### Comprehension

*Open a blank document
in your word processor and record your answers to these questions.
Copy each question into the journal and type your response below
in a different color or font. Be sure to write in complete sentences
and express your ideas so that others can understand. Save your
work and keep this window open throughout this study so you can
easily record other questions and answers. *
**Questions**

- What foods do you
eat that use grain directly? indirectly?

- Do you think that
the production of grain is increasing or decreasing?

- What factors do
you think affect the amount of grain being produced each years?

- How do you think
changes in grain suply might influence your life?

- What effect does
new methods of agriculture have on grain supply?

- What effect does
increased urbanization have on grain supply?

- How do you think
mathematics can be used to study grain supply?

**Acquisition**

**Mathematical Information**: For a brief review of each concept,
click on the appropriate link.

**Information/Assumptions
**

Some of the factors you might have mentioned that influence the
amount of produced are: the amount of land cultivated (area) and
the amount of grain produced per fixed area of land (yield). In
this module we examine each factor separately and then combine
them to create one function to describe the total world grain
supply. We will use metric units in this study. Area is measured
in hectares; one hectare is 10,000 square meters or about 2.47
acres. When we write "ton" we will be referring to metric
ton; one metric ton is 1000 kilograms or about 2205 pounds. Complete
each section below; be sure to save the results for each factor.

**Objectives To Determine**:

- area of land cultivated
for grain
- average yield per
hectare
- world grain supply
- when supply reaches
a maximum

**Application**

Retrieve the functions Y(t) and A(t) you derived previously. We
will use them to predict how much grain will actually be available.
Here Y(t) is the yield, measured in tons per hectare, and A(t)
is the area cultivated, measured in billion hectares, both for
t years from the year 2000.

**Assumptions**

- The current trend
for per-hectare grain yield continues.
- The current trend
for land cultivated continues.

**Questions**

Use the functions Y(t)
and A(t) to construct a function to describe the total amount
of grain produced in year t. What are the units? Graph the function
from now through 1980 to 2075.

Estimate the total
grain supply for 2000 and for 2040. Identify the corresponding
points on the graph.

Predict when the grain
supply will be 2000 billion kilograms and label the corresponding
points on the graph.

Determine the rate
of change of the grain supply for the indicated years. Discuss
the significance of the sign for each. a) 2000 b) 2045 d) 2060

Is the grain supply
increasing at a faster rate in 2000 or in 2020? Explain your answer
in terms of the graph of the function as well as numerically.

Use the derivative
to decide when the grain supply is increasing and when it is decreasing.

**Reflection**:
Assessing the method, solution and implications

Do you think that the
functions used are appropriate? If not, what might be better?

Do you think that the
solution is reasonable? Why?

Do you think that the
supply function is accurate? Do you think that it is reasonable
to assume that this trend will continue? If not, what do you think
might happen?

Would it be appropriate
to construct an alternative model?