- Level: International Baccalaureate
- Subject: Maths
- Word count: 2374
This assignment aims to develop a mathematical model for the placement of lines on a fishing rod
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Introduction
MATHS ASSIGNMENT
FISHING ROD – MODELLING TASK
This assignment aims to develop a mathematical model for the placement of lines on a fishing rod by investigating different methods (matrix methods, polynomial functions, and technology) that model a given set of data and discovering the equation that best models the data.
Leo has a fishing rod with length 230cm.
The given data about his rod is:
Guide number: 1, 2, 3, 4, 5, 6, 8
Distance from tip (cm): 10, 23, 38, 55, 74, 96, 120, 149
Define suitable variables, discuss parameters / constraints.
x (independent variable): Guide number from the tip of the rod (1, 2, 3, 4, 5, 6, 7, 8)
y (dependent variable): Distance from the tip in centimeters (10, 23, 38, 55, 74, 96, 120, 149)
Constraints on x: Whole number; Positive number; Greater than or equal to 1; Smaller than or equal to 8 (may not need this last requirement cause we may add guide?)
Constraints on y: Real number; Positive number; Less than length of fishing rod (230cm); Space accommodated for reel and handle further limits the space between the guides
Since x and y cannot be negative, the plotted graph will be limited to the first quadrant.
Using technology, plot the data points on a graph.
(insert graph)
The scatter plot resembles a curve and probably represents a part of a quadratic function.
Let f(x) = y.
Notice that:
f(2) – f(1) = 15
f(3) – f(2) = 17
f(4) – f(3) = 19
And the difference between these are all 2.
Middle
a(4)³ + b(4)² + c(4) +1d = 55
which is
1 a + 1 b + 1 c + 1 d = 10
8 a + 4 b + 2 c + 1 d = 23
27 a + 9 b + 3 c + 1 d = 38
64 a + 16 b + 4 c + 1 d = 55
As we carry on, we will find that the cubic equation is y=0x3+x²+10x-1, which is the same as the quadratic equation we found earlier in the previous section.
Therefore, we pick:
(1, 10), (4, 55), (6, 96), (8, 149)
Form 4 equations:
a(1)³ + b(1)² + c(1) +1d = 10
a(4)³ + b(4)² + c(4) +1d = 55
a(6)³ + b(6)² + c(6) +1d = 96
a(8)³ + b(8)² + c(8) +1d = 149
which is
1 a + 1 b + 1 c + 1 d = 10
64 a + 16 b + 4 c + 1 d = 55
216 a + 36 b + 6 c + 1 d = 96
512 a + 64 b + 8 c + 1 d = 149
Transform the equations into matrices:
1…1…1...1...|10
64…16…4…1...|55
216…36...6...1…|96
512…64...8…1...|149
Take -64 x row 1 + row 2 → new row 2
Take -216 x row 1 + row 3 → new row 3
Take -512 x row 1 + row 4 → new row 4
1…1…1…1…|10
0…-48…-60…-63…|-585
0…-180...-210...-215…|-2064
0...-448...-504...-511...|-4971
New row 2 = old row 2 / -48
1…1…1…1…|10
0…1…5/4…21/16…|195/16
0…-180…-210…-215…|-2064
0…-448…-504…-511…|-4971
-row 2 + row 1 → new row 1
180 x row 2 + row 3 → new row 3
448 x row 2 + row 4 → new row 4
1…0…-1/4…-5/16…|35/16
0…1…5/4…21/16…|195/16
0…0…15…85/4…|519/4
0...0…56…77…|489
Divide row 3 by 15 to get new row 3
1…0…-1/4…-5/16…|35/16
0…1…5/4…21/16…|195/16
0…0…1…17/12…|173/20
0…0…56…77…|489
1/4 row 3 + row 1 → new row 1
-5/4 x row 3 + row 2 → new row 2
-56 x row 3 + row 4 → new row 4
1…0…0…1/24…|-1/40
0…1…0…-11/24…|55/40
0…0…1…17/12…|173/20
0…0…0…-7/3…|23/5
Divide row 4 by -7/3
1…0…0…1/24…|-1/40
0…1…0…11/24…|55/40
0…0…1…17/12…|173/20
0…0…0…1…|-69/35
-1/24 row 4 + row 1 → new row 1
11/24 x row 4 + row 2 → new row 2
-17/12 x row 4 + row 3 → new row 3
1…0…0…0…|2/35
0…1…0…0…|33/70
0…0…1…0…|801/70
0…0…0…1…|-69/35
a = 2/35, b = 33/70, c = 801/70 and d = -69/35
Conclusion
(1, 10), (4, 48), (8, 124)
Form 3 equations:
a(1)²+b(1)+c=10
a(4)²+b(4)+c=48
a(8)²+b(8)+c=124
which is
a + b + c = 10
16a + 4b + c = 48
64a + 8b + c = 124
Conducting a similar process as before:
1…1…1…|10
16…4…1…|48
64…8…1…|124
We get the quadratic function as y=(19/21)x²+(57/7)x+(20/21)
Limitations to my model:
The quadratic model for Leo’s fishing rod is not universal and is probably only meant for a 230cm rod.
Limitations to polynomial models in general:
- Polynomial models have poor interpolatory properties. High-degree polynomials are notorious for oscillations between exact-fit values.
- Polynomial models have poor extrapolatory properties. Polynomials may provide good fits within the range of data, but they will frequently deteriorate rapidly outside the range of the data.
- Polynomial models have poor asymptotic properties. By their nature, polynomials have a finite response for finite x values and have an infinite response if and only if the x value is infinite. Thus polynomials may not model asymptotic phenomena very well.
- While no procedure is immune to the bias-variance tradeoff, polynomial models exhibit a particularly poor tradeoff between shape and degree. In order to model data with a complicated structure, the degree of the model must be high, indicating that the associated number of parameters to be estimated will also be high. This can result in highly unstable models.
^ Summarize as necessary
Source: http://en.wikipedia.org/wiki/Polynomial_and_rational_function_modeling
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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