course Mth 173
Simulated data for water flow:clock t
5.3
10.6
15.9
21.2
26.5
31.8
depth
63.7
54.8
46
37.7
32
26.6
Exercise with potato temperature:
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.
Points chosen – (20,60),(30,49),(40,41)
Write the equations that result from the assumption that the appropriate mathematical model is a quadratic function y = a t^2 + b t + c.
60 = a (20^2) + b(20) + c, or
400a + 20b + c = 60
49 = a (30^2) + b(30) + c, or
900a + 30b + c = 49
41 = a (40^2) + b(40) + c, or
1600a + 40b + c = 41
Eliminate c from your equations to obtain two equations in a and b.
400a+20b=60
900a+30b=49
You appear to have 'eliminated' c by removing it.
To eliminate a variable you have to subtract a multiple of one equation from a multiple of another.
Solve for a and b.
500a+10b=-11
500a=-11-10b
A=(-11-10b)/500
400((-11-10b)/500)+20b=60
-8.8-8b+20b=60
-8.8+12b=60
12b=68.8
B=86/15
A = (-11-10(86/15))/500
A = -41/300
1600(-41/30) + 40(86/15) + c = 41
C = 91/3
You've used the method of substitution to solve for a and b. This is a valid method.
However you also need to understand the method of elimination, as explained in the worksheets.
Write the resulting model for temperature vs. time.
Y = (-41/300)x^2 + (86/15)x + 91/3
Make a table for this function:
Time (minutes) Model Function's Prediction of Temperature
0 30.3
10 74
20 90.3
30 79.3
40 41
50 -24.6
60 -117.66
70 -238
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 30.3 64.7
10 75 74 1
20 60 90.3 30.3
30 49 79.3 30.3
40 41 41 0
50 35 -24.6 59.6
60 30 -117.66 147.66
70 26 -238 264
Find the average of the deviations.
74.695 degrees Celsius
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).
At some points the deviation is very small and the function fits the data well. However, at most points the deviation is large and the function fails to fit the data.
Carefully read and understand the outlines and summaries of the modeling process as given below, as it applies to the examples discussed so far. Be sure to note the outline to be memorized (see the end of the page or click on the link To Be Memorized).
Except for the elimination step your process is good.
However see my notes and be sure you know how to use elimination to solve systems of this nature.