Review Topics

## A VILLAGE and ITS RESOURCES

This Study is an investigation of availability of normal resources for a growing village. It is divided into three parts, Population, Food, Coal and Electricity, and Water.

### Comprehension

1. How large is the community or city that you live in or near?
2. Is your community or city growing? If so, how fast?
3. What are some commodities that you depend on daily?
4. How much land does your community or city occupy?
5. Is most of the surrounding land agricultural, recreational, or industrial?
6. What is your source of food?
7. What is your source of water?
8. What is your source of electricity?

### Objective

To determine the function which can be used to predict the population of the Village

### Assumptions

1. There are 10,000 residents at the beginning of the year 2000.
2. The population is growing by 300 people each year.
3. This annual increase remains constant in the future.

### Application

Write the linear function M(t) which can be used to predict the population of the Village t years from the year 2000.

## FOOD

### Objective

Determine:
1. area of land cultivated for growing grain for the Village;
2. total grain supply for the Village;
3. per-capita grain consumption for the US;
4. total demand for grain for the Village; and
5. the number of years before the the Village demand is greater than its supply.

### Assumptions

1. The current trend for per-hectare grain yield continues (see World Grain Yield).
2. In 2000, the community has 1500 hectares of land used for grain production.
3. For each additional person added to the village population, .04 hectare of land is taken out of cultivation for development.
4. The current decrease in land cultivated for grain for the Village continues.
5. The current increase in the Village population continues.
6. Each person requires188 kilograms of grain each year.

### Application

1. How much land will be taken out of cultivation for development t years from now?
2. Write the function G(t) which estimates the amount of land will be used for cultivation t years from now?
3. Use the world grain yield per-hectare function Y(t) (from the food module) and your answer to #2 to write a function S(t) which describes the number of kilograms of grain produced for the Village in the year 2000 + t.
4. Use the derivative to estimate the rate of change in grain production at the beginning of the year 2020. At the beginning of the year 2040.
5. When will the grain production reach a maximum? What is the maximum production?
6. Recall the linear Village population function M(t). Use this function and Assumption #6 to write the function D(t) which can be used to predict the Village demand for grain t years from now.
7. Predict when the amount of grain consumed will reach the amount of grain produced, or when supply equals demand.
8. Sketch graphs of both the demand and supply functions on the same coordinate system, and interpret the point of intersection.
9. Suppose the people of the Village decide in 2000 that they will store excess grain for use once demand reaches supply. How many additional years of food will this provide?

## COAL and ELECTRICITY

### Objectives

Determine:
1. the amount of coal used in the U.S.;
2. U.S. share of world coal supply;
3. The Village share of U.S. coal supply;
4. total amount of coal used for electricity for the Village; and
5. the number of years before the Village's coal for electricity runs out.

### Assumptions

1. The per-capita usage of coal for electricity remains constant at the year 2000 level of .861 ton per year.
2. Each person in the U.S. uses same amount of electricity.
3. The U.S. uses 17% of the world coal supply.
4. The world coal supply in the year 2000 is estimated to be 8.471x106 million tons.
5. The U.S. uses approximately 85% of its total coal supply for electricity.

### Application

1. Use the Village population function M(t) and assumptions #1 and 2 to write the function C(t) which can be used to predict the rate of coal usage for electricity for the entire Village at time t, t = 0 in 2000.
2. Use the integral to write the function TC(t) which can be used to estimate the total coal used for electricity for the Village x years from now.
3. Use assumptions #3 and 4 to determine the U.S.'s share of world coal.
4. You may have derived the linear function used to predict the U.S. population in a previous Study. This function is P(t) = 2.303t + 273.739 million people in 2000 + t. Use this to find the year 2000 population of the U.S.
5. Use assumption #2 to figure the Village's share of the present U.S. coal. This will be the percentage determined by the ratio of the year 2000 Village population to the U.S. population for the same year. (Use five decimal places for the ratio.)
6. Use assumption #5 to find the amount of coal the Village uses for electricity.
7. Now, use your answers to #3 and #5 to estimate the number of years before the Village will have no more coal to generate its electricity.

## PETROLEUM and GASOLINE

### Objectives

Determine:
1. the amount of petroleum used in the U.S.;
2. U.S. share of world petroleum supply;
3. the Village share of U.S. petroleum supply;
4. the total amount of petroleum used for gasoline for the Village; and
5. the number of years before the Village's petroleum for gasoline runs out.

### Assumptions

1. The per-capita usage of petroleum for gasoline remains constant at the year 2000 level of 2.012 tons per year.
2. Each person in the U.S. uses same amount of gasoline.
3. The U.S. uses 30% of the world coal supply.
4. The world coal supply in the year 2000 is estimated to be 0.172x106 million tons.
5. The U.S. uses approximately 25% of its total coal supply for electricity.

### Application

1. Use the Village population function M(t) and assumptions #1 and 2 to write the function L(t) which can be used to predict the rate of petroleum usage for gasoline for the entire Village at time t, t = 0 in 2000.
2. Use the integral to write the function TL(t) which can be used to estimate the total amount of petroleum used for gasoline for the Village x years from now.
3. Use assumptions #3 and 4 to determine the U.S.'s share of world petroleum.
4. You may have derived the linear function used to predict the U.S. population in a previous Study. This function is P(t) = 2.303t + 273.739 million people in 2000 + t. Use this to find the year 2000 population of the U.S.
5. Use assumption #2 to figure the Village's share of the present U.S. petroleum. This will be the percentage determined by the ratio of the year 2000 Village population to the U.S. population for the same year. (Use five decimal places for the ratio.)
6. Use assumption #5 to find the amount of petroleum the Village uses for gasoline.
7. Now, use your answers to #3 and #5 to estimate the number of years before the Village will have no more petroleum to produce its gasoline.

## WATER (extra credit)

### Objectives

Determine:
• U.S. water usage;
• per-capita U.S. water usage;
• total water used by the village of The Village;
• when The Village water supply will begin to diminish; and
• when their water source will be totally depleted.

### Assumptions

1. The per-capita usage of water remains constant at the year 2000 level of 172 gallons per day.
2. Each person in the U.S. uses same amount of water.
3. The Village obtains its water from a reservoir, called Lonesome Lake.
4. Lonesome Lake is fed by the Nizhoni River that runs consistently all year at the rate of 15 million gallons per day.
5. There are also other towns and villages downstream from the reservoir, so the Village cannot use up all the water from the river. A government regulation states that the Village can only withdraw 20% of the streamflow, thus allowing a minimum of 80% to continue downstream.
6. The capacity of the reservoir is 7.5 billion gallons.
7. Lonesome Lake remains full until it starts to become depleted from overuse.

### Application

1. Use the Village population function M(t) and assumptions #1 and 2 to write the function MW(t) which can be used to predict the rate of water usage for the entire village of The Village at time t, t = 0 in 2000.
2. With a growing population and a fixed reservoir capacity, it is suspect that adequate water will be not be available in the future. To estimate this date, we note that Lonesome Lake will begin to drop when the rate at which water that enters the lake equals the rate at which it leaves. The amount of water flowing into the reservoir is the streamflow of the Nizhoni River. The amount flowing out will be the amount required by regulation together with the amount withdrawn by the Village. Therefore, to find out when Lonesome Lake starts to drop, it is necessary to solve the equation Rate In = Rate Out.
3. Use assumptions #4 and 5 and the function MW(t) to write this equation and then solve it to estimate when Lonesome Lake will begin to drop.
4. We will estimate how long it will take for it be become totally dry once it begins to drop. This prediction can be obtained by solving the equation Reservoir Capacity + Total Water In - Total Water Out = 0. Here, "total water in" and "total water out" should begin at the time the reservoir starts to empty. Use assumptions #4, 5, 6, and 7 and the function MW(t) to write this equation and then estimate when Lonesome Lake will be totally dry.

### Reflection

1. Do you think the models derived here are an accurate representation of a community?
2. How do you think this might generalize to a country? The world?
3. Can you think of other resources/commodities that you might add to the ones studied here?
4. What do you think that these problems of depletion of resources could be solved?
5. How did mathematics help in the analysis of data in this study?
6. How could mathematics be used in finding solutions?