- Level: International Baccalaureate
- Subject: Maths
- Word count: 1461
The following data in table #1 describes the flow rate of the Nolichucky River in Tennessee between 27th of October 2002 and 2nd of November 2002,The following data in table #1 describes the flow rate of the Nolichucky River in Tennessee between 27th of O
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Introduction
The following data in table #1 describes the flow rate of the Nolichucky River in Tennessee between 27th of October 2002 and 2nd of November 2002, taken from the web site:
water.usgs.gov/pub/wri934076/stations/03465500.html.
And in this portfolio we are going to investigate the relationship between the flow of water and time in order to have understanding of the amount of flow of water and the change of the rate of flow.
Table #1:
The original data for the flow rate of Nolichucky River in Tennessee between 00:00 27th of October 2002 and 00:00 2nd of November 2002 (the time is measured in hours past midnight, and the flow rate is measured in cubic feet per second [cfs]):-
Time / (Hours) | Flow / (cfs-1) |
0 | 440 |
6 | 450 |
12 | 480 |
18 | 570 |
24 | 680 |
30 | 800 |
36 | 980 |
42 | 1090 |
48 | 1520 |
54 | 1920 |
60 | 1670 |
66 | 1440 |
72 | 1380 |
78 | 1300 |
84 | 1150 |
90 | 1060 |
96 | 970 |
102 | 900 |
108 | 850 |
114 | 800 |
120 | 780 |
126 | 740 |
132 | 710 |
138 | 680 |
144 | 660 |
Graph #1: Graph of the original data for the Flow vs. time:
The line of best fit is the best approximation of a function for a set of data. However, our original data for the flow rate of Nolichucky River in Tennessee seems to be divided into two functions, one function is increasing, while the other function is decreasing. And in such case we can use the idea of a
Middle
- Best Fit #2 exponential that is represented by the black portion on graph #2 is an exponential function with equation:
Y1b = 338.129 e 0.030 ×
Graph #3: Best fit lines for the decreasing part of the Data
- Best Fit #1 polynomial represented by the black portion on graph #3 is a polynomial function of power 7 with equation:
Y2a= (-7.298×10-11×7) + (1.183×10-7×6) + (-5.665×10-5×5) + (0.013×4) + (-1.624×3) + (114.783×2) + (- 4316.0849×) + (69060.761)
- Best Fit #2 exponential represented by the gray portion on graph #3 is an exponential function with equation:
Y2b = 3163.562 e -0.0116x
And so as we can see from the graphs above, the polynomial function are more suitable to be used as best fit functions from the exponential functions.
From this we can say that the function (polynomial) for the increasing part of the original data for the flow rate of the Nolichucky River in Tennesse:
Y1= (-1.762×10-11 ×7) + (3.116×10-5 ×6) + (- 0.00214×5) + (0.0718×4) + (-1.229×3) + (10.44×2) + (- 30.774×) + (440.834)
While the function (polynomial) for the decreasing part is:
Y2= (-7.298×10-11×7) + (1.183×10-7×6) + (-5.665×10-5×5) + (0.013×4) + (-1.624×3) + (114.783×2) + (- 4316.0849×) + (69060.761)
Graph #3: The lines of best fit for all of the original data (the decreasing and the increasing part):
Conclusion
Graph #4: Graph of the derivatives of the two functions:
And as we can see from graph #4 the amount of flowing water was increasing in the time periods from 02:00 of 27th of October until 06:00 of 29th of October and from 23:00 of 1st of November until 00:00 of 2nd of November. And at those times the dams of the river where opened, which explains the increase in the flow of water.
And in order to find the average values for the flow of water, we can use the Mean Value Theorem for Y1 and Y2 , and then we add the two values and divide them by 2.
= 858.83 cfs -1
=1047.39 cfs -1
Therefore,
cfs-1
And again we have two time periods at which the flow rate was equal to this value, and we can find them by putting 953.11 cfs-1 equal to Y1 and Y2
Y1 = 953.11 = (-1.762×10-11 ×7) + (3.116×10-5 ×6) + (- 0.00214×5) + (0.0718×4) + (-1.229×3) + (10.44×2) + (- 30.774×) + (440.834)
x= 36.1 hours (12:10 on 28th of October)
Y2 = 953.11= (-7.298×10-11×7) + (1.183×10-7×6) + (-5.665×10-5×5) + (0.013×4) + (-1.624×3) + (114.783×2) + (- 4316.0849×) + (69060.761)
x= 96.35 hours(01:20 on 31st of October)
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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