Write-up V
EMT 668

BOILING WATER PROBLEM
by
Brian Seitz and Beth Richichi

This is to be an exploration of boiling water temperatures. The process was to to bring 2 cups of water to a boil and then transfer the water to a separate cup for cooling. The initial temperature of the boiling water was recorded as 212 degrees fahrenheit and then temperatures of the cooloing water were taken every minute for 30 consecutive minutes. The following is a chart for time (column 1) and temperatures observed (column 2). Following that is the graph of the same data.

Graph of time (x-axis) vs. temperatures observed (y-axis)

The next step was to attempt to construct a formula to fit our plotted data. Dr. Wilson had previously explained to us that a exponential equation would be used for this problem, and using Mocrosoft Excel, the process was rather easy. Generating a function with Excel saves time and effort. We first thought that my equation was in the form of y= a * e^x + c. We needed our intial value to equal 212 and after 300 minutes, we hypothesized that the temperature would be approximately equal to the room temperature of 69 degrees. The formula we tried was (212-69)*(e^-.06*temp) + 69. The -.6 was a random choice. To check the validity of our equation, we took the square of the difference for each time, summed the squares and divided by 31 ( data points). This statistic turned out to be over 250 for the given equation. Now, our job was to play with the values until a graph similar to ours was generated and the statistical value was as close to 0 as possible.

After many attempts, the statistic could not be brought down lower than about 120. We had difficulty with the initial few values because the drop in the first minute was so great, 19 degrees. After the first minute, the numbers dropped more consistently. So, we decided to omit the first minute in our equation and begin with 193 degrees in our equation. The equation that dropped the statistic to the lowest value was (193-69)*e^(-.038 * temp) + 69. The statistic equaled 10.23. The following is a graph of the time (column 1), temperature recorded (column 2), the equation values (column 3), the difference between the equation values and the actual temperature recorded (column 4) and the squares of the difference (column 5).

Table for Cooling Water

Graph of time (x-axis) vs. temperature (y-axis)
Series 1 (purple) is observed values
Series 2 (pink) is predicted values

The next graph is a graph of the time vs. the predicted values. The three extra plotted points are the predicted temperatures for 45 minutes, 60 minutes, and 300 minutes. After 45 minutes, the equation predicts that the temperature will be 91.4 degrees, and after 60 minutes the temperature will equal 81.7 degrees fahrenheit. The equation was constructed so that the final value would equal the room temperature at the begining of the lab and as seen, it does. The temperature after 300 minutes is 69 degrees fahrenheit.

Graph of time (y-axis) vs. predicted values (x-axis)
y =(212-69)*EXP((-0.038)*TIME) + 69

To sum up, this exploration of boiling water is an excellent exercise to experiment with Excel as an effective mathematical tool. The spreadsheet software completes the computations for a person immediately and allows one to freely adjust formulas to fit results. In this particular problem, the spreadsheet effectively calculated a statistic to evaluate our formula's accuaracy. This process would have been tedious without the technological tool.

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