**Acquisition**

Mathematical Topics

The mathematical topics required for this study are listed in
the menu to the left. Click on the topics if you need to learn
more or refresh your memory.

Information/Assumptions

Since the mid thirties, carbon dioxide concentration has increased.
An increase in carbon dioxide concentration corresponds to an
increase in average global temperature, and this in turn leads
to increased ocean level. The exact mechanisms are very complicated
and there is real disagreement among scientist as to exactly
what the effect is. In this module we will make certain assumptions
and see what consequences we can predict. You can then choose
other assumptions and see what outcomes would follow. Since
1979, scientists have generally agreed that a doubling of atmospheric
carbon dioxide increases the earth’s average surface temperature
by 1.5-4.5°C (3-8°F) (http://www.epa.gov/global warming). There
is less agreement about how this changes ocean levels, but we
will use a "moderate" assumption that a 3°C increase in global
temperature raising ocean levels approximately 0.3 meters (about
one foot).

Assumptions

1. The trend in atmospheric CO2 concentration derived in the
previous part continues. If you did not complete the first part,
click here to retrieve the function
which describes the trend.

2. Doubling the pre-industrial
level of CO2 corresponds to a 3°C increase in average global
temperature

3. The relation between
CO2 concentration and temperature increase is linear.

4. A 3°C increase
in average global temperature corresponds to a 0.3 meter (about
1 foot) increase in ocean level.

5. The relation between
temperature increase and rise in ocean level is linear.

Objectives:

to predict the change in global temperature from pre-industrial
time; and

to predict ocean level change from pre-industrial time.

Application

It is estimated that an increase in carbon dioxide concentration
corresponds to a change in global temperature and a change in
global temperature corresponds to a change in ocean level. The
relations can be shown with this diagram, where CO2:

Recall (from module
I) that according to our model carbon dioxide concentration
was at the pre-industrial level of 280 ppm in 1935 (t = -65).
This will be our base level for global temperature and ocean
level:

Objective 1 First
we will write a linear function to describe global temperature
change as a function of atmospheric carbon dioxide concentration
where the temperature change is 0°C when the carbon dioxide
concentration is at the pre-industrial level of 280 ppm, that
is, since 1935. Use a (C,GT) coordinate system, where

C = CO2 concentration
in PPM,

and

GT = global temperature
change since 1935.

GT measures how much
the temperature changes from the pre-industrial times. According
to our model for CO2 concentration, the concentration was at
the pre-industrial level of 280 ppm in 1935 and so we assume
that T = 0 and C = 280 in 1935.

1. To write the equation
we need two points.

a. The first point
comes from the pre-industrial level which was 280 PPM and corresponding
global temperature change of 0°C. Write the coordinates of that
point, using a (C,GT) coordinate system.

b. The second point
comes from our ”doubling” assumption: a doubling of the concentration
corresponds to an average global temperature increase of 3°C.
Write the coordinates of that point.

2. Write the equation
of the line through the two points you determined in #1.

3. Interpret the
slope of the line in #2.

Now we write global
temperature change as a function of time.

4. Replace C in your
equation by the function C(t) you derived previously. Call this
function G(t). What does this new function describe?

5. What is the slope
of this function?

6. What does this
slope describe?

Objective 2 Now we
will write a linear function to describe change in ocean level
as a function of change in temperature (that is, if you know
how much the temperature changes, predict the corresponding
change in ocean levels. Use a (OL,GT) coordinate system, where

OL = change in ocean
level since 1935 (in meters)

and

GT = global temperature
change since 1935.

Here we assume that
OL = 0 and GT = in 1935.

1. To write the equation
we need two points.

a. The first point
comes from the 1935 level which was an ocean level change of
0 meters and corresponding global temperature change of 0°C.
Write the coordinates of that point, using a (C,T) coordinate
system.

b. The second point
comes from assumption #4: a 3°C increase in average global temperature
corresponds to a 0.3 meter (about 1 foot) increase in ocean
level.

2. Write the equation
of the line through the two points you determined in #1.

3. Interpret the
slope of the line in #2.

Now we write ocean
level change as a function of time.

4. Replace GT in
your equation by the function G(t) you derived previously. What
does this new function describe?

5. What is the slope
of this function?

6. What does this
slope describe?

Reflection

Revisiting the Problem Although most scientists agree that there
is a relationship between increased carbon dioxide concentration
and rising global temperature, the relationship is complicated
and there is not agreement on the amount of increase.

Scenario 2 Now assume
that doubling of atmospheric carbon dioxide from the pre-industrial
era level of 280 PPM increases the earth’s average surface temperature
by 1.5°C. (This is the “low” assumption.) Redo the problem with
this assumption.

Scenario 3 Now assume
that doubling of atmospheric carbon dioxide from the pre-industrial
era level of 280 PPM increases the earth’s average surface temperature
by 4.5°C. (This is the “high” assumption.) Redo the problem
with this assumption.

Additional Questions

1. What assumptions
went into your model(s)?

2. How valid do you
think these assumptions are?

3. What events would
change your prediction(s)?