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However, with the equation produced above, despite the discrepancies, it represents an extremely accurate model of the cooling coffee. Thus I have fulfilled my objective of determining an equation to model the cup of cooling coffee through implementing the laws of logarithms and simultaneous equations. Through this investigation, I was able to apply math to a real life situation outside of the classroom environment. Experimentally Determined Values of the Temperature of the Coffee (℃) Over a Fixed Period of Time (Two Hours) Through a Series of Five Minutes Intervals After the data was collected in the tables and averaged out between each trial, as shown above, the next stage logically was to graph the results and produce a graph of my own. The scatter plot below was produced using Microsoft Excel, with the time in minutes (over 5 minute intervals) – in the graph the scale was deduced to 20 minute intervals - plotted on the X axis and the temperature (℃) plotted on the Y axis. The graph below thus indicates the experimentally determined values of the temperature of the coffee over a series of two hours (with the result The line, as depicted through the graph, can be seen to be accounted for small uncertainties in the line created by real world factors. The equation for the line of best fit was generated: Bourne, Murray. “3. The Logarithm Laws.” Intmathcom RSS, www.intmath.com/exponential-logarithmic-functions/3-logarithm-laws.php.

If you need assistance with writing your essay, our professional essay writing service is here to help! Essay Writing Service According to online publications, integration was used through Newton’s law of cooling, and the fact that is directly proportional to the difference between the temperature of the soup T(t) and the ambient temperature Tc . Our academic experts are ready and waiting to assist with any writing project you may have. From simple essay plans, through to full dissertations, you can guarantee we have a service perfectly matched to your needs. View our services As can be seen below, the data are basically identical, with the only difference being that the equation generated values intersect the y-axis at a lower value. The asymptote is the same, with the mean error of the differences in values being 0.11℃.

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Due to the temperature of the room being 24℃, the entire graph was translated up the y axis by a degree of 24, and thus there was a translation constant: and 35.4 degrees. I then used two results from the data to create a pair of two variable equations where Although seemingly accurate, I wanted to further explore and delve into the actual math of Newton’s law of cooling, as stated in the introduction, which states that: It can be seen from the graph that the actual data (orange) tends to be slightly lower than the temperature projected by the original data. I believe that this could

Feldman, Joel, et al. “CLP-1 Differential Calculus.” Plimpton 322, www.math.ubc.ca/~CLP/CLP1/clp_1_dc/sec_newtonCooling.html. is always equal to 24.5, there could have been major inaccuracies as as the coffee cooled down, the air surrounding it would have immediately warmed up as the heat diffused away from the cup (Murray, 2012). This then explains why the differences of the first and third equation were so great, as between the two, a lower ambient temperature was always predicted, thus inaccurately assuming that the rate of change was faster. I can safely revise that 1.41 hours will be the proper time before my coffee is undrinkable. As precise as this value is however, it only represents how long I can leave my coffee out in a room with 24℃ temperature. A second investigation would be interesting to see if an equation could be produced that takes into account the surface area of the cooling body and the changing ambient temperature. As indicated through the graph, the two lines are basically identical, and the equation produced greatly resembles the first equation produced. Figure 1.7: Original data compared to T ( t ) = 24 + ( 55 . 9 ) e – 0 . 0279 t

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To answer my original research question, “How long can I revisit my coffee before it becomes undrinkable?”, I can achieve this by using the second equation: To translate this equation to represent the original, it requires a translation on the x axis (k) and a gradient much smaller 1(a), so it can be assumed that: