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13 Nov 2019
(5 points) As part of a liability defence (see the Wikipedia page on Liebeck v. McDonald's for a similar case), lawyers at Tim Hortons have hired you to determine the temperature of a cup of Tim Horton's coffee when it was initially poured. However, you only have measurements of the coffee's temperature taken after it has been purchased. According to Newton's Law of Cooling, an object that is warmer than a fixed environmental temperature will cool over time according to the following relationship T(t) = E + (Tinit-E) where E is the constant environmental temperature, and T is the temperature of the object at time t. The object has initial temperature Tinit Below you are given a data set measured from a purchased cup of coffee. The external temperature of the room is 20 °C. The temperature of the coffee T, is given for several ti, where t, is the time in minutes since the coffee was poured Transform the solution T(t) by putting the exponential term on one side and everything else on the other and taking natural logs of both sides to get: In(T(t) _ E) = ln(Tinit-E)-kt Now transform the data below in the same way so that you can use linear least squares to estimate the unknown parameters Tini and k. Fit the b + axi, ie., find the values of a and b which minimizefa, b)-il (yi)-(D+ axi))2 transformed data to a line yi = t_i (in minutes) 2 4 6 10 T_i (in °C) 92.4450 86.685489.806881.678381.9015 81.3594 81.920080.9616 75.1080 Use the computed coefficients a and b to calculate the following quantities What was the initial temperature Tini of the coffee when it was poured? 93.8019 What is the time constant k?0.0092784 /min
(5 points) As part of a liability defence (see the Wikipedia page on Liebeck v. McDonald's for a similar case), lawyers at Tim Hortons have hired you to determine the temperature of a cup of Tim Horton's coffee when it was initially poured. However, you only have measurements of the coffee's temperature taken after it has been purchased. According to Newton's Law of Cooling, an object that is warmer than a fixed environmental temperature will cool over time according to the following relationship T(t) = E + (Tinit-E) where E is the constant environmental temperature, and T is the temperature of the object at time t. The object has initial temperature Tinit Below you are given a data set measured from a purchased cup of coffee. The external temperature of the room is 20 °C. The temperature of the coffee T, is given for several ti, where t, is the time in minutes since the coffee was poured Transform the solution T(t) by putting the exponential term on one side and everything else on the other and taking natural logs of both sides to get: In(T(t) _ E) = ln(Tinit-E)-kt Now transform the data below in the same way so that you can use linear least squares to estimate the unknown parameters Tini and k. Fit the b + axi, ie., find the values of a and b which minimizefa, b)-il (yi)-(D+ axi))2 transformed data to a line yi = t_i (in minutes) 2 4 6 10 T_i (in °C) 92.4450 86.685489.806881.678381.9015 81.3594 81.920080.9616 75.1080 Use the computed coefficients a and b to calculate the following quantities What was the initial temperature Tini of the coffee when it was poured? 93.8019 What is the time constant k?0.0092784 /min
Collen VonLv2
27 Apr 2019