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Fishing Rod

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Introduction

Min Hua Ma

5-1-09

IB SL MATH

Internal Assessment

Type II

IB SL Math Internal Assessment: type II

This assignment is an investigation to find different methods that model a given set of data. By using matrix methods, polynomial functions, and technology to find different equations, we can discover which equation best models the data. Leo has a fishing rod that has a length of 230cm, and the given data about his rod is:

Guide #

1

2

3

4

5

6

7

8

Distance from tip (cm)

10

23

38

55

74

96

120

149

        To begin the investigation, we began by plotting the given points in a scatter plot:

image03.png

        The first method is using matrices to find a quadratic function. A quadratic equation is: ax2+bx+c=y, since there is only 3 variables (a, b, c) we can only create a 3 by 3 matrix and a 3 by 1 matrix. So I choose the first six data and separated them into two sets of three to make these equations to model the data.

image04.pngimage04.png = image16.pngimage16.png

image26.pngimage26.pngimage35.pngimage35.pngimage05.pngimage05.png

To solve for  image06.pngimage06.png for the first set of three data, we take image07.pngimage07.png x image05.pngimage05.png and get:image08.pngimage08.png Using that information, a quadratic equation can be created: 1x2+10x-1. The model with the actual points is shown by this graph from a graphing calculator:

image09.png

This graph shows that the equation 1x2

...read more.

Middle

8

23

55

96

149

Now plug each of the x’s into the cubic function (ax3+bx2+cx+d). Then put each group of data into a 4 by 4 matrix and set the y values into a 4 by 1 matrix.

For Odds:

image14.pngimage14.png x image15.pngimage15.png = image17.pngimage17.png

                                                [A]                [x]       [B]

For evens:

image18.pngimage18.png x image15.pngimage15.png =image19.pngimage19.png

                                                [A]                 [X]      [B]

At this time, use the process image07.pngimage07.png x image20.pngimage20.png to solve for [X]. After doing so:

Odd: [X]=image21.pngimage21.png                            Even: [X]=image22.pngimage22.png

Then put each of the variables into a cubic function format and generate a graph for each equation.

The odd: f(x)=.0416666667x3 +.625x2 +10.9583333x-1.625

image23.png

This graph shows that it touches all the data points, except when x=6. This equation models the given data by almost passing every point.

The even: f(x)= .0625x3 +.375x2 +12x -3

image24.png

This graph illustrates that all, but the sixth point is touched by this function. Although this function is different from the odd cubic function, they both go through the same points and not the sixth. The differences of these two graphs are only the different numbers written for the cubic function.

        Next, we need to create a polynomial function that passes through all of the particular data. To do this, I decided to create a polynomial function by using matrices.

...read more.

Conclusion

See below:image02.pngimage02.pngimage01.pngimage00.pngimage00.pngimage01.png

1

2

3

4

5

6

7

8

9

10

23

38

55

74

96

120

149

178

By adding a ninth guide, it will affect the all the functions created. The cubic equations would not be as precise since there are an odd number of points. The polynomial equation must begin with ax8instead of ax7.

        Then the assignment gave us a new set of data to compare with the quadratic model from the first part of this investigation. This time Mark has a fishing rod that is 300cm.

Guide number (from tip)

1

2

3

4

5

6

7

8

Distance from tip (cm)

10

22

34

48

64

81

102

124

The quadratic model fits this data by only going through the first two points, and misses the rest. The quadratic model from the first part of the investigation doesn’t really model this new set of data. To find how to modify the quadratic equation, I used technology to figure out what the QuadReg was and subtracted the first quadratic model by this QuadReg:

1.244047619x2 + 8.458333333x + .8392857143        Original function

-         .9345238095x2 + 7.720238095x + 2.05351429    New function

                         .3095238095x2 + .738095243x   -1.214228547

So, the quadratic has to modify by .3095238095x2 + .738095243x   -1.214228547 to fit the data created by the 300cm rod. The regression of the function 9345238095x2 + 7.720238095x + 2.05351429, is .9997802355. The LinReg of this data is written y=(16.13)*x+(-11.95 ) with the regression (.99).

image34.png

The limitations to this model is that there is can’t be an infinite number of guides and rod have only have certain lengths.  

Ma

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

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