite all sources of information used in your project. In the appendix that follows, you cite your sources for all data and quantitative information used in the project. So, you don’t need to include those sources again here. In the bibliography section, include any remaining sources that you used.
5. Appendix—In this part of the report you should copy your list of quantitative information and data from exercise 3 for reference purposes.
Note: MicroSoft Word has an Equation Editor which allows you to create mathematical formulae quite easily. If you haven’t used it before, let me know and I’ll show you how to use it. You can also go to the Math/Writing Center for assistance in writing your report.
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Math 102 Project 4 Worksheet Name _________________________________________________________ (1) Write a paragraph describing your topic of interest and why this topic is important. (2) Write down ONE carefully chosen question that you would like to explore for this project.
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(3) The question that you have chosen to explore will require you to find/gather data or quantitative information. Use internet, library, or personal resources to obtain this information. On a separate sheet of paper, make a complete list of all data values or quantitative information that you have obtained and will be using in your project. Be sure to cite the sources for all of the information.
(4) Summarize the mathematical work that you intend to do to explore your question. In other
words, how do you intend to use the data or quantitative information that you listed in part (3) to explore your question?
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Math 102 Project 4 Worksheet ****************COMPLETED EXAMPLE******************* Name _________________________________________________________ (1) Write a paragraph describing your topic of interest and why this topic is important. The topic that I am interested in exploring is home efficiency. In my home, we have a dishwasher, refrigerator, washing machine, clothes dryer, and many other appliances that consume water and energy. Furthermore, many of the items that I buy and use at home required water and energy. For example, foods, toiletries, baby supplies, toys, and many other items require energy and water to produce. We also heat, cool, and light our home that adds to my family’s energy consumption. This topic is important to me because, first, water and energy consumption cost money and my family lives on a tight budget. Second, water is a limited resource and should not be wasted. Third, energy production—which is primarily coal driven—is harmful to our atmosphere. (2) Write down ONE carefully chosen question that you would like to explore for this project. Well, I’ve come up with a few questions, but I don’t know which questions are likely to work well for my project. Can you give me some advise on these questions? Are there any that you would recommend? 1. What is the difference in energy consumption between different electric-powered
household appliances? 2. Many of our appliances at home are getting older and will need to be replaced. If
we purchase water-efficient appliances, how much will we need to spend, how much of a cost savings can we expect, and how long will it take for the savings to make-up for the cost of buying new appliances? How would this be different if we purchased appliances that were less water efficient?
3. Should I use cloth diapers or disposable diapers? 4. What is my family’s energy footprint? 5. Why do Americans use more energy than people living in other countries? 6. Are hybrid cars better than Hummers?
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(3) The question that you have chosen to explore will require you to find/gather data or quantitative information. Use internet, library, or personal resources to obtain this information. On a separate sheet of paper, make a complete list of all data values or quantitative information that you have obtained and will be using in your project. Be sure to cite the sources for all of the information.
(4) Summarize the mathematical work that you intend to do to explore your question. In other
words, how do you intend to use the data or quantitative information that you listed in part (3) to explore your question?
I decided to explore question 2 and focus on dishwashers. The data that I listed in part (3) included prices and water consumption amounts for a range of new dishwashers that I saw when I visited SEARS and Best Buy, estimated number of loads that we do each week, and the amount my family pays per gallon of water consumed at home. There are several variables in this problem: how much water a dishwasher uses per load, the amount my family is charged per gallon of water (this amount has been increasing over the past few years), the time period, and the total cost of this water to my family over this time period. I’m going to use unit analysis to create a function that has the first three of these variables as inputs and the final variable as the output. This will allow me to estimate the amount of money my family will spend on water over any period of time that I want for each of the dishwashers. By adding in the original price of each dishwasher to my function, I will be able to estimate the total cost to my family for each dishwasher over any period of time. I will use this function to draw graphs of dishwasher costs over time. From the graphs, I will be able to see how long it will take until the less-water-efficient dishwashers’ costs exceed the more-water-efficient dishwashers’ costs.