Fishing Rods Internal Assessment

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Fishing Rods

SL Type II

Henry Deng

143439

Block B

Leslie

An efficient fishing rod requires numerous guides for the line or else it will get tangled. Companies must find a strategic position for them or else the rod will become obsolete. As a result, a mathematical model can help companies predict where to place these line guides in order for a successful launch. Imagine customers launching a fishing rod that gets caught up before landing in the water! Not a very pleasant experience, is it? In this portfolio, I will be finding an appropriate model that can mathematically show the placement of line guides on a fishing rod. To do so, I will be using a given diagram, whereas: a fishing rod has an overall length of 230cm. The distances for each of the guides from the tip of the fishing rod is shown in a table on the back, but I will be copying it out, so one can refer to it while reading.

Ninth guide on the tip, (0,0)

Before beginning, I will be identifying any parameters or constraints in relation to the fishing rod. First of all, it is common sense that a negative number of guides cannot exist in reality, just like there cannot be negative numbers of apples, and as a result, negative distances are neglected. As a result, the graph must be limited to the first quadrant because the second, third, and fourth ones contain negative numbers which don't exist in a fishing rod. Similarly, we are limited in our calculations to the fact that all fishing rods have a certain length; thus, the addition of infinite guides is not possible. The ninth guide on the tip will be included in the calculations because it is technically part of the rod. If we want to model an accurate situation, it is evident that the fishing line must pass by the ninth guide one way or another in order for the mechanism to work. In this portfolio, the variables I will use are:

g  = Guide number (from tip)

X = Distance from tip (cm)

Now, that I have listed some possible restraints and variables of the model, I will be plotting the data points on graph to provide a pictoral representation of the general shape of the fishing rod.

Although we know the general shape of the fishing rod and the location of the guides, companies need a mathematical model that can help predict the most efficient placement of other guides. By using matrix methods, I will describe a way to model this situation using a quadratic function and a cubic function. To those who are not familiar with matrices, they are a rectangular table of numbers, or more generally, any abstract quantity that can be added or multiplied. Matrices can be used to depict linear equations, trace various transformations and to record data that depend on multiple parameters. They can also be used to solve a systems of equation by using this general formula:

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A-1AX = A-1B, whereas A-1 is the inverse of the matrix A.

To begin the model, it is crucial to realize the general equation of a quadratic function which is:

0 = Ax² + Bx + C , which will be referred as X = Ag² + Bg + C in the portfolio whereas g is equal to the guide number (from tip) and X is equivalent for the distance from tip (cm).

Now, that I have established the premises of what is necessary in order to create  a mathematical model that represents the placement of guides on ...

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