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## Streamflow Temperature

### Part 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.

Comprehension

Observations of temperature trends, visualization of a model

1. Write down average monthly temperatures for the area in which you live. You can estimate these or consult a website for accurate data.
2. 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.
3. What kind of function would best fit the points on your graph?
4. Would the function be different for different parts of the world?
5. Why do you think it might be important to know about the average temperatures?
6. How does temperature affect precipitation?
7. Sketch a graph of the function that might describe average monthly temperatures in your area for one year.
8. How would mathematics be useful in the analysis of temperature changes and the resulting effects?

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 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 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
1. Round answers to one decimal place.
2. Make points from the average temperature data provided; the first coordinate will be the mid-point of the month and the second coordinate will be the average temperature for that month. Plot these points on the applet screen.
3. 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.
4. What will be average temperature be on May 15? On August 15? On April 6? On September 21?
5. When will the daily average temperature be 50 degrees? When will it be freezing?
6. On what day will the average temperature be the warmest?
Reflection

Examine the model and its implications
1. Do you think that this model would work for long periods of time, i.e. decades or centuries? Why or why not?
2. Could this function be used to predict the temperature at a specific time on a given day?
3. What are some properties of the quadratic function that make it appropriate to use in this situation?
4. What do you see as advantages to the use of this function? Disadvantages?
5. Do you know other functions that might also be appropriate for use in this study?  