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Math Portfolio: Fishing Rod

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Introduction

image00.png Ismar Hota

Math Portfolio: Fishing Rod

The table shown below shows the distance for each of the line guides from the tip of the fishing rod.

x

Guide number (from tip)

1

2

3

4

5

6

7

8

y

Distance from tip (cm)

10

23

38

55

74

96

120

149

The information in the above table can be presented on a XY graph; where guide number is shown on X axis and the distance from the tip is shown on the Y axis. (See graph below)

image01.png

From this graph one can come to a conclusion that the points represent a part of a quadratic function. A matrix method ( image07.png ) will be used to define parameters of the quadratic function ( image17.png ) using a given points (table above).

The matrix method:

Points: (1,10); (2,23); (3,38)

a + b + c = 10

4a + 2b + c = 23

9a + 3b + c = 38

image20.png        =>image21.png

From matrix result it follows:

a = 1

b = 10

c = -1

When plugged back to the image17.png, the result is quadratic function

image05.png

image22.png

From the graph it is visible that the first five points fits the quadratic function exactly however afterwards the points don’t correspond the curve.

...read more.

Middle

23

38

55

74

96

120

149

image03.png

Distance from tip (cm)

10.5417

22.7321

37.4107

54.5774

74.2321

96.375

121.006

148.125

This allows for a conclusion to be made that the function image04.pngbetter fits the model than the image05.png that was proved using the matrix method. Although the first one lacks correspondence with the all of the points it is to a very slight value and can be said that it gives a more reasonable result. Even though image05.png corresponds exactly for the first five points afterwards the correspondence difference grows to a bigger value and the reliability of this function is low (see graph below).


image06.png

x

Guide number (from tip)

1

2

3

4

5

6

7

8

y

Distance from tip (cm)

10

23

38

55

74

96

120

149

image03.png

Distance from tip (cm)

10.5417

22.7321

37.4107

54.5774

74.2321

96.375

121.006

148.125

image05.png

Distance from tip (cm)

10

23

38

55

74

95

118

143

To calculate the guide for point 9 we will have to substitute 9 into the equation(s) image05.pnggiving the value of 170. The second function image04.pnggives the value of 177.7321

image08.png

image06.png

The implication with this is that we cannot with certainty tell where the best place for the 9th point is. This is the limit

...read more.

Conclusion

Points: (1,10); (3,34); (5,64)

a + b + c = 10

9a + 3b + c = 34

25a + 5b + c = 64

image16.png        =>image18.png

Here when using the matrix method we find out that the value for a mustn’t equal zero because then the function that is a result of that is a linear one which will make the curve completely unreliable and for that reason the values that were used for determining the equation are the 1st 3rd and 5th.

image19.png

This means that by guessing and trying method one can come to a more reliable result, which will correspond more closely using the matrix method.

Conclusion:

This method is probably used by the factories that make fishing rods. They could use this to manufacture a long lasting rod that would be able to catch many heavy fish and be resistant to breaking or damages. I can see this be used in building of many items that have a spring or other like objects where their endurance needs to be tested. This model seems like a reasonable method to be used by mechanical engineers when observing items so they can be ready to hit the market as a final product.

...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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