2089 Words9 Pages

Rationale:
When examining heating and cooling curves in Physics in the past, it is interesting to study the form the graphs take- how the body of liquid does not constantly increase or decrease in temperature and how the graphs are not always smooth. Developing upon this idea, I wanted to carry out an experiment which would find an appropriate mathematical model for the cooling curve of hot water which would then be cross-checked with Newton’s Law of Cooling to study how accurately the model represents the curve. This experiment involved modelling, graphing quadratic and exponential equations, logarithms and calculus. The conclusion arrived at was that the exponential model was the most appropriate model for the cooling curve and*…show more content…*

Since the exponential model’s error sum is approximately 348 sq. units less than that of the quadratic model’s error sum, which is a significant difference, we can conclude that the exponential model is a better model to use for the cooling curve of hot water. Newton’s Law of Cooling: Newton’s Law of Cooling states that the cooling rate is proportionate to the difference between the temperature of the hot water and the surrounding temperature. ddt=-k(-r) cooling rate=-k (temperature of hot water-room temperature) Using the same integration method as 1ax+bdx=1alnax+b+c and k dx = k1 dx = kx+c; ddt=-k(-r) Cross multiply into: d-r=-kdt Integrate both sides: d-r=-kdt which would equal to: ln (-r )=-kt+c when t=0, =0 where 0 is the initial temperature. ln (0-r )=-k(0)+c ln (0-r)=c Substituting c=ln (0-r ) in the equation ln (-r )=-kt+c, ln (-r)=-kt+ln(0-r) Ifc=logea ec=a and if log e=ln,then c=ln a*…show more content…*

They are mostly used in the fields of natural sciences and math but are also applied in areas like economics. I now have a better and more clear understanding of how to carry out bivariate analysis- such as the application of regression lines and least square regressions. Except for the common example of population growth in exponents, I previously never understood how concepts like integration and logarithms could be used in real-life situations but after applying them to a very common situation like the rate of cooling, I can see that the concepts are used to explore situations in everyday life. It also helped me grasp the four areas of mathematics better and showed me that different topics in mathematics are not necessarily ‘separate’ and can be used and applied together to achieve one, and the most appropriate, solution to a problem. I would be interested to find out how more areas in mathematics can be used together and other ways in which math can be applied in real-life because it really gives a better understanding of the concepts we learn in

Since the exponential model’s error sum is approximately 348 sq. units less than that of the quadratic model’s error sum, which is a significant difference, we can conclude that the exponential model is a better model to use for the cooling curve of hot water. Newton’s Law of Cooling: Newton’s Law of Cooling states that the cooling rate is proportionate to the difference between the temperature of the hot water and the surrounding temperature. ddt=-k(-r) cooling rate=-k (temperature of hot water-room temperature) Using the same integration method as 1ax+bdx=1alnax+b+c and k dx = k1 dx = kx+c; ddt=-k(-r) Cross multiply into: d-r=-kdt Integrate both sides: d-r=-kdt which would equal to: ln (-r )=-kt+c when t=0, =0 where 0 is the initial temperature. ln (0-r )=-k(0)+c ln (0-r)=c Substituting c=ln (0-r ) in the equation ln (-r )=-kt+c, ln (-r)=-kt+ln(0-r) Ifc=logea ec=a and if log e=ln,then c=ln a

They are mostly used in the fields of natural sciences and math but are also applied in areas like economics. I now have a better and more clear understanding of how to carry out bivariate analysis- such as the application of regression lines and least square regressions. Except for the common example of population growth in exponents, I previously never understood how concepts like integration and logarithms could be used in real-life situations but after applying them to a very common situation like the rate of cooling, I can see that the concepts are used to explore situations in everyday life. It also helped me grasp the four areas of mathematics better and showed me that different topics in mathematics are not necessarily ‘separate’ and can be used and applied together to achieve one, and the most appropriate, solution to a problem. I would be interested to find out how more areas in mathematics can be used together and other ways in which math can be applied in real-life because it really gives a better understanding of the concepts we learn in

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