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Cell["\<\
1-031-T-Mma-CoolIt-TeacherVerstion
Modeling Cooling Water in Environment\
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We presume nothing here, i.e. we have unknown environmental temperature, \
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to see if we can get these even without observations,e.g.,  one would assume \
TE and T0 cold be observed.\
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We see we obtain values for k, TE, and T0. These are quite reasonable and the \
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Cell["Collection of data with changing environmental temperature.", \
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Cell["\<\
Here we collected data in our home office and we note that the environment' s \
temperature DOES NOT stay constant. Indeed, the office is warming up, no \
doubt due to the author' s body mass, the computer generated heat, and the \
warming outside the office due to rising sun.

So now we will  model first a changing (increasing) environmental temperature \
TE(t) as a function of time and incorporating this into our cooling model.

data contains sets of observations first coordinate is time in minutes, \
second coordinate is temperature of environment in degrees Fahrenheit, and \
third coordinate is temperature of water in beaker in degrees Fahrenheit. NB: \
 We did not record the temperature of the environment at time 0 and add in \
that data point later. \
\>", "Text",
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Cell["\<\
Here is the set of environment temperatures with time of observations.\
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Cell["\<\
We use the best fitting parameters for a model of the environment' s \
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Cell["\<\
We plot the temperature of the environment offered by our fitted model.\
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Cell["\<\
We see that our environmental temperature mode, TModel(t), is quite good at \
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\>", "Text",
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Cell["\<\
We use our changing environmental temperature model TModel(t) in our Newton\
\[CloseCurlyQuote]s Law of Cooling model, 

                       T\[CloseCurlyQuote](t) \[Equal] - k (T(t) - TModel(t) \
),     with T(0) = 103.1.\
\>", "Text",
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Cell["\<\
We seek to determine best fitting value of k using sum of square errors \
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 and our observed water temperature at each of the times of observations, \
WaterTemp[[i,2]].\
\>", "Text",
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Here we find the value of the parameter k, whose units are 1/time, which \
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\>", "Text",
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Cell["We put this value of \[LineSeparator]k into our new model.", "Text",
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We compare our new model and the data for the temperature of the water in \
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We offer a smaller sample for students to use in their work. We do this \
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