Fishing rods type 2 portfolio

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Math Portfolio Type II: Fishing Rods


A fishing rod can only work well if there are numerous "guides" along the blank of the rod to guide the fishing lines. Having numerous "guides" along the blank of the rod provides greater castability, greater hooking power, and most importantly prevents the line from twisting and tangling up. A fishing rod without the "guides" would have little use since the lines will most likely tangled up before it even enters the water.

In this portfolio, I would have to develop a mathematical model for the effective placement of line guides on a fishing rod. A mathematical model for the placement of line guides on a fishing rod has many applications in the real world. One of the main benefits would be helping fishing rod companies in placing their "guides" in the most optimal positions so the rod can function better than their competitors. The fishing rod that I would have to develop a model on would be Leo's fishing rod. Leo's fishing rod is 230cm long and has eight guides along it with a ninth at the very tip of the rod. The table below shows the values for the distance for each of the line guides from the tip of the fishing rod on Leo's rod.

Table 1: The relationship between the # of guides and the distance from tip (cm)

Part A: Variables and Parameter/Constraints

Before a mathematical model can be made, the variables for the model must first be chosen. The Distance from tip (cm) is dependent on the guide number and therefore is the dependent variable and should have the variable of Y and be on the Y axis in a graph. This means that the Guide number (from tip) should be the independent variable and have the variable of X and be on the X axis in a graph. At the same time, by looking at the data there can also be multiple parameters/constraints that should be addressed before a model can be made. First of all for variable X, it can be implied that the number of guides must be 0 > since a negative number of guides cannot exist in the real world. At the same time, the distance from tip (cm) or the Y variable must also be 0 > since a negative distance also doesn't exist. On top of that, the rod length of 230 cm also affects the Y variable which means the distance from tip must be < 230 cm. This also means that the X variable is also affected and that the number of guides is finite and not infinite. All in all, these parameters and constraints mean that all of the values in a graph would be in the first quadrant.

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{ X E l K l | 0 < X < 1 0 } * The maximum # of guide depends on the model used

Graph 1: The relationship between the # of guides and the distance from tip (cm)

Part B: Determining a quadratic model that fits the situation using systems of equations

A quadratic equation in its standard form is ax^2 + bx + c where a, b, and c are the coefficients and "a" in the equation can never be 0.

To find a quadratic equation to model this fishing rod situation, ...

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