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course Math 163
This is a resubmission. Isubmitted this assignment 2 days ago but have not received a response. Submitted 09/14/10 @8:27
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.
On my graph, I selected the following three points: (10,75), (40,41), (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.
The three equations from the points noted above are:
75=100a+10b+c; 41=1600a +40b+c; and 26=5625a+75b+c
Eliminate c from your equations to obtain two equations in a and b.
Solve for a and b.
Write the resulting model for temperature vs. time.
My resulting equation model for temperature vs. time is
Y=.0108t^2-1.675t+90.67
Substituting the listed times below, I found the predicted temperatures:
Make a table for this function:
Time (minutes) Model Function's Prediction of Temperature
0 90.67
10 75
20 61.49
30 50.14
40 40.95
50 33.92
60 29.05
70 26.34
Sketch a smooth curve representing this function on your graph.
In plotting these points on my graph, I found the deviation listed below:
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 90.67 4.33
10 75 75 0
20 60 61.49 -1.49
30 49 50.14 -1.14
40 41 40.95 .05
50 35 33.92 1.08
60 30 29.05 .95
70 26 26.34 -.34
Find the average of the deiations.
When dealing with deviation you use the absolute value; therefore, the average deviation for my model is .7214
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).
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).
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Summary of Modeling Process, Version 1
The Modeling Process for the Flow Depth vs. Time Phenomenon
The modeling process can be broken into a number of steps. The general process as described here for the flow situation can be applied to a wide range of real-world situations. The details encountered in different situations, such as the nature of the observations, the type of mathematical function used, the number of data points selected, etc., might vary from one situation to another. However, the overall steps will be pretty much the same as those outlined here.
The steps are as follows:
Orient
For the flow situation, we visualize the flow and we sketch a predicted graph. This step helps to focus our attention on what we are about to observe. It helps 'get the wheels turning'.
Observe
For the flow situation, we observe depth vs. time. We do this by marking the cylinder at various depths and using a timer to determine when each depth is attained.
Organize Data
Organize the observed data into table(s). Note the patterns in the numbers on the table(s) and try to visualize the corresponding graph(s).
Graph
Graph the observed data, with the dependent variable on the y axis, the independent variable on the x axis.
Postulate
Postulate a model (we postulate a quadratic model for flow depth vs. time).
Select Representative Points
Pick an appropriate number of representative data points (three for the quadratic model used with the flow depth phenomenon, one for each parameter a, b and c in the model depth = a t^2 + b t + c). Note that these points are to represent the data set as well as possible, and need not be actual data points.
Obtain an equation for each selected point
Substitute the coordinates of the data point into the model. In the flow depth vs. time situation we obtain for each point an equation with a, b and c as unknowns (e.g., 900 t^2 + 30 t + 1 = 4900).
Solve the system of equations
Use whatever techniques are required to solve the system of equations. Usually these techniques will involve eliminating variables. Computerized algebra programs will also be used. For the flow depth vs. time situation we might obtain parameter values like a = .01, b = -2, c = 100.
Substitute parameters
Substitute the values of the parameters into the general form of the model to get the specific model. For the flow depth vs. time situation, we substitute the values for a, b, and c into the general form depth = a t^2 + b t + c to obtain a specific model ( e.g., if a = .01, b = -2 and c = 100, we get depth = .01 t^2 - 2 t + 100).
Graph the model
Graph the model along with the data points and compare the model with the data. The model will usually be a straight or curved line, obtained by graphing the function. The model will compare well with the data when all the data points lie close to the graph of the function, and when there is no apparent pattern to how the data points deviate from the function.
Quantify the comparison
Determine how much each data point deviates from the prediction of the model. This deviation appears on the graph as the vertical distance from the data point to the graph of the function. We then average the magnitudes of these deviations. In the flow depth vs. time situation we evaluate the model function for the time of each data point, and compare the resulting depth prediction to the observed depth by taking the difference between prediction and observation. We finally average the deviations to obtain a number for the 'average closeness' of the model and the observations.
Pose and answer questions
Use the model to determine such things as the value of the dependent variable given the value of the independent variable, or vice versa. For the flow depth vs. time situation, the model permits us to determine the depth for a given time (just substitute the time for t and we get the depth), or the time at which a given depth occurs (just substitute the desired value of the depth to get an equation which can be solved using the quadratic formula).
Do the science: relate the mathematics to the real world.
The step is actually optional in this course, since we are concerned with the mathematics and not the science. It is included here for completeness, and because often the science is pretty simple and enhances our understanding of the mathematics. What science does is tries to figure out why the model is or is not appropriate to the situation. Science might use the model to speculate on the underlying mechanism or process. The science of the flow depth vs. time situation relates pressure to the velocity of the escaping fluid in order to explain the model.
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Summary of Modeling Process, Version 2
Orient
Predictions, speculations.
Observe
Set up and take data.
Organize Data
time depth
30 49
60 16
90 1
Graph
Postulate
quadratic function depth = a t^2 + bt + c ????
Select Representative Points
Obtain an equation for each selected point
400 a^2 + 20 b + c = 61
1600a^2 + 60 b+ c = 13
8100a^2 + 90 b+ c = 2
Solve the system of equations
a = .01, b = -2, c = 100
Substitute parameters
depth = .01t^2 - 2 t + 100
Graph the model
Quantify the comparison
time observed values prediction of model deviation
0 95 97 -2
10 75 74 +1
. . . . . . . . . . . .
. . . . . . . . . . . .
Average of deviations = 2.43
Pose and answer questions
Do the science: relate the mathematics to the real world.
Pressure is proportional to depth.
Energy conservation implies that velocity is proportional to square root of pressure.
Thus dy/dt = k `sqrt(y).
Therefore velocity is quadratic in time.
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Summary of Modeling Process, Version 3
A. Obtain and Represent Data
A1. Orient
A2. Observe
A3. Organize Data
A4. Graph
B. Obtain a Model
B1. Postulate
B2. Select Representative Points
B3. Obtain an equation for each selected point
B4. Solve the system of equations
B5. Substitute parameters
C. Validate and Use the Model
C1. Graph the model
C2. Quantify the comparison
C3. Pose and answer questions
C4. Do the science: relate the mathematics to the real world.
To Be Memorized
The above outline of the model is to be memorized. At any time during the course, on any test or quiz, you might be asked to write down this model and illustrate the meaning of one or more of the steps.
This outline will be more or less followed throughout the course.
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