Cooled Data:

A Long Study of Watching Boiling Water Cool

In this assignment, I took a cup of boiling water and measured its temperature every minute for 30 minutes. I am going to attempt to come up with a function that will model the data. I did the experiment twice to see how much of a difference there would be depending on the size/shape of the container used. I first used a pan that had a diameter of 6.5 inches and a depth of 0.5 inches giving a volume of approximately 16.6 cubic inches and a surface area of 33.2 square inches. Next, I used a mug that had a diameter of 3 inches and a depth of 2 inches giving a volume of approximately 14.1 cubic inches and 7.1 square inches.


First, let's look at the data from the pan of boiling water:

We can use Newton's Law of Cooling to find an equation that fits the data.

Newton's Law of Cooling states:


t = time in minutes

k = constant


In the data above, the room temperature was 70 degrees. In order to find the constant, k, I took two of my corresponding data values, plugged them into the equation, and solved for k. I then took the average of those two k-values to get a more accurate value of k. Therefore, my equation is:

Now, I am going to use this function to predict the temperature after 45, 60, and 300 minutes.

T(45) = 77

T(60) = 72

T(300) = 70


Click here to see an Excel file showing my temperature reading compared to the temperature yielded in the formula. Then, you will see the square of each difference for each time in its own column. Then, those squares are summed and divided by 31 (there are 31 data points from t = 0 - 30) giving the standard deviation to be 7.16 (which seems to be very high -- I was hoping for a better fit).

Just for fun, I used Maple to find a polynomial of degree 6 to fit my data. I used seven data points that correspond to t = 0, 5, 10, 15, 20, 25, & 30 minutes and used the "interp" command to get the following polynomial:

The polynomial looks pretty messy but it is a good fit for several data points.

The main problem with this polynomial is that as time gets larger, my temperature should approach 70. This polynomial apparently curves upward above t = 30 because I got extremely large values for the temperature when plugging in 35, 40, and 45 minutes. For example, this polynomial gives a temperature of 686 degrees for time = 45 minutes which is obviously incorrect. It was fun to try!


The following table and chart accompany the experiment when done with a coffee mug as described. I wanted to see how much of a difference there would be when a different container was used.


From the data, we can see a significant difference in the time it took for the water to cool in the mug compared to the pan. We can deduce that the larger the surface area of the water, the quicker it cools.