Mathematical Modeling of the Rate of Coffee Cooling - Report Example

MATHEMATICAL MODELING OF THE RATE OF COFFEE COOLING. SUMMARY. Investigation of cooling rate of coffee over time is a mathematical modeling cases which can be solve using different models of mathematics. However, the most preferred method of solving the case without been biased is linear modeling which is elaborative and an easier way of concluding the final results. In conclusion, after the investigation of hot water coffee it was found to cool at a faster rate that the milk coffee, this can result to the conclusion that milk in the coffee holds temperature or heat for a long time compared to water. INTRODUCTION. An experiment was carried out to investigate the cooling of coffee of only hot water and that of a mixture of milk and water in a mathematical model. The investigation was carried out and data recorded with time intervals. Texas instruments CBL interface and an attached handheld TI-84 plus silver edition were used with a goal of creating a model that describes the process. After investigation of the cooling process and recording all the observation by the students, the following table was formulated. The table above shows the time coffee took to reach varying temperatures. The data from the experiment were then used to represent the experiment graphically as shown in the graph below where both observations where draw on the same axis. The horizontal axis (x axis) on the graph represent the time in seconds while the vertical axis (y axis) represent temperature in degrees Celsius. Using various mathematical models that include; Quartic model, linear model, quadratic model, cubic model and the logistics model, the data was used to determine the coffee that cooled first. After running the data in quartic model, the equation implied that at time 0, that is x=0, the coffee had an initial temperature of 43.355°C. A unit increase in time decreases the temperature with a marginal gap. The coefficient of determination (R2) shows that 69.71% of cooling temperature is explained by time. The data have no correlation meaning that there is no association between coffee temperature and time it takes to cool. Under the linear model, the equation y=42.884-0.0056456­X implies that at time zero, the coffee temperature was 42.884°C. This means that a unit change in the time reduces the temperatures of coffee by 0.0056456. However, 68.438% of cooling temperature variation is explained by time. The correlation coefficient (r) shows that the association is negatively correlated where temperature decrease as time increases. Basing on quadratic model, the equation implies that at time zero, coffee temperature is 43.886°C, which was the initial temperature of coffee before it was subjected to the experiment. A unit increase in time reduces the temperature of coffee by 0.004. However, there is no association between temperature and time. 69.546% explain the best fitness of model, that is, the variation of temperature and time. Cubic model equation implies that a unit increase in cooling time of coffee reduces the temperature of the same coffee from the initial 43.6760°C to 43.669°C. The model has no correlation and 69.151% explain the goodness of fit of the model. Logistic model initial coffee temperatures were 43.417°C. A decrease in the temperatures of coffee by 0.006 is as a result of an increase in a unit of time. The model has no correlation and the goodness of fit is explained by 69.151% of the model. In conclusion, the best model type to use in this kind of experiment of describing cooling rate of coffee is linear since its shows the correlation between the two variables time and temperature, linear model will also provide a single slope with unbiased fit of data and minimum errors. Basing the model on hot mixture of water and milk, the linear model equation y=47.12-0.71317X shows that the initial temperature when time was zero was 47.12°C, with a unit increase in time, the temperatures decrease by 0.71317. The model is best fit with 96.146%, and there is a positive correlation showing a stronger association between time and temperature. Quartic model shows an initial temperature of 48.406 with which a very little margin of 0.017 reductions in temperature due to a unit increase in time. The model is best fitted at 99.234% and there is no correlation. Basing the model on hot water coffee only, the linear model equation is y= 38.647-0.0041596X. It implies that at the initial time of zero, the temperature was 38.647°C. This also indicates that a unit increase in time shows a decrease in the temperature by 0.004596. There is also a very strong association between the variables and the model is best fit at 99.9%. Still under Quartic model the initial temperatures were 38.105°C, a unit increase in time does not affect the temperatures of the coffee. The graph below shows a linear model and Quartic model of coffee without milk respectively. The models for single coffee combination, that is, hot water coffee and a mixture of water and milk in a separate model running shows the models do not march an earlier conjure of the combine graphs. Under liner model the initial hot water coffee temperature was 38.647 °C and that of hot water and milk was 47.12°C. After 800 seconds, the temperature of mixture of hot water and milk was 41.415°C and that of hot water only was 35.319°C. When t is very large, the temperature of the coffee tends to zero. At no one time do the two types of coffee reach the same temperature since according to the graph, there is no any particular time the graphs intercept. The time it takes the water to reach a temperature 1°C above room temperature is 9050.6 seconds according to the equation of the linear model. It does not take a half longer after the difference between the starting temperature and room temperature is cut in half. The coffee that cools faster is the hot water coffee where after 2400 seconds, the coffee has a temperature of 28.664°C while that of mixture of milk and water has a temperature of 30°C after the same seconds. Using the linear model, compare the two equations of the each data temperatures after five minutes as follows, For the hot water coffee, y = 38.647-0.0041596 (300) =37.399 °C. For the hot water and milk coffee, y =47.12-0.007137 (300) = 44.98°C. This implies that the coffee that can be spilt on your lap safely without causing any harm is the hot water coffee. It is better for the drinker to wait for ten minutes before adding the cream so as to get a result with higher temperatures. This is because, adding the cream and waiting for the 10 minutes will result to the cream catalyzing the coffee to cool at a faster rate. REFERENCES. Aris R., 1978. “Mathematical modeling techniques.” Bender E., 1978. “An introduction to mathematical modeling.” Morris E., 1998. “Multivariate statistics.” Read More