- Level: International Baccalaureate
- Subject: Maths
- Word count: 3036
Math IA Type II _ Modelling the Length of a Fishing Rod
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Introduction
Mariya LupandinaSummative. IA Type II: Fishing RodsMay 14, 2012
An efficient fishing rod requires numerous guides to ensure that the line casts easily and does not tangled. A mathematical model can be used to help predict the optimal placement of guides in order to achieve a successful launch. In this investigation, an appropriate model will be found that can mathematically show the placement of the line guides. We have been provided with Leo’s fishing rod, which has an overall length of 230 cm and eight line guides, plus a guide at the tip of the rod. The distance for each line guide from the tip of Leo’s fishing rod is shown in Table 1, below. This investigation will develop mathematical models for the placement of the line guides using a quadratic, a cubic, and a polynomial function, as well as one other free function. To graph these function we will use the TI-Nspire Student Software, and the TI-Nspire CX will be used for calcualtion. The developed quadratic model will be further tested by applying it to Mark’s fishing rod.
Table 1. The number of guides and there respective distances 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 |
Ninth guide on the tip (0,0)
Prior to beginning the investigation it is pertinent to define the suitable variables and identify any parameters or constraints in relation to the fishing rod. In this investigation our independent variable, defined as x, is the guide number (from the tip); and our dependent variable, defined as y, is the distance from the tip. The distance from the tip will be measured in centimeters; the distance is therefore a continuous variable since it is uncountable.
Middle
Using the GDC we are able to find the following values:
By approximating, to three significant digits, the following are the values for the four unknowns: a≈0.0571, b≈0.486, c≈11.3, and d≈ -1.86; resulting in a cubic model function of:
y ≈ 0.0571x3+0.486x2+11.3x-1.86
OR
g(x) ≈ 0.0571x3+0.486x2+11.3x-1.86
The comparison of the cubic model function with the original data points can be seen in Graph 2.
Graph 2. Comparison of quadratic model and cubic model with the original data points
Distance from Tip (cm)
Guide Number (from tip)
From Graph 2, above, it can be seen that the quadratic function only passes through the center of three points making the quadratic model far from a desired model function, and the cubic model passes through the center of four points, making it more accurate than the quadratic model. Generally, the quadratic and cubic models are fairly accurate since they pass through almost all of the points, however neither is the desired function for modeling the optimal placement of guides on a fishing rod because they do not pass through the center of each point. Graph 3 confirms this, as we can see that the quadratic function (f(x)) does not even pass through the 6th and 7th guides, and the cubic function (g(x)) hardly passes through both points.
Graph 3. Zoom-in of points 6 and 7 from Graph 2
Guide Number (from tip)
Distance from Tip (cm)
By analyzing the previous two models we notice that the quadratic model, which has three unknown constant, passes properly through three points and the cubic model, which has four unknown constants passes properly through four points.
Conclusion
Graph 9. Quadratic model, ƒ(x)=1.29x2+8.29x+0.429, versus Mark’s fishing rod.
Guide Number (from tip)
Distance from tip (cm)
The quadratic model function does not fit the new data very effectively. It only passes through two points and completely misses the rest, as its exponential increase is greater. In order to understand the changes that are needed to improve the model we can use a quadratic regression on the GDC to find a line of best fit for Mark’s fishing rod.
The result is as follows:
And the improved quadratic model function is:
f(x)=0.935x2+7.72x+2.05
One limitation of the quadratic model is that it only applies to fishing rods similar to Leo’s, as we saw with Mark’s set of data, even though Mark’s fishing rod was not very different. Another limitation of the quadratic model is that the model function must be adjusted every time a guide is added to the fishing rod. According to the new, improved quadratic model function for Mark’s fishing rod, f(x)=0.935x2+7.72x+2.05, if a ninth and tenth guide were added to Mark’s fishing rod, they would be placed 147 cm and 172 cm from the tip, respectively. If we were to create a regression of the data, we would obtain values different from the ones above, the new set of values is below.
The new modified function for Mark’s fishing rod is:
f(x)=0.917x2+7.87x+1.83
Another limitation of the quadratic model is that it does not take into account the length of the fishing rod. A fishing rod with a different length would require more guides, which would result in new data points. Different data points would result in a new curve, which the original model will not fit.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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