course Mth 163
1/21/10 2:54pm
Exercises:Here are some data for the temperature of a hot potato vs. time:
Time (minutes) Temperature (Celsius)
0 95
10 75
20 60
30 49
40 41
50 35
60 30
70 26
Graph these data below, using an appropriate scale:
Pick three representative points and circle them. (20,60), (50,35), (70,26)
Write the equations that result from the assumption that the appropriate mathematical model is a quadratic function y = a t^2 + b t + c.
400a+20b+c=60
2500a+50b+c=35
4900a+70b+c=26
Eliminate c from your equations to obtain two equations in a and b.
2100a+30b= -25
2400a+20b= -9
Solve for a and b.
A=0.00767, b= -1.3704, c=84.34
Write the resulting model for temperature vs. time. Y= 0.00767t^2 -1.3704t +84.34
Make a table for this function:
Time (minutes) Model Function's Prediction of Temperature
0 84.3
10 71.4
20 60
30 50.13
40 41.8
50 34.99
60 29.73
70 25.995
Sketch a smooth curve representing this function on your graph.
Expand your table to include the original temperatures and the deviations of the model function for each time:
Time (minutes) Temperature (Celsius) Prediction of Model Deviation of Observed Temperature from Model
0 95 84.3 -10.7
10 75 71.4 -3.6
20 60 60 0
30 49 50.13 +1.13
40 41 41.8 +0.8
50 35 34.99 -0.01
60 30 29.73 -0.27
70 26 26 0
Find the average of the deviations. Average deviation: -1.58
Comment on how well the function model fits the data. (Note: the model might or might not do a good job of fitting the data. Some types of data can be fit very well by quadratic functions, while some cannot).
The function model fits the data very well.
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Your work is good. However the Query at the end of the assignment will ask you questions about these assigned exercises, and they don't need to be submitted separately.