Bourne, Murray. “3. The Logarithm Laws.” Intmathcom RSS, www.intmath.com/exponential-logarithmic-functions/3-logarithm-laws.php. This section of the investigation caused some mayhem, as I had a lot of issues within this particular area of the report. I had produced an equation and had found it difficult to solve it. 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

and 35.4 degrees. I then used two results from the data to create a pair of two variable equations where

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 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℃. 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. If you need assistance with writing your essay, our professional essay writing service is here to help! Essay Writing Service 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:

Experimentally Determined Values of the Temperature of the Coffee (℃) Over a Fixed Period of Time (Two Hours) Through a Series of Five Minutes Intervals 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:The final equation produced seems to produce a graph that matches the original data, but it can be seen with the results that the rate of cooling in the first 50 seconds of cooling is underpredicted. 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: verifyErrors }}{{ message }}{{ /verifyErrors }}{{

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: It can be seen that the second equation, which was produced using lines of best fit, produced the most accurate model of the coffee cooling down.This is most likely due to the fact that the lines of best fit take into account the slight discrepancies in the real results that were caused by the environment. Since the third and first equation rely on the fact that 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 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

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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. Feldman, Joel, et al. “CLP-1 Differential Calculus.” Plimpton 322, www.math.ubc.ca/~CLP/CLP1/clp_1_dc/sec_newtonCooling.html. 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. According to online publications, integration was used through Newton’s law of cooling, and the fact thatAs 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 is directly proportional to the difference between the temperature of the soup T(t) and the ambient temperature Tc . 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.