Argen Hicks
EMT 668
Summer 1996

This is a write-up of Assignment 12 #6. For this exploration, I boiled water and recorded temperatures each minute for the next 30 minutes as it cooled. Perhaps because of the high altitude in extreme northeast Georgia, or perhaps because of an inaccurate thermometer, the water boiled at 195° F. The data was entered in a ClarisWorks spreadsheet.

Although there were drops of more than 1° per minute early on, it did appear that a drop of 1° per minute could be expected after about 23 minutes. If that were the case the temperature could be expected to be 110° after 45 minutes and 95° after 60 minutes. The spreadsheet showed that.

Here is the first graph of the data I had collected using a line chart.

I tried matching an equation of a line to the data. The closest match (I thought) was with the equation y = 195 - (2.75x). Even that was not very close, the reason being that the temperature actually drops off quickly to begin with, approaches a steady change and then evens off. A line has a more constant drop. That graph is shown below.

The next bit of data I applied was the room temperature at the time of the experiment; it was 66° F. Assuming that the water would not become cooler than the room temperature, I used the spreadsheet to predict the time the water would be expected to reach 66°. Given the data, that time would be after approximately 89 minutes. I graphed that data using a line graph for time values up to 150. The last temperatures, of course, remained 66° and probably would even after 300 minutes.

A scatter graph of the data looked very similar:

Then, I tried an x-y chart, which turned out to be very similar to the line graph.

Now, if the data did not take into consideration the room temperature, then the graph might continue to fall. I set up the data that way in the spreadsheet to see what would happen. An X-Y graph shows the data looking more like a "typical" line with the temperature reaching 0° at 155 minutes.

After doing this graph, I decided I did not like my choice of equation. So I compared several times and values for different equations:

The first two times are actual times data was recorded. The next two and the last are ones given in the assignment to predict, and 89° is when the temperature was predicted to reach room temperature. The * represents the room temperature. The equation y=195-1.5 seems to fit better. Here is the graph:

Here is what happened when I took the sum of the squares of the differences between the three equations and the data:

What appears to happen in the first column is that y = 195 - 2.75x would be the closest fit with the data. For the first 30 minutes that may be so, but over time (300 minutes), y = 195 - 1.5x may be the better fit. Of course, I was looking at time values over 30 minutes, which is what caused me to finally choose the latter equation. This statistic could be used to refine the model for the data.

When I graphed the log of the data points, the graph looks, in shape, like the graph of
y = logax.

Here is another one with the scales reversed:

These log graphs seem better models than the graph of the lines because they show better the gradual changes in temperature in the data. I made several attempts to find an exponential equation to model the data without success. The one I liked best can be found in the Write-up #5 folder under "Exp. graphs, 2": y = [1 / (3^(x-10)] + 66.

The End.