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.

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.

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.

*the guide at the tip of the rod is not counted

**n is the finite value that represents the maximum number of guides that would fit on the rod.

Neither of the highlighted values are analyzed in this investigation, they are only here for the purpose of defining the limits of the variables.

The first step in this investigation is to graph the points in the table above (excluding highlighted points) to see the shape of the trend that is created as more guides are added to the rod.

From this scatter plot of the points, we can see that there is an exponential increase in the distance from the tip of the rod as each subsequent guide is added to the rod.