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## Precipitation - Rain and Snow

### Part 2: Streamflow Prediction

This module is the second 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. A model for temperature was developed in the first module in this series. In this module, we will construct a model for average precipitation in the region.

Comprehension

Observations of precipitation trends, visualization of a model

Questions

1. Write down average monthly precipitation amounts for the area in which you live. You can estimate these or consult a website for accurate data. If it snows in your area, then record this as water from snow melt, the ratio is about 10:1, snow to water.
2. During what period of the year is precipitation primarily in the form of snow?
3. What units could be used to measure precipitation?
4. Draw a coordinate system with months listed on the horizontal axis and precipitation on the vertical axis.
Then plot your average precipitation on this system.
5. What kind of function would best fit the points on your graph?
6. Would the function be different for different parts of the world?
7. Why do you think it might be important to know about the average precipitation amounts?
8. How does temperature affect precipitation?
9. Sketch a graph of the function that might describe average monthly precipitation in your area for one year.
10. Do you think that average precipitation is cyclic, i.e., does it repeat its pattern year after year?
11. How would mathematics be useful in the analysis of precipitation?

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
We will measure precipitation in inches of water, even if the precipitation is snow. (Note: We specify "inches of water" to distinguish between inches of snow; 10 inches of snow only produce approximately 1 inch of water.) The function that describes precipitation 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 the function is, in general, cyclic (repeats 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 precipitation for any month, we use the mid-month value for t. For example, for average precipitation for the month of May, we use t = 4.5. The table below shows the t values to use to estimate averages for each month of the year.

 Jan. Feb. Mar. Apr. May Jun. Jul. Aug. Sep. Oct. Nov. Dec. .5 1.5 2.5 3.5 4.5 5.5 6.5 7.5 8.5 9.5 10.5 11.5

We will fit data provided with a sine function, denoted by P(t). Although the data provided is for average monthly precipitation, we will assume that P(t) gives the rate of precipitation at time t, 0 < t < 12. Thus, even though it is unrealistic, we are assuming that it is always raining in this part of the country; this assumption allows us to easily define a precipitation function.

Average monthly precipitation is, on a short term basis, a cyclic phenomenon repeating year after year. If a region gets 3 inches of rain in August as an average over many years, then a good guess is that the same region will get close to 3 inches in August of next year and the year after also. The function you derive in this section will be defined for one year but copies of its graph could be placed end-to-end to extend the domain for as many years as desired. However, the domain of this function will be only one year, i.e., from t = 0 to t = 12, or 0 < t < 12, where t denotes the time of year.

Finally, use the midpoint of any month to approximate the average precipitation for that month.

Assumptions

• The average maximum precipitation of 3 inches occurs at the end of May (t = 5).
• The average minimum precipitation of 1 inch occurs at the end of November (t = 11).
• Average monthly precipitation is cyclic with period 12 months.

Application

Apply mathematical knowledge and Tool Chest applets to analyze precipitation.

Questions
1. Make points from the data provided: the first coordinate will be time t and the second coordinate will be average precipitation for that time. Plot these points using the applet.
2. Let P(t) denote the average monthly precipitation in month t. Use the information provided in the Assumptions to find the sine function that fits the data points.
3. Graph the function P(t) on the applet screen with the data points showing one year, January through December. Check to be sure the graph goes through the points.
4. Use the function to determine the rate of precipitation at each of the times. A) t = 1.4 B) t = 11
5. Use the function to determine the average precipitation in: A) April; B) August; C) December
6. When will the rate of precipitation be 2 inches per month?

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. What do you see as advantages to the use of this function? Disadvantages?
3. What are some properties of the sine function that make it particularly appropriate to use in this situation?
4. Do you know other functions that might also be appropriate for use in this study?  