# (or not...)

### By: Lauren Wright and Andy Tyminski

In this experiment, we took a pot of boiled water and measured its initial temperature (time t = 0) and then recorded temperature readings for the next 30 minutes. You would think that this would be an easy task for two graduate students...

After measuring the temperatures, we entered them on a spreadsheet and ATTEMPTED to use Newton's Law of Cooling to model the data and predict the temperature of the water after 45 minutes, 60 minutes, or 300 minutes.

First, we will show you the data that we collected with a room temperature of 74 degrees Fahrenheit.

 Minute Temperature Minute Temperature Minute Temperature 0 212 1 204 11 155 21 136 2 192 12 153 22 135 3 186 13 151 23 134 4 180 14 149 24 132 5 175 15 147 25 131 6 171 16 145 26 130 7 168 17 143 27 129 8 164 18 142 28 128 9 162 19 139 29 127 10 159 20 139 30 125

The formula for Newton's Law of Cooling is:

with t = time in minutes, = temperature of the air, = initial temperature, and k = experimental constant.

First, we set out finding what our k would be. We did this by calculating the value of k at five different data points, namely T(6), T(11), T(19), T(26), and T(29). Then, we took the average of those five k values to get an estimate of our experimental constant.

The k that we used in Newton's formula is k = 0.042901.

Here is a table of the values that were returned.

 Minute Temperature Minute Temperature Minute Temperature 0 212 1 206.204859462488 11 160.085646064217 21 130.05496264225 2 200.653078735481 12 156.470585069981 22 127.701003320969 3 195.334438222578 13 153.007334122942 23 125.445895630749 4 190.239147486782 14 149.68951814659 24 123.285488418732 5 185.357827228474 15 146.511029777924 25 121.215804854627 6 180.681492020207 16 143.46601812515 26 119.233035110248 7 176.201533766522 17 140.548877997494 27 117.333529346465 8 171.909705858361 18 137.754239587282 28 115.51379099466 9 167.798107992885 19 135.076958585322 29 113.77047032032 10 163.859171630773 20 132.512106711368 30 112.10035825692

As you can see, these values differ greatly from what we recorded. Here is a graph of the two sets of data.

So, the next step in our experiment was to see exactly how bad the error was.

We took the square of the difference for each time, summed the squares, and divided by the number of data points. This gave us an average error of approximately 60.3 degrees. For illustration, take a look at the graph of the residuals...

So, this experiment was a complete disaster using Newton's Law of Cooling. We could attribute this to many factors - a faulty thermometer or human error are the most likely causes (I mean, ANDY was in charge of reading the thermometer. So, anything is possible.)