## Investigation of a Water Temperatures Function Using Excel

#### by Jennifer Roth

I took a cup of boiling water (taken off the heat source), and measured its initial temperature (time=0), and then recorded the temperature each minute for 30 minutes. In doing so, I compiled the following data.

In which column A contains the time in minutes, and column B contains the recorded temperature.
If we graph this relationship, we get the following picture.

Now we will try to create a function that will model this data.
It looks similar to an exponential function, so we will experiment with that to try to create a function that will model our data. I experimented with different exponential functions and the closest I could come, was the function y = (208-76)exp(-time*.03)+76. I calculated a measure of error between my model, and my observed data. This measure of error is equal to the sum of the square of the differences between the model and the observed. My statistic is fairly high, but this is the best I could refine it. When I plug in my function, I get the following figures in Column C, as compared to my observed data is Column B.

when I graph the two models, I get the following picture.

Series 1 is my observed data, and Series 2 is the modeled data.

Using my model:

If I predict the value for the temperature after 45 minutes I get 110.22 degrees.
If I predict the value for the temperature after 60 minutes I get 97.17 degrees.
If I predict the value for the temperature after 300 minutes I get 76.01 degrees.
I observed the room temperature to be 76 degrees, so my function is somewhat modeling my observed data, since it is levelling off at 76.