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.
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).
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.