#$&*
course MTH 163
This is my partial work for assignment 2. I will wait for feedback to complete the rest of it because I am not sure what I am doing wrong.
When submitting your work electronically, show the details of your work and give a good verbal description of your graphs.One very important goal of the course is to learn to communicate mathematical thinking and logical reasoning. If you can effectively communicate mathematics, you will be able to effectively communicate a wide range of important ideas, which is extremely valuable in your further education and in your career.
When writing out solutions, self-document. That is, write your solution so it can be read without reference by the reader to the problem statement. Use specific and descriptive statements like the following:
• Using the depth vs. clock time data points (0, 13), (3, 12), (10,10), (25,8), (35, 6), (52, 3), (81, 1), we obtain a model as follows . . .
• Using the depth vs. clock time data points (3, 12), (25, 8) and (52,3) we obtain the system of equations . . .
• From the parameters a = -1.3, b = 12 and c = 15 we obtain the function . . .
• Comparing the predicted depths at clock times t = 0, 3, 10, 25, 35, 52, 81 with the observed depths we see that . . .
Here are some data for the temperature of a hot potato vs. time:
Time (minutes) Temperature (Celsius)
0 118
20 118
40 91.96194
60 81.85297
80 73.29324
100 66.04533
120 59.90819
140 54.7116
Graph these data below, using an appropriate scale:
The graph starts at the top left and decreases as it goes towards the right hand side of the graph.
Pick three representative points and circle them.
The three points that I chose were (20, 118), (40, 91.96194) and (60, 81.85297)
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=118
1600a+40b+c=91.96194
3600a+60b+c=81.85297
Eliminate c from your equations to obtain two equations in a and b.
(3600a+60b+c=81.85297)-( 1600a+40b+c=91.96194)=(2000a+20b=-10.85297)
(1600a+40b+c=91.96194)-(400a+20b+c=118)=(1200a+20b=-26.03806)
Solve for a and b.
-1(1200a+20b=-26.03806)-1
2000a+20b=-10.85297
800a=15.92909 a=0.01991
1200(0.01991)+20b=-26.03806 b=-2.49650
400(0.01991)+20(-2.49650)+c=118 c=159.966
Write the resulting model for temperature vs. time.
Y=0.01991t^2-2.49650t+159.966
Make a table for this function:
Time (minutes) Model Function's Prediction of Temperature
0 159.966
20 118
40 91.96194
60 81.85297
80 87.67
100 109.416
120 147.09
140 200.692
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
0 118 159.966 41.966
20 103.9006 118 14.0994
40 91.96194 91.96194 0
60 81.85297 81.85297 0
80 73.29324 87.67 14.37676
100 66.04533 109.416 43.37067
120 59.90819 147.09 87.18181
140 54.7116 200.692 145.9804
Find the average of the deviations.
43.37113
Self-Critique: I understand that I have not done this correct considering my temperatures from my model increase. I thought that I had done everything right and I have reworked it and reworked it and every time I get the same thing. If you can let me know what I am doing wrong, I would really appreciate it, as I am really at a loss of what I’ve done wrong.
@& Your process is fine. However I believe the t = 20 temperature is 103.9, not 118.
So your equation
400a+20b+c=118
appears to be the culprit here. *@
1. If you have not already done so, obtain your own set of flow depth vs. time data as instructed in the Flow Experiment (either perform the experiment, as recommended, or E-mail the instructor for a set of data).
Complete the modeling process for your own flow depth vs. time data.
Y=0.05385t^2-2.41660t+89.39981
Use your model to predict depth when clock time is 46 seconds, and the clock time when the water depth first reaches 14 centimeters.
92.18281
Self-Critique: Again, I feel as if I am doing every single step right. I did it the same way that I did in the first round, however, I still don’t get an accurate answer. I am so confused.
Comment on whether the model fits the data well or not.
No. My deviation is 78.18281.
2. Follow the complete modeling procedure for the two data sets below, using a quadratic model for each. Note that your results might not be as good as with the flow model. It is even possible that at least one of these data sets cannot be fit by a quadratic model.
Data Set 1
In a study of precalculus students, average grades were compared with the percent of classes in which the students took and reviewed class notes. The results were as follows:
Percent of Assignments Reviewed Grade Average
0 .9258546
10 1.36495
20 1.750933
30 2.090227
40 2.388479
50 2.650655
60 2.881118
70 3.083704
80 3.261786
90 3.418326
100 3.555931
Determine from your model the percent of classes reviewed to achieve grades of 3.0 and 4.0.
Determine also the projected grade for someone who reviews notes for 80% of the classes.
Comment on how well the model fits the data. The model may fit or it may not.
Comment on whether or not the actual curve would look like the one you obtained, for a real class of real students.
Data Set 2
The following data represent the illumination of a comet by a certain star, reasonably similar to our Sun, at various distances from the star:
Distance from Star (AU) Illumination of Comet (W/m^2)
1 1280
2 320
3 142.2222
4 80
5 51.2
6 35.55556
7 26.12245
8 20
9 15.80247
10 12.8
Obtain a model.
Determine from your model what illumination would be expected at 1.6 AU from the star.
At what range of distances from the star would the illumination be comfortable for reading, if reading comfort occurs in the range from 25 to 100 Watts per square meter?
Analyze how well your model fits the data and give your conclusion. The model might fit, and it might not. You determine whether it does or doesn't.
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Self-critique (if necessary):
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Self-critique rating:
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#$&*
@& You know what you're doing. You just put the t = 0 temperature into the equation for the t = 20 clock time. That of course threw your model off.
See my note.*@