Cooling Water
by Jamila K. Eagles
a. Take a cup of hot water and measure its initial temperature(time=0)
and then record the temperature readings each minute for 30 minutes. Make
note of the room temperature...
b. Enter the data on a spreadsheet and construct a function that will model
the data.
c. Using the function predict the temperature after 45 minutes, 60 minutes,
or 300 minutes.
d. Calculate a measure of the error between your model and the observerd
data by taking the square of the difference for each time, and dividing
the sum of the differences squared by the number of data points.
We begin by taking the temperature of 1 cup of water after
it begins to boil. Then chec its temperature each minute for 30 minutes.
The data is shown in the potion of the spreadsheet below.
The following is the graph of the water as it cools over time.
Next I tried to create a funtion that would model this data. I came
up with
Temp=137*exp^(-.037*t) +75.
Where did I get this function? First I knew that my room temperature was
about 75 degrees, so I used 75 as my constant and since we began at 212
degrees, I subtracted 75 from 212 and got 137 . I substituted these numbers
into the general equation of the form Temp= N*exp(-xt)+C. Now the question
of x has to be considered. Using the spreadsheet, I tried several values
of x to see which value would come closest to looking like my initial graph.
I also used the measure of error to guide refinement of my function to model
the data. The closest fit was the value x=0.37. The pink graph represents
the function to model my data.
Using this function I then predicted the temperature of the water at
45, 60, and 300 degrees.
Notice that after 300 minutes our temperature is about 75 degrees. When
water is cooling, the temperature will never be below our room temperature.
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