• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

FISHING RODS

Extracts from this document...

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. 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 1. Number of guides, with respective distances from the tip, on Leo’s 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

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

                   g= Guide number (from tip)    and      X= distance from tip (cm)

One constrain of the data presented in Table 1 is that there cannot be negative number of guides or negative distances from the tip. As a result the plotted graph is limited to the first quadrant as seen below from    Graph1. Also the placement of the guides (distance) from the tip does not follow a regular pattern.

...read more.

Middle

Equation 2: 38 = A (27) + B (9) + C (3) + D

Equation 3: 74 = A(125) + B (25) + C (5) + D

Equation 4: 149 = A (512) + B (64) + C (8) + D

Then,                         image06.pngimage07.pngimage03.pngimage08.png

image09.pngimage06.pngimage07.pngimage03.pngimage09.pngimage08.png

By using technology the answer is:-

image07.pngimage03.pngimage10.png

By approximation, A= 0.0571, B= 0.486, C= 11.3, D=-1.86, resulting in a cubic model function of

                                           X= 0.0571g3 +0.486g2 +11.3g -1.86

Graph 3. Plot of the cubic model function together with original data points

image11.png

The cubic model fits better than the quadratic one since it passes through 4 points precisely which is one more than the quadratic model making it more accurate. By analysing the previous two models one can notice that the quadratic model which had three unknowns constants passed through three points properly and the cubic model which had four unknown constants passed through 4 points properly. So if a model function needs to pass through all the points it is necessary to have a function with eight unknowns and that can be solved by the matrix method using 8 set of equations. Hence, a 7th degree polynomial with 8 unknowns enables the formation of 8 equations that can be solved by the matrix method to result in a model that passes through all data points. The general expression for a 7th degree polynomial is

                         X= Ag7 +Bg6 +Cg5+ Dg4+ Eg3 + Fg2 + Gg + H

Equation 1: 10 = A(1)7 +B(1)6 +C(1)5+ D(1)4+ E(1)3 + F(1)2 + G(1) + (1)

Equation 2: 23 = A(2)7 +B(2)6 +

...read more.

Conclusion

Table 3. Number of guides, with respective distances from the tip, on 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 the quadratic model fits the new data it can be plotted as follows:-

 Graph 6. Quadratic model, X = 1.21g² + 8.93g-0.143, versus marks’s fishing rod

image20.png

The quadratic model does not fit very well to the new data. It only passes through the first 2 points and missing the rest. In order to see the changes that are needed to improve the model the GDC can be used to find a quadratic best fit line for Mark’s fishing rod. The result is

image21.png

The improved quadratic function is

                X = 0.93g² + 7.72g +2.05

One of the limitations of the model function is that it has to be modified every time an addition guide is added to the fishing rod to make it accurate. According to the improved quadratic function, X = 0.93g² + 7.72g +2.05 , if a ninth and tenth guides are to be added to Mark’s fishing rod, they would be placed at a distance of 147 cm and 172 cm from the tip respectively. Then if the GDC is asked to give the best fit line, it would not give the one that is seen above. Instead it will modify it and comes with a new set of values as follows:-

image22.png

The new modified function for Mark’s fishing rod with ten guides is:-

                           X = 0.91g² + 7.87g +1.83

...read more.

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

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Stopping distances portfolio. In this task, we may develop individual functions that model the ...

    To get a better equation, we will use 3 of the points to find a quadratic equation fitting these points. If we use the points (48,23), (80,53), and (112,96) in the quadratic equation ax� + bx + c = y, we get: ax� + bx + c = y a(48)� + b(48)

  2. A logistic model

    (?3.0 ? 10?5 )(u ) ? c 2.5 ? (?3.0 ?10?5 )(1?104 ) ? c ? c ? 2.5 ? (?3.0 ? 10?5 )(1? 104 ) ? 2.8 Hence the equation of the linear growth factor is: r ? ?3.0 ?10?5 u ? 2.8 {8} n n Using equations {1} and {2}, one can find the equation for un+1: un?1 ?

  1. Tide Modeling

    for the average of the data exactly at the crest and through. At the other points there is a difference between the points. The difference oscillates from a difference of 0.087 to a high 0.725. Therefore in order to have a precise measurement the best degree of accuracy should be to the nearest whole number.

  2. Creating a logistic model

    to: y = Putting this in a graph, we have: Model for Growth Rate = 2.3 When we have a new initial growth rate, the ordered pairs i.e. (un, rn) have now changed to: (60000, 1) (10000, 2.3) To find the linear growth factor, we form the two equations: 2.3 = m(10000)

  1. Fishing rods type 2 portfolio

    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)

  2. In this investigation, I will be modeling the revenue (income) that a firm can ...

    As we know the price (P) already from the linear demand equations, we simply need to multiply everything by Q to get the functions for the revenue. Calculations: Quarter One: 1. Revenue Equation: R=PQ 2. P=-0.75Q+6 3. R=(-0.75Q+6)(Q) Quarter Two: 1.

  1. Investigating transformations of quadratic graphs

    and same value of c (the y-coordinate). In the given equation, b is -4 and c is 5. Therefore the point of the vertex is found at (-b, c) = (4, 5). The following graph proves the aforementioned statement: 4a. Express x2 ? 10x + 25 in the form (x ? h)2 h = 5 b.

  2. SL Math IA: Fishing Rods

    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)

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work