• Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

This assignment aims to develop a mathematical model for the placement of lines on a fishing rod

Extracts from this document...

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.

...read more.

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

...read more.

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

image03.png

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:

  1. Polynomial models have poor interpolatory properties. High-degree polynomials are notorious for oscillations between exact-fit values.
  2. 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.
  3. 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.
  4. 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

...read more.

This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.

Found what you're looking for?

  • Start learning 29% faster today
  • 150,000+ documents available
  • Just £6.99 a month

Not the one? Search for your essay title...
  • Join over 1.2 million students every month
  • Accelerate your learning by 29%
  • Unlimited access from just £6.99 per month

See related essaysSee related essays

Related International Baccalaureate Maths essays

  1. Tide Modeling

    Unfortunately the models are not able to show the oscillation of heights that the excel graph presents. It rather shows a perfectly periodical graph that probably would not exist in nature under normal circumstances. After a more extensive research it was possible to notice that the model is only accurate

  2. Virus Modelling

    to die (after there are 1012 particles): y=1000000�100.0496x y=1000000 x=Time 1000000000000=1000000�100.0496x 1000000=100.0496x Log101000000=0.0496xLog1010 (Log1010 = 1) 6=0.0496x x=120.97 (2 d.p.) 120.97hrs = 120hrs 58mins 12secs 120.97hrs is not how long it will take for the patient to die. As I mentioned above it would take 26.56hrs before the immune response would begin and so that has to be added.

  1. Math IA Type 1 In this task I will investigate the patterns in the ...

    It is given that parabola intersects the lines y=mx+d and y=nx+e. Therefore in order to find D, their intersections must be found. Their intersection can be found be equation the function of the parabola and the function of the line.

  2. A logistic model

    One can write two ordered pairs (1?104 , 2.3) , (6 ?104 , 1) . The graph of the two ordered pairs is: 5 IB Mathematics HL Type II Portfolio: Creating a logistic model International School of Helsingborg - Christian Jorgensen Plot of population U n versus the growth factor r 2,5 2,4 2,3 2,2 2,1 2 1,9 1,8

  1. Analysis of Functions. The factors of decreasing and decreasing intervals (in the y ...

    The relation of periodicity, however, holds for any change to x, so it can also be accepted the idea that polynomial functions are periodic functions with no period. In the case of maximums and minimums for any case of the functions there would be always absolute maximums and minimums.

  2. Population trends. The aim of this investigation is to find out more about different ...

    The following model is a logarithmic model, the normal curve is very curved and isn't at all what the data looks like in a graph. It starts increasing disproportionally, it then curves and starts to decrease the increase per year of population.

  1. Parabola investigation. The property that was investigated was the relationship between the parabola and ...

    Name it y Now use the formula x-((0.01y)+0.01). The real value of D would be got by the end of this operation. In the above conjecture, the difference is just taken away from the conjecture based value of D. An example is shown below. Take the graph y = 1.27x 2 ? 6x + 11 for example.

  2. SL Math IA: Fishing Rods

    First, by using the x-values 1 and 8 in both sets of data, we will have a broad range of all of the data that is being represented in the final equation after the values of the coefficients are averaged.

  • Over 160,000 pieces
    of student written work
  • Annotated by
    experienced teachers
  • Ideas and feedback to
    improve your own work