As can be depicted through the above tables, the results as collected did not vary so much; and the overall data set remained fairly the same. Nevertheless, I averaged out each data set for each specific time slot between the three trials using the formula:

As can be seen in the above figure, the graph shows an exponential curve, between the values of temperature of the coffee and the time in minutes. verifyErrors }}{{ message }}{{ /verifyErrors }}{{ 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. 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: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: 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 Experimentally Determined Values of the Temperature of the Coffee (℃) Over a Fixed Period of Time (Two Hours) Through a Series of Five Minutes Intervals

Bourne, Murray. “3. The Logarithm Laws.” Intmathcom RSS, www.intmath.com/exponential-logarithmic-functions/3-logarithm-laws.php. The graph thus holds a negative correlation, so logically it would mean that the power of the function must be negative.Figure 1.6: Graph displaying the original data versus the original data graphed according to the equation T = 24 . 5 + 54 . 5 e – 0 . 0274 t Feldman, Joel, et al. “CLP-1 Differential Calculus.” Plimpton 322, www.math.ubc.ca/~CLP/CLP1/clp_1_dc/sec_newtonCooling.html. 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.