Leaky Faucet: A Mathematical Model

Objective: This data-based demo illustrates how linear functions can arise from real problems.  This demo makes use of a graphing calculator for data analysis.

Level:  College Algebra, Precalculus or any course in which modeling is incorporated.

Prerequisites: It would be helpful for students to have a knowledge of linear equations, however,  the demo could be used to motivate the study of linear functions.  Students should have basic graphing calculator skills.

Materials:

• paper cups
• a sharp instrument for making a hole in the cup (needle, pin, sharp pencil)
• graduated cylinder, calibrated in mm or ounces
• watch or clock with second hand
• a blank table to record data
• graphing calculator
Instructor's Notes:
• Instructor selects two student assistants.
• One assistant will hold the cup over the graduated cylinder, fill the cup with water, makes a hole in the bottom of the cup, and holds her finger over the hole until the instructor starts the experiment.
• When the student starts the drip, the instructor calls five second intervals and the second assistant records the amount of water collected in the cylinder.
• After the cup is empty, the data that was collected is analyzed graphically.
• In this sample data, the TI-83 was used to analyze the data.  Instructions for calculator input/analysis may be different if a different calculator is utilized.
Sample Data:

 Time 0 5 10 15 20 25 30 35 40 45 50 55 60 Amount(mm) 0 7 15 21 28 34 42 50 56 63 70 75 81

Calculator Input (TI-83) for Sample Data:

In the STAT menu, choose EDIT and enter the data in L1 and L2.

Scatterplot of Sample Data:

In the STATPLOT menu, turn on Plot 1, choose a scatterplot using L1 and L2.  The window settings for this data should be

Pressing GRAPH will show the following scatterplot.

• Many students are surprised to see that the data is linear.  Questions about why it is linear are appropriate.
• Discuss how the model can be used to predict how much water a leaky faucet might waste in 1 hour, 2 hours, 24 hours.
• Depending on the data collected, the scatterplot may give the appearance of being "perfectly" linear.  This scatterplot does give that appearance.  One way to demonstrate that not all points lie on a line is to have the calculator graph the regression line on top of the scatterplot.  Even in a class where linear regression isn’t a topic, you can use the command LinReg L1,L2, Y1 to get a regression line and discuss the attributes of the line and the data without going into the details of linear regression.
• If this demo is used as a lab project students will probably observe that different experiments produce different mathematical models because the slope of the data line is dependent upon the size of the hole.  Before telling the students why the slopes are different, ask them to speculate on why results from different experiments are not the same.  You may instruct one group to compare their experiment with that from another group.

This graph shows that not all points are included on the line and how our model isn’t a 100% guarantee of accuracy.

Credits:  This demo was submitted by
Dr. Kathleen Cage Mittag
Department of Mathematics and Statistics
University of Texas at San Antonio

Dr. Sharon Taylor
Mathematics and Computer Science Department
Georgia Southern University

and is included in Demos with Positive Impact with their permission.

Originally posted by LFR on 20 December, 1999 counter information was lost.

DRH  Last updated 1/30/2005

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