Exploration of Water Level Data:
AN ALGEBRA 2 PROJECT

Data Source: http://www.swf-wc.usace.army.mil/index2.htm

| Guiding Question | Project Requirements | Data Table | Rubric |

Question: How are piecewise defined functions used as mathematical models of real-world data?

Project Requirements:

1. Using the Canyon Lake, TX data of water levels for the month of August 2002 [Click for data tables], plot the data as a scatter plot. Let the independent variable be t, time, with August 1 = 1, August 2 = 2, and so on. Let the water height in feet be a function of time, H(t).
    
2. Enter the data into a spreadsheet or graphing calculator and find a linear regression model for the entire data set. Record the model's equation and the value of r, the coefficient of correlation. [If using a spreadsheet, such as Excel, the program will only calculate the square of r. Convert this to r.]
    
3. If | r | is very close to 1, the data is a good fit with a linear pattern. You should notice that the data does not fit a simple linear pattern. A better model would be a linear equation for the first, nearly horizontal portion of the data and then a second linear equation for the more steeply sloped section. Therefore, calculate a pair of regression lines for the data, one for the first part of the data and one for the second part. Round any "messy decimals" to some appropriate level of accuracy. Record the coefficients of correlation (note whether you used the calculator or a spreadsheet). Explain your decision where to "cut off" the data.
    
4. Express your two regression models as one piecewise defined function, H(t).
    
5. Find the intersection point of the two pieces of your piecewise defined function. Explain the method you used to find this information.
    
6. Compare the actual data and output from the piecewise mathematical model by creating a table with the following information.
   

   Time, t	Actual Water Height, A(t)	Predicted height, H(t)
   1
   7
   14
   21
   28
   31

   
   Explain the significance of the information in the last column.
    
7. Find the "average rate of change" in the water level at the dates Aug 5, 10, 15, 20, 25, and 30. [Use the actual data.]
    
   The average rate of change, m, of a function over the interval [a, b] is given by . This formula is called a "difference quotient."
    
   To find the average rate of change in the water level at the different dates, use symmetric difference quotients by using the (x, y) pairs immediately before and after each date. For example, to find the average rate of change in water level for August 5, use the data pairs (4, 943.8) and (6, 943.7). Indicate the units of measure.
    
8. At what date does the water level appear to be changing at the greatest rate? … the slowest rate?
    
9. The water level seems to be falling as time goes on. Use your prediction model, H(t), to predict when the level falls below a level of 900 feet. On what date would this occur? Discuss the accuracy of this prediction and show the method you used to perform your calculation.
    
10. Observations: What did you learn as a result of this activity? What aspects did you find most interesting, most difficult, least difficult?
    

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