Newton's Law Of Cooling

Law of Cooling
Amy (Dohyun) Kim _10I Let’s say a 90.0 °C cup of coffee was placed in a refrigerator and after 4 minutes the temperature was measured to be 83.2 °C and in another 4 minutes it was measured to be 77.0 °C. In order to find out the initial temperature of the refrigerator and the time it takes for the coffee to cool down to 40.0°C, we will research about the Newton’s Law of Cooling rate and find a suitable model that describes the change in temperature over time. According to http://www.ugrad.math.ubc.ca, the solution for Newton’s Law of Cooling is
T(t)=T_a+(T_0-T_a ) e^(-kt)
T (t) = Temperature of the coffee at time t (°C) t=time (in minutes)
T0=Initial temperature of the coffee (°C)
Ta=Temperature of the refrigerator

We can insert this into the formula in order to know the value of Ta and k:
{█(83.2=T_a+(90.0-T_a ) e^(-4k)@77.0=T_a+(90.0-T_a ) e^(-8k) )┤
By using Wolfram Alpha, the values happen to be Ta= 12.9 and k=0.0231
{The values are rounded up to have 3 sig figs since the numbers that are presented in the information/introduction has 3 sig figs. (Except for the sample values below)}
This means that the refrigerator temperature is ≈12.9°C
If we organize the exact formula for the change of temperature in this experiment, it would

Since the coffee and the refrigerator have a different temperature, the coffee temperature will decrease and will try to reach the same temperature with its surrounding. When we look at the sample values, we can see that the coffee’s temperature actually reached the same temperature as the refrigerator which is about 12.9°C and it stays constant no matter how much time pasts. Therefore, this experiment was an exponential case.
Additionally, in order to know the amount of time it takes for the coffee to be 40.0°C, we could insert the number in the equation and calculate it, or find the value by using the graph above.
The function would look like
40.0=12.9+(90.0-12.9) e^(-0.0231t)
Again by using Wolfram Alpha, t ≈45.3 minutes which means that it will take about 45.3 minutes for the coffee to reach 40.0°C. To wrap up, through this investigation, we were able to investigate some of the properties of an exponential model. In addition, we were able to learn what a Newton’s Cooling Law is and how that is related to an exponential model.
Works Cited
"Other Differential Equations." University of British Columbia. Undergrad Mathematics Lab, www.ugrad.math.ubc.ca/coursedoc/math100/notes/diffeqs/cool.html. Accessed 1 Dec. 2017.
Martin, David, et al., editors. Mathematics for the International Student Mathematics HL (Core) Third Edition. PDF