- Level: International Baccalaureate
- Subject: Maths
- Word count: 1974
FISHING RODS
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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. 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.
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,
By using technology the answer is:-
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
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 +
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
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
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:-
The new modified function for Mark’s fishing rod with ten guides is:-
X = 0.91g² + 7.87g +1.83
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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