- Level: International Baccalaureate
- Subject: Maths
- Word count: 3755
Fishing Rods
Extracts from this document...
Introduction
Alex Knights
Year 11 IB Maths – Portfolio Type II
Fishing Rods
A fishing rod requires guides for the line so that it does not tangle and so that the line casts easily and efficiently. In this investigation, a mathematical model will be developed using matrix methods, polynomial functions, and technology to calculate functions from the given data points of two fishing rods of lengths 230cm and 300cm. This model will further determine the placement of the line guides on the fishing rod.
In mathematics, a function is a relation between a given set of elements and another set of elements that associate with each other, algebraically and graphically. In this investigation, an approach using matrices will be attempted to calculate functions for the given data and then be plotted to verify the results. Furthermore, there will be an effective use of technology, using Graphic calculator and excel, so as to minimize errors and flaws.
The first investigation is of Leo’s fishing rod:
Leo has a fishing Rod with overall length 230cm. The table below gives the distance for each of the line guides from the tip of the 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 |
Firstly, before a mathematical model can be formulated, we must outline and define the variables and constraints associated with the values given above.
Independent variable (x): The guide number from the tip of the fishing rod – let this equal g
Dependent variable (y): The distance from the tip of fishing rod – let this equal d
Parameters/Constraints:
- The distance from tip for each guide number does not follow a particular pattern. Hence it is difficult to achieve a function that satisfies all of the points on Table 1.
- Model of a real life situation so there must be space for a reel and to hold the rod – limits the space the guide’s can have between each other
- The overall length of the fishing rod – it cannot be negative or too long as this will reduce efficiency
Middle
Using the 10 different data points, that represent the spread of the data, 5 equations can be formed:
The three equations can then be transformed to form a 5 x 5 and a 5 x 1 matrix with the corresponding coefficients:
We will call this first matrix - Matrix A
We will call the second matrix – Matrix B
We will now take the inverse of Matrix A and multiply it with Matrix B to get Matrix C, using the graphics calculator. Matrix C will contain the a,b,c ,d and e values of the quadratic equation in a 5x1 Matrix.
=
Therefore the respective values of the unknowns a,b,c,d and e are a= -0.005, b=0.138, c=0.033 d=12.262 and e=-2.429. These values can then be subbed into the original equation of the quartic to form the quartic function modelling the situation:
Using Excel, we can now take this new function of and plot it on an axes along with the graph of the original data points to evaluate the accuracy of the function in modelling the situation.
Green Line: Original Data Points
Red Line:
In the graph above, it can clearly be seen that the function accurately models the original data points given when Leo has a rod of overall length 230cm. As the red line representing the function of is not visible behind the graph of the original data points, this demonstrates the near complete accuracy of the function in modelling the situation. Using a graphics calculator, the data points of the function can be found and then compared with the original data points to assess the differences between them.
Guide number (from tip) | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 |
Distance from tip (cm) –Original Data Points | 10 | 23 | 38 | 55 | 74 | 96 | 120 | 149 |
Distance from tip (cm) - | 9.999 | 23.251 | 37.975 | 54.699 | 73.831 | 95.659 | 120.35 | 147.96 |
As seen in the table above, the values found from the function
Conclusion
Throughout this investigation, we have assessed various models of quadratic, cubic and quartic function, found through both matrix methods and technology to model the situation’s of both Leo and Marks fishing Rods. From the different values we have found for each equation, it is possible to determine that the most accurate graph that could be found is that of an eighth order function given we were given 8 data points. However, as this was not possible with a graphics calculator, by determining the values of quadratic, cubic and quartic functions, we were able to assess that the function found using technology, and with the highest power, the quartic regression equation, was the most accurate.
As the investigation requires an application to a real life situation, our method carried out allows for various conclusions to be drawn as the placement of guides on a fishing rod. For Leo’s rod of 230cm, our quadratic functions found modelled the situation, but with a degree of error that meant it could not be deemed totally accurate in relation to the data points given. As our next functions, the cubic and quartic modelled the situation with a lesser degree or error we can conclude that as the power of the polynomial increases, the accuracy of the function increases. Secondly, it can also be seen that by adding a ninth guide, the implications could be positive in nature, easier cast and less tangle, or negative, decreased ability to cast and increased difficulty for the fisherman in the use of the rod. Our final conclusion is drawn from our inability to apply the quadratic function of Leo’s rod to fit the data of Mark’s. This shows that when the length of the fishing rod is changed, so too must the equation be changed.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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