                  Review Topics Home Algebra with Equations Precipitation Snowmelt and Streamflow

## 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.

1. What kind of function would most accurately describe temperature during the months from April through September?
2. Would the function be different for different parts of the world?
3. What function would accurately describe temperature over only one day?
4. Why is it important to know about the average temperatures over a period of time?
5. How does temperature affect precipitation?

## 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

Round answers to one decimal place.

1. 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.

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

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. Could we use this function to predict average daily temperatures during the other months (January, February, March, October, November or December)?
4. What do you see as advantages to the use of this function? Disadvantages  