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# fishing rods portfolio (SL maths)This portfolio deals with Leo's fishing rod which has an overall length of 230 cm together with eight guides that are placed a certain distance from the tip

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Introduction

FISHING RODS SL TYPE II

This portfolio deals with Leo's fishing rod which has an overall length of 230 cm together with eight guides that are placed a certain distance from the tip of the fishing rod as shown in Table 1.

Table 1

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

The task therefore is to develop mathematical models for the placement of the line guides on the fishing rod using quadratic, cubic, polynomial and one other free function. In addition, the quadratic model function that is developed will be further tested by applying it to Mark's fishing rod which has an overall length of 300 cm and eight guides. Table 2, shows the distances for each of the line guides from the tip of Mark’s fishing rod.

Table 2

 Guide Number (from tip) 1 2 3 4 5 6 7 8 Distance from tip (cm) 10 22 34 48 64 81 102 124

Before beginning the process of formulating different mathematical models, it is possible to mention certain constraints as well as variables. The two variables in this modeling are

Middle

Furthermore, the   variable, is the Independent Variable, and the   variable is the Dependant Variable. Table 1, which gives the distances for each of the line guides from the tip of Leo’s fishing rod, is plotted on the graph below:

I will consider the variables as a sequence, and from there I shall calculate the polynomial equation modeling this situation.

First I will find the pattern in the sequence. To find the pattern, I will list the numbers, and find the differences for each pair of numbers. That is, I will subtract the numbers in pairs (the first from the second, the second from the third, and so on), like this:

10        23        38        55        74        96        120        149

13              15               17               19               22                24                29

Since these values, the "first differences", are not all the same value, I'll continue subtracting:

10        23        38        55        74        96        120        149

13              15               17               19               22                24                29

2        2        2        2        2          5

Since these values, the "second differences", are all the same value, then I can stop. It isn't important what the second difference is (in this case, "2"); what is important is that the second differences are the same, because this tells me that the polynomial for this sequence of values is a quadratic.

Conclusion

y = 1.214x2 + 8.738x + 0.464. In order to find the distance of the guide from the tip, I will substitute x with 9 in the equation. 9 is the guide number. I get y (distance from the tip) to be equal to 177.44 cm, which is approximately equal to 178 cm. Adding a ninth guide, forms a curve which is shown below:

If I compare this curve, with the original curve of 8 guides, we can see that the new curve is slightly steeper than the second curve.

Hence, I consider that the implication of adding a ninth guide to the fishing rod is a good one. This is because the last guide (9th guide) is closest to the reel. It allows the line to come off the pool much easier. Basically to sum it up, the more guides the more line control, and hence the farther and smoother the cast, giving a smoother retrieve.

Table 2, shows the distances for each of the line guides from the tip of Mark’s fishing rod.

Table 2

 Guide Number (from tip) 1 2 3 4 5 6 7 8 Distance from tip (cm) 10 22 34 48 64 81 102 124

The graph below shows how well my quadratic model fits this new data:

As we can see, Mark’s fishing rod’s set of data points forms a less steep curve than Leo’s fishing rod.

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

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