- Level: International Baccalaureate
- Subject: Maths
- Word count: 1121
IB Fish
Extracts from this document...
Introduction
Alice Wang
Fishing Rod Lab
Table 1. 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 |
Dependent variable: The distance from the tip of fishing rod
Independent variable: The guide number from the tip of the fishing rod
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.
Graph 1. Leo’s fishing rod
Consider general form of cubic and quadratic equations (where a,b,c,d are coefficients and where a is never zero):
y=ax2 +bx+c (quadratic)
y=ax3 +bx2 +cx+d (cubic)
10= a+ b + c
23= 4a+ 2b + c
38= 9a+3b+c
Put into matrix:
First, to clear the first 0 of the column, the first row has to be multiplied by 9 and subtracted from the second row.
To clear the second 0 of the first column, the second row has to be multiplied by 3 and be subtracted to the third row.
To clear the second column’s last row to a 0, the second row has to multiplied by ½.
Middle
Table 2. Data points of Cubic Function
1 | 9.869 |
2 | 23.186 |
3 | 38.043 |
4 | 54.818 |
5 | 73.889 |
6 | 95.634 |
7 | 120.43 |
8 | 180.69 |
Thus, the equation of the cubic equation is y =.063x3 +.392x2 +11.7x-2.286
Cubic Function Graph (above)
Quadratic Function (above)
The main difference between quadratic and cubic functions is that a quadratic equation is a polynomial fit using three points while a cubic function uses four or more points and is a polynomial regression. With a higher degree for a polynomial function, there is a less degree of accuracy, especially when the guide number increases. The quadratic model is more precise than the quadratic function when the guide number reached 7 and 8.
I used the function to find a function which passes through every data point in Table 1. With the 7th degree in a polynomial, it will pass through every data point of Leo’s fishing rod.
Subsitution of Table 1’s x and y values to is as followed:
Conclusion
Table 5. 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 |
The quadratic model of y=x2 +10x-1 fits the new data adequately. Although the fit of the quadratic function is not perfect, it closely represents the data provided.
Using Excel, the line of best fit was considered y= 0.9345x + 7.7202x + 2.0536 as a polynomial function. The quadratic model of y=x2 +10x-1 and the line of best fit are extremely similar (demonstrated on the graph below). The narrower function is the line of best fit and the wider function is the quadratic model. My quadratic model could be more vertically compressed by multiplying the value of a (from y=a2+bx+c) by a positive integer. The limitations of the quadratic model is that it is only suited for fishing rods that are similar to Leo’s. If the distance from tip of each guide number of a fishing rod is substantially different that Leo’s fishing rod, the quadratic model of y=x2 +10x-1 will not work. In this case, Mark’s distance from tip of each guide number was similar to Leo’s. Thus, the quadratic model worked for Mark’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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