Modelling before drinking it, or making a cup

Modelling a cup of cooling tea My inspiration to investigate this came from my indisputable favoritism of calculus over other topics, however I had to connect this to a real life situation. I had to find something physical and relevant to justify that what I was learning was real. It was difficult at first to find something related to the perplexing methods of calculus that were applicable to real life, but I finally settled on something I consume routinely and have a fervent admiration for. TeaThere are times where I have woken up to a cup of tea next to my bed and fallen asleep again before drinking it, or making a cup of tea and simply forgetting to drink it, and then having to consume a cold, significantly more undesirable beverage. Naturally I started thinking about how tea cooled, or anything cooled for that matter, and from prior knowledge I knew that the difference between the cooling body and the ambient temperature (room temperature) changed the rate at which the body cooled. Large differences between temperatures cause a large rate of change and smaller differences between the cooling body and the ambient temperature cause a smaller rate. From common sense I knew that tea couldn’t go below room temperature, which means the tea would approach room temperature as time goes on, eventually staying at a constant temperature which would only change if the room temperature changed. This sounds like an exponential decay pattern, which could be graphed. I aimed to model this rate of exponential decay and produce an equation which would allow me to see how much time I had before my tea became cold. To gather the data required to produce this graph I had to measure a cup of tea cooling down. This was easy to do by making a cup of tea and then using a thermometer, checking the temperature of the tea every 5 minutes for 2 hours. I started recording the temperature after the tea was completely made, so after the sugar was dissolved and the tea bag was taken out. The next thing to do was to graph the results. I used microsoft excel to plot a scatter graph with Temperature as the Y axis and Time as the X axis.It can be determined that:The power of the exponent is negative- Since there is exponential decay, the graph has a negative correlation, so the exponent must always be negativeWhen time (t)= 0, the temperature is 71.7°- The initial temperature of the tea before any cooling has taken place is 71.7°The room temperature is 20°- The temperature of the tea can never go below 20°From my observations I needed to construct a formula which could model this. Choosing e meant that I could find the constants needed to solve the equations later. Using temperature (T) and time (t), I made the equation for exponential decay:There is a translational constant because the graph was shifted up by 20°, making the equationThis produced the graphThis is not the correct formula, as the the graph of this formula doesn’t represent the original graph. To represent the original graph, it requires a translation on the x-axis (k), and a gradient smaller than 1 (a). This gives the equationTo make the formula more approachable, I rearranged and used natural logs.Then I plugged in two data points from my results to create a set of two variable equations.I used t=0 and t=50Subtracting equation A from equation B:However since there is exponential decay, a should be negativeSolving for k:Plugging these values into the original equation gives Graphing this givesThe graph seems to roughly represent the original data, however it can be seen that approximately the first 50 seconds seem to represent a faster decay, however it can be seen on the original graph that it is slower.To find the exact differences, I decided to put each temperature into the equation, plotting each data point in excel to get the following graph