1: Streamflow Prediction
This module is the first
of three which are designed to lead to the prediction of streamflow
for a river in a particular region. Temperature obviously affects the
type of precipitation we get, and the amount of precipitation affects
the amount of water in the river. In this module, we will construct
a model for average temperatures in the region.
Observations of temperature
trends, visualization of a model
- Write down average monthly
temperatures for the area in which you live. You can estimate these
or consult a website for accurate data.
- Draw a coordinate system
with months listed on the horizontal axis and temperature on the vertical
axis. Then plot your average temperatures on this system.
- What kind of function
would best fit the points on your graph?
- Would the function be
different for different parts of the world?
- Why do you think it might
be important to know about the average temperatures?
- How does temperature affect
- Sketch a graph of the
function that might describe average monthly temperatures in your
area for one year.
- How would mathematics
be useful in the analysis of temperature changes and the resulting
Learn or review mathematical concepts and skills to study temperature
change. See the menu at the left.
We will use regression to
fit these data with a quadratic function. Functions will be of the variable
t, 0 < t < 12, where t represents monthly time of
year. For example, t = 5 corresponds to the end of May, t = 4.5 corresponds
to mid-May and t = 9.3 corresponds to the point in time which is .3
into the month of October. Also, since these functions are cyclic (repeat
year after year) t = 0 and t = 12 correspond to exactly the same time,
the very beginning of January and the very end of December (midnight
on New Year's Eve).
In order to approximate the
average temperature for any month, we use the mid-month value for t.
For example, for average temperature for the month of May, we use t
= 4.5. The table below shows the t values to use to determine averages
for each month of the year. Following this, relevant temperature information
- The minimum average temperature
for the year is 0 degrees F and occurs at the end of January.
- The maximum average temperature
for the year is 64 degrees F and occurs at the end of July.
- Average annual temperature
is cyclic with period 12 months.
- It snows when the temperature
is 32° or below, and it rains when the temperature is above 32°.
Apply mathematical knowledge
and Tool Chest Applets to data provided to analyze average temperatures
Round answers to one decimal place.
- Make points from the
average temperature data provided; the first coordinate will be
the time of year t, and the second coordinate will be the temperature
at that time. Plot these points on the applet screen.
- Write the expression
that defines a sine function A(t) that fits the given data. Graph
the function on the same screen as the points to check to see if
the graph goes through the points; show an appropriate domain and
range. A(t) will be the average temperature at time t, 0 < t
<12. Use this function to answer the remaining questions.
- What will average temperature
be on May 15? On September 21?
- When will the average
temperature be 50 degrees? When will it be freezing?
- During what period
of the year will precipitation be in the form of snow? What period
will it be rain?
- On what day will the
average temperature be the warmest?
Examine the model and its implications
- Do you think that this
model would work for long periods of time, i.e. decades or centuries?
Why or why not?
- Could this function
be used to predict the temperature at a specific time on a given
- What are some properties
of the sine function that make it appropriate to use in this situation?
- What do you see as advantages
to the use of this function? Disadvantages?
- Do you know other functions
that might also be appropriate for use in this study?