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SL Math IA: Fishing Rods

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Introduction

Math Summative: Fishing Rods

Fishing Rods

A fishing rod requires guides for the line so that it does not tangle and so that the line casts easily and efficiently. In this task, you will develop a mathematical model for the placement of line guides on a fishing rod.

The Diagram shows a fishing rod with eight guides, plus a guide at the tip of the rod. Leo has a fishing rod with overall length 230 cm. The table shown below gives the distances for each of the line guides from the tip of his 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

Define suitable variables and discuss parameters/constraints.

Using Technology, pot the data points on a graph.

Using matrix methods or otherwise, find a quadratic function and a cubic function which model this situation. Explain the process you used. On a new set of axes, draw these model functions and the original data points. Comment on any differences.

Find a polynomial function which passes through every data point. Explain you choice of function, and discuss its reasonableness. On a new set of axes, draw this model function and the original data points. Comment on any differences.

Using technology, find one other function that fits the data. On a new set of axes, draw this model function and the original data points. Comment on any differences.

Which of you functions found above best models this situation? Explain your choice.

Use you quadratic model to decide where you could place a ninth guide. Discuss the implications of adding a ninth guide to the rod.

Mark has a fishing rod with overall length 300cm. The table shown below gives the distances for each of the line guides from the tip of 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

How well does your quadratic model fit this new data? What changes, if any, would need to be made for that model to fit this data? Discuss any limitations to your model.

Introduction:

Fishing rods use guides to control the line as it is being casted, to ensure an efficient cast, and to restrict the line from tangling. An efficient fishing rod will use multiple, strategically placed guides to maximize its functionality. The placement of these will depend on the number of guides as well as the length of the rod. Companies design mathematical equations to determine the optimal placement of the guides on a rod. Poor guide placement would likely cause for poor fishing quality, dissatisfied customers and thus a less successful company. Therefore it is essential to ensure the guides are properly placed to maximize fishing efficiency.

In this investigation, I will be determining a mathematical model to represent the guide placement of a given fishing rod that has a length of 230cm and given distances for each of the 8 guides from the tip (see data below). Multiple equations will be determined using the given data to provide varying degrees of accuracy. These models can then potentially be used to determine the placement of a 9th guide. Four models will be used: quadratic function, cubic function, septic function and a quadratic regression function.

To begin, suitable variables must be defined and the parameters and constraints must be discussed.

Variables:

Independent Variable:

Let x represent the number of guides beginning from the tip

Number of guides is a discrete value. Since the length of the rod is finite (230cm) then the number of guides is known to be finite.

Domain = , where n is the finite value that represents the maximum number of guides that would fit on the rod.

Dependent Variable:

Let y represent the distance of each guide from the tip of the rod in centimetres.

The distance of each guide is a discrete value.

Range = Parameters/Constraints:

There are several parameters/constraints that need to be verified before proceeding in the investigation. Naturally, since we are talking about a real life situation, there cannot be a negative number of guides (x) or a negative distance from the tip of the rod (y). All values are positive, and therefore all graphs will only be represented in the first quadrant. The other major constraint that must be identified is the maximum length of the rod, 230cm. This restricts the y-value as well as the x-value. The variable n represents the finite number of guides that could possibly be placed on the rod. While it is physically possible to place many guides on the rod, a realistic, maximum number of guides that would still be efficient, is approximately 15 guides.

 Guide Number (from tip) 0* 1 2 3 4 5 6 7 8 n** Distance from Tip (cm) 0 10 23 38 55 74 96 120 149 230

Middle

9. Add (-1 * row 2) to row 1 After all of the row operations, matrix A has become the identity matrix and matrix B has become the values of matrix X (a, b, c). Therefore we have determined that the quadratic equations given the points {(1,10), (6,96), (8,149)} is .

Averaging of the Two Equations

The next step in finding our quadratic function is to average out our established a, b, and c values from the two sets data.   Therefore we have finally determined our quadratic function to be: Rounded to 4 sig figs, too maintain precision, while keeping the numbers manageable.

Data points using quadratic function

 Guide Number (from tip) 1 2 3 4 5 6 7 8 Quadratic values Distance from Tip (cm) 10 22 37 54 74 97 122 149 Original - Distance from Tip (cm) 10 23 38 55 74 96 120 149

New values for the distance from tip were rounded to zero decimal places, to maintain significant figure – the original values used to find the quadratic formula had zero decimal places, so the new ones shouldn’t either.

After finding the y-values given x-values from 1-8 for the quadratic function I was able to compare the new values to the original values (highlighted in green in the table above). We can see that the two values that are the exact same in both data sets is (1,10) and (8,149) which is not surprising since those were the two values that were used in both data sets when finding the quadratic function. Another new value that was the same as the original was (5,74). All other new data sets have an error of approximately ±2cm. This data shows us that the quadratic function can be used to represent the original data with an approximate error of ±2cm. This function is still not perfect, and a better function could be found to represent the data with a lower error and more matching data points.

Cubic Function:

Conclusion

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

To see how well my quadratic model fits this new data, they must be both plotted on the same graph, seen below. My quadratic model for Leo’s fishing rod correlates with Mark’s fishing rod data for the first few values and then diverges as the number of guides increases by growing at a higher exponential rate. The difference between Leo and Mark’s eighth guide from the tip of their respective rods is 25cm, yet both men’s first guides start the same distance from the tip of their rods. The quadratic function used to model Leo’s fishing rod does not correlate well with Mark’s fishing rod data.

Changes to the model must be made for it to fit this data. The best way to find a model for Mark’s data would be to go through the same steps that we went through to determine the first quadratic formula that model’s Leo’s fishing rod. By doing so, specific values that better represent Mark’s fishing rod data could be used to establish a better fitting function.

The main limitation of my model is that is was designed as a function for Leo’s data specifically. It was created by solving systems of equations that used solely Leo’s fishing rod for data. Consequentially, the quadratic model best represented Leo’s fishing rod, which had a maximum length of 230cm, with differently spaced out guides. There were many differences between Leo and Mark’s fishing rods (such as maximum length and guide spacing) that caused my original quadratic model to not well represent Mark’s data.

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

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