Home
/
Calculus
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
 How large is the community
or city that you live in or near?
 Is your community
or city growing? If so, how fast?
 What are some commodities
that you depend on daily?
 How much land does
your community or city occupy?
 Is most of the surrounding
land agricultural, recreational, or industrial?
 What is your source
of food?
 What is your source
of water?
 What is your source
of electricity?
POPULATION
Objective
To determine the function
which can be used to predict the population of the Village
Acquisition
Assumptions
 There are 10,000 residents
at the beginning of the year 2000.
 The population is
growing by 300 people each year.
 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:
 area of land cultivated
for growing grain for the Village;
 total grain supply
for the Village;
 percapita grain consumption
for the US;
 total demand for grain
for the Village; and
 the number of years
before the the Village demand is greater than its supply.
Acquisition
Assumptions
 The current trend
for perhectare grain yield continues (see World Grain Yield).
 In 2000, the community
has 1500 hectares of land used for grain production.
 For each additional
person added to the village population, .04 hectare of land is
taken out of cultivation for development.
 The current decrease
in land cultivated for grain for the Village continues.
 The current increase
in the Village population continues.
 Each person requires188
kilograms of grain each year.
Application
 How much land will
be taken out of cultivation for development t years from now?
 Write the function
G(t) which estimates the amount of land will be used for cultivation
t years from now?
 Use the world grain
yield perhectare 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.
 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.
 When will the grain
production reach a maximum? What is the maximum production?
 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.
 Predict when the amount
of grain consumed will reach the amount of grain produced, or
when supply equals demand.
 Sketch graphs of both
the demand and supply functions on the same coordinate system,
and interpret the point of intersection.
 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:
 the amount of coal
used in the U.S.;
 U.S. share of world
coal supply;
 The Village share
of U.S. coal supply;
 total amount of coal
used for electricity for the Village; and
 the number of years
before the Village's coal for electricity runs out.
Acquisition
Assumptions
 The percapita usage
of coal for electricity remains constant at the year 2000 level
of .861 ton per year.
 Each person in the
U.S. uses same amount of electricity.
 The U.S. uses 17%
of the world coal supply.
 The world coal supply
in the year 2000 is estimated to be 8.471x106 million tons.
 The U.S. uses approximately
85% of its total coal supply for electricity.
Application
 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.
 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.
 Use assumptions #3
and 4 to determine the U.S.'s share of world coal.
 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.
 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.)
 Use assumption #5
to find the amount of coal the Village uses for electricity.
 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:
 the amount of petroleum
used in the U.S.;
 U.S. share of world
petroleum supply;
 the Village share
of U.S. petroleum supply;
 the total amount of
petroleum used for gasoline for the Village; and
 the number of years
before the Village's petroleum for gasoline runs out.
Acquisition
Assumptions
 The percapita usage
of petroleum for gasoline remains constant at the year 2000 level
of 2.012 tons per year.
 Each person in the
U.S. uses same amount of gasoline.
 The U.S. uses 30%
of the world coal supply.
 The world coal supply
in the year 2000 is estimated to be 0.172x106 million tons.
 The U.S. uses approximately
25% of its total coal supply for electricity.
Application
 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.
 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.
 Use assumptions #3
and 4 to determine the U.S.'s share of world petroleum.
 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.
 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.)
 Use assumption #5
to find the amount of petroleum the Village uses for gasoline.
 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;
 percapita 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.
Acquisition
Assumptions
 The percapita usage
of water remains constant at the year 2000 level of 172 gallons
per day.
 Each person in the
U.S. uses same amount of water.
 The Village obtains
its water from a reservoir, called Lonesome Lake.
 Lonesome Lake is fed
by the Nizhoni River that runs consistently all year at the rate
of 15 million gallons per day.
 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.
 The capacity of the
reservoir is 7.5 billion gallons.
 Lonesome Lake remains
full until it starts to become depleted from overuse.
Application
 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.
 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.
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.
 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
 Do you think the models
derived here are an accurate representation of a community?
 How do you think this
might generalize to a country? The world?
 Can you think of other
resources/commodities that you might add to the ones studied here?
 What do you think
that these problems of depletion of resources could be solved?
 How did mathematics
help in the analysis of data in this study?
 How could mathematics
be used in finding solutions?
Return
to Earth Math Home Page

