Assignment Twelve:

The Cooling Function

The purpose of this exploration is to determine a cooling function that best represents the given data.

The given data is that of the cooling of boiling water over a period of thirty minutes. That is, time versus temperature.

The first step was to enter the data into an excel spreadsheet in order to create a graph of time vs temperature.

 Time (min) Temp (Deg F) Time (min) Temp (Deg F) Time (min) Temp (Deg F) 0 212 11 163 21 143 1 205 12 161 22 141 2 201 13 159 23 140 3 193 14 155 24 139 4 189 15 153 25 137 5 184 16 152 26 135 6 181 17 150 27 133 7 178 18 149 28 132 8 172 19 147 29 131 9 170 20 145 30 130 10 167

Once this was done then the next step was to try various functions as theoretical curves that best fit the data. This was done by 'sight' initially. Each time the graph was made anew and the curve was added. The first curve tested was the linear function. This option was eliminated merely by the 'sight' test. The fit was poor.

The next graph used was a exponential. This fit somewhat better than the linear but it too didn't seem to fit best. The last one was the polynomial curve. The raw data seemed to mimic this type of curve best and was the most promising to the investigators. However, when the data underwent the number crunching, the temperature increased in value as time continued. The temperature was predicted after 45, 60 and 300 minutes. The temperature reached beyond 5,000 degrees F after 5 hours.

-Click Polynomial or Chart 6 to see this graph-

This led us to leave the polynomial function and research the exponential curve. This fit much better. The data made sense after it was applied as well as the predictions.

-Click on Chart 5 to see the exponential graph-

Next, a calculation of the measure of the error between the model and the observed dat was performed. The square of the difference of each time, summing this information and dividing by the number of data points. A refinement of the funcion was accomplished.

 Best Fit Cooling Function Y = 199.29exp^(-0.0155*x)+/-17.92705304 Error = 17.92705304

Click on Sheet 1 to see the data - the raw data, error and the theoretical curve.