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course Mth163
6/6 8
ExercisesWhen 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 . . .
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 click on Randomized Problems and select Precalculus then Simulated Data for Flow Experiment , which can be accessed at http://164.106.222.236/interactivepro).
Complete the modeling process for your own flow depth vs. time data.
Use your model to predict depth when clock time is 46 seconds, and the clock time when the water depth first reaches 14 centimeters.
Comment on whether the model fits the data well or not.
Even though you probably understand the process at this point, it can be challenging to get through these problems without making mistakes. An error on one step can throw the entire problem off, and result in a model that doesn't work at all.
I used the first simulated Flow experiment Depth (CM) vs. Time (seconds) with points at (5.3, 63.7), (10.6, 54.8), (15.9, 46), (21.2, 37.7), (26.5, 32), (31.8, 26.6)
I used points (5.3, 63.7), (15.9, 46), and (26.5, 32) to create three simultaneous equations and eliminating variables. Using the quadratic equation y = ax^2 + bx + c
From these points I found a = .016 b = -1.999 and c = 73.846
I graphed the observed data which displayed a parabola opening upward flowing from the left to right. The depth is decreasing as well as the rate.
Predicting the depth at 46 seconds.
Y = .016(46^2) - 1.999(46) + 73.846 , Y = 33.856 - 91.540 + 73.846 , Depth (Y) = 15.748 cm at 46 seconds
Predicting the time when the depth reaches 14 cm
14 - 14= (.016)x^2 - 1.999(x) + 73.846 - 14, 0 = (.016)x^2 -1.999(x) + 59.846
using the quadratic formula
[-(-1.999) +- sqrt(-1.999^2 - 4(.016)(59.846)) / 2(.016), 1.999 +- sqrt(3.996 - 3.830) / .032, 1.999 +- .407 /.032
1.999 + .407 / .032 = 75.201, 1.999 - .407 / .032 = 49.750
At a depth of 14 cm the clock time is predicted to be 49.750 seconds
The formula is a good representation of the graph with a deviation of .098
Flow experiment Simulated data
Time Seconds Observed Depth Cm Prediction y=.016(x^2) -1.999(x) + 73.846
0 73.846 73.846 0.000
3 67.993
5.3 63.7 63.701 0.001
10 55.456
10.6 54.8 54.454 -0.346
15.9 46 46.107 0.107
21.2 37.7 38.658 0.958
25 33.871
26.5 32 32.109 0.109
31.8 26.6 26.458 -0.142
35 23.481
46 15.748
52 13.162
58 11.728
64 11.446
70 12.316
76 14.338
81 16.903
88 21.838
94 27.316
100 33.946
106 41.728
112 50.662
118 60.748
124 71.986
130 84.376
136 97.918
&#Very good responses. Let me know if you have questions. &#