Aras Ibrahim
Module Leader: Geoff Grenyer Student Number: 100303438
Module Code: 5HX500 (BEng) (Hons) Civil and Infrastructure Engineering
1. 20% the depth of a river flowing in a rectangular channel 5m. Wide is measured over a period of 11 hours following a storm. The tables show depth of the river in mm.
Time T hours
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Depth D mm
325
330
326
320
345
393
420
451
490
6
6.5
7
7.5
8
8.5
9
9.5
10
10.5
11
532
582
600
626
633
637
645
641
a) Using excel plot a graph of this data, state the function representing the 5th order polynomial line of best fit and show this on the graph. (For accuracy the line of best fit constants should be expressed to 6 decimal places).
b) Comment on the differences between the actual data and the line of best fit function including largest differences.
The purpose of plotting the line of best fit is to simply spot the errors of the plotted data on the graph. When plotting line of best fit on the graph, the order of function has an effect to the accuracy of the line or curve. When increasing the order of the line of best fit it will reduce the error that is created from the actual data. Therefore there will be a very small error when setting line of best fit with an order of 5. The data are almost equally above and below the curved line (Looking at the graph at x axis, at 3 the data is below the curved line and almost equal to 6.5 where the data is above the curved line and so on. Increasing the decimal places of coefficient will result of decreasing the error, depending on the correlation of data. The results seems to be too accurate as the (R2) values are very close to 1.
c) Differentiate the best fit function and find the rate (mm/hour) at which the water is rising after 4.5 hours
y = 0.057610x5 - 1.557842x4 + 13.411917x3 - 35.500034x2 + 26.491269x + 325.752341
dydx=0.28805x4-6.231368x3+40.235751x2-71.000068x+26.491269
y=0.28805(4.5)4-6.2313684.53+40.2357514.52-71.000068(4.5)+26.491269
y=72.050015 mm/h
d) Determine the maximum river depth suggested by the best fit function and the time at which this occurs and compare these values with the information suggested by the data.
Ø From the graph the Maximum depth is (650 mm)
Ø From the graph (Time) where the Maximum depth occurs (9 h)
Ø Equating the differential of the line of best fit equation to (ZERO)
Ø 0=0.28805x4-6.231368x3+40.235751x2-71.000068x+26.491269
Ø Thefore applying Newton Raphson's Method: x2=x1-fx₁f'x₁
f'x1...
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