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IB Fish

Extracts from this document...

Introduction

Alice Wang

Fishing Rod Lab

Table 1. Leo’s fishing rod

Guide number (from tip)

1

2

3

4

5

6

7

8

Distance from tip (cm)

10

23

38

55

74

96

120

149

Dependent variable: The distance from the tip of fishing rod

Independent variable:  The guide number from the tip of the fishing rod

Parameters/Constraints: The distance from tip for each guide number does not follow a particular pattern. Hence it is difficult to achieve a function that satisfies all of the points on Table 1.

Graph 1. Leo’s fishing rod

image10.png

Consider general form of cubic and quadratic equations (where a,b,c,d are coefficients and where a is never zero):
y=ax
2 +bx+c (quadratic)
y=ax
3 +bx2 +cx+d (cubic)

10= a+ b + c

23= 4a+ 2b + c

38= 9a+3b+cimage11.png

Put into matrix:

image21.png

image21.pngimage00.pngimage01.png

First, to clear the first 0 of the column, the first row has to be multiplied by 9 and subtracted from the second row.

image36.pngimage04.png

To clear the second 0 of the first column, the second row has to be multiplied by 3 and be subtracted to the third row.

image05.png

image37.png

To clear the second column’s last row to a 0, the second row has to multiplied by ½.

image38.png

...read more.

Middle

Table 2. Data points of Cubic Function

1

9.869

2

23.186

3

38.043

4

54.818

5

73.889

6

95.634

7

120.43

8

180.69

Thus, the equation of the cubic equation is y =.063x3 +.392x2 +11.7x-2.286

image14.png

Cubic Function Graph (above)

image15.png

Quadratic Function (above)

The main difference between quadratic and cubic functions is that a quadratic equation is a polynomial fit using three points while a cubic function uses four or more points and is a polynomial regression. With a higher degree for a polynomial function, there is a less degree of accuracy, especially when the guide number increases. The quadratic model is more precise than the quadratic function when the guide number reached 7 and 8.

I used the functionimage16.png to find a function which passes through every data point in Table 1.  With the 7th degree in a polynomial, it will pass through every data point of Leo’s fishing rod.

Subsitution of Table 1’s x and y values toimage16.png is as followed:

image17.png

image18.png

image19.png

image20.png

image22.png

image23.png

image24.png

image25.png

...read more.

Conclusion

image32.png

Table 5. Mark’s fishing rod

Guide number (from tip)

1

2

3

4

5

6

7

8

Distance from tip (cm)

10

22

34

48

64

81

102

124

image33.png

The quadratic model of y=x2 +10x-1 fits the new data adequately. Although the fit of the quadratic function is not perfect, it closely represents the data provided.

image34.png

Using Excel, the line of best fit was considered y= 0.9345ximage29.png + 7.7202x + 2.0536 as a polynomial function. The quadratic model of y=x2 +10x-1  and the line of best fit are extremely similar (demonstrated on the graph below). The narrower function is the line of best fit and the wider function is the quadratic model. My quadratic model could be more vertically compressed by multiplying the value of a (from y=a2+bx+c) by a positive integer. The limitations of the quadratic model is that it is only suited for fishing rods that are similar to Leo’s. If the distance from tip of each guide number of a fishing rod is substantially different that Leo’s fishing rod, the quadratic model of y=x2 +10x-1 will not work. In this case, Mark’s distance from tip of each guide number was similar to Leo’s. Thus, the quadratic model worked for Mark’s fishing rod.

image35.png

...read more.

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