Streamflow
Temperature
Part
1: Streamflow Prediction
In
this module we will construct a model for average temperatures
for spring through fall.
Comprehension
Observation
of temperature trends, visualization of method of study.
 What kind
of function would most accurately describe temperature during
the months from April through September?
 Would the
function be different for different parts of the world?
 What function
would accurately describe temperature over only one day?
 Why is
it important to know about the average temperatures over a period
of time?
 How does
temperature affect precipitation?
Acquisition
Mathematical
Topics
Learn or review
mathematical concepts and skills to study temperature change.
See the menu at the left.
Information
The table
below gives the average temperature for the indicated month in
a certain location in the southwest.
Month 
April

May

June

July

Aug

Sept

Average
Temp (degrees F) 
25

36

53

61

61

47

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 midMay 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
midmonth value for t. For example, for the average temperature
for the month of May, we use t = 4.5. The table below shows the
values to use to determine averages for each month of the year.
Jan.

Feb.

Mar.

Apr.

May

June

July

Aug

Sep.

Oct.

Nov.

Dec.

0.5

1.5

2.5

3.5

4.5

5.5

6.5

7.5

8.5

9.5

10.5

11.5

The function
will be denoted A(t); A(t) will be the average temperature at
time t, 3 < t < 10.
Application
Apply mathematical
knowledge and Tool Chest Applets to data provided to analyze average
temperatures
Questions
Round answers
to one decimal place.
1. Make points
from the average temperature data provided; the first coordinate
will be the midpoint of the month and the second coordinate will
be the average temperature for that month. Plot these points on
the applet screen.
2. Use regression
to fit the given data with a quadratic function A(t). Graph the
function showing an appropriate domain and range. Use this function
to answer the remaining questions.
3. What will
be average temperature be on May 15? On August 15? On April 6?
On September 21?
4. When will
the daily average temperature be 50 degrees? When will it be freezing?
5. On what
day will the average temperature be the warmest?
Reflection
Examination
of the model
 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 day?
 Could we
use this function to predict average daily temperatures during
the other months (January, February, March, October, November
or December)?
 What do
you see as advantages to the use of this function? Disadvantages