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

Fishing Rods Internal Assessment

Extracts from this document...

Introduction

Fishing Rods

SL Type II

Henry Deng

143439

Block B

Leslie

An efficient fishing rod requires numerous guides for the line or else it will get tangled. Companies must find a strategic position for them or else the rod will become obsolete. As a result, a mathematical model can help companies predict where to place these line guides in order for a successful launch. Imagine customers launching a fishing rod that gets caught up before landing in the water! Not a very pleasant experience, is it? In this portfolio, I will be finding an appropriate model that can mathematically show the placement of line guides on a fishing rod. To do so, I will be using a given diagram, whereas: a fishing rod has an overall length of 230cm. The distances for each of the guides from the tip of the fishing rod is shown in a table on the back, but I will be copying it out, so one can refer to it while reading.

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)

Before beginning, I will be identifying any parameters or constraints in relation to the fishing rod. First of all, it is common sense that a negative number of guides cannot exist in reality, just like there cannot be negative numbers of apples, and as a result, negative distances are neglected. As a result, the graph must be limited to the first quadrant because the second, third, and fourth ones contain negative numbers which don't exist in a fishing rod.

...read more.

Middle

Next, isolate the variables by multiplying  image01.png with its inverse which will result in the identity matrix. What is done on one side will result in the other side, so the final equation will look like this:

image12.png

After that, matrix “A” turns into the identity matrix and the inverse of matrix “A” is used to multiply with matrix “B”. Using technology, the answer is:

image22.png

 ∴ A = 1.21, B = 8.93, C = -0.143, rounded to three significant figures

Let's revisit the general quadratic equation and substitute these three values in.

The quadratic model of the guide placements of a fishing rod can be represented as:

image17.png

On the next page, there will be a comparison of the quadratic model with the data points.

For a cubic model, I will use the same methods except, use the general equation of a cubic equation:

image23.png

This time, 4 ordered pairs will be required and by using the same method as the previous model, I chose them in an interval of two (intervals of three does not work). The start guide is the first ordered pair, and then two up; the third guide, two up again; the fifth guide, and lastly, the end guide.

Substitution:

Equation 1: 10 = A (1) + B (1) + C (1) +D

Equation 2 : 38 = A (27) + B (9) + C (3) + D

Equation 3: 74 = A (125) + B (25) + C (5) + D

Equation 4: 149 = A (512) + B (64) + C (8) + D

Now, to represent these equations in matrix form:

image24.png

Using the same methods to isolate the unknown variables (A,B,C,D), I will multiply each side with the inverse of matrix “A”. Using technology, the answer is:

image25.png

∴ A = 0.0571, B = 0.486, C = 11.3, D = -1.

...read more.

Conclusion

th powered model is definitely the most accruate.

As stated above, both a quadratic and cubic model was solved by using matrix methods. Here's a comparison of the quadratic and cubic regression equations in relation to the model functions.

image17.png  Model Function – Quadratic

image18.pngRegression – Quadratic

image19.png Model Function – Cubic

image20.pngRegression – Cubic

As one can see, these two equations do not differ greatly as expected because essentially, a regression is a best fit line which should be equivalent to a model that passes through most of the points. Of course, using technology would result in a more accurate answer, but the use of matrix methods is crucial if one does not have knowledge about regressions.

After exploring a fishing rod with eight guides, it would be interesting to guess where  would the placement of a ninth guide be (excluding the tip) and measure the effects of this phenomenon. By using my quadratic model, it's quite simple to find out where a ninth guide could possibly be. Since it's just a general idea, the use of complicated functions are not needed; thus, a quadratic model should be sufficient.  

Let's revisit the quadratic model:

image17.png

As mentioned before, the variables: X is the distance from tip (cm) and g is the guide number (from tip). So, to find the ninth guide, all we need to do is substitute g with the number nine.

image21.png, which ends up to around 178cm.

Therefore, if a ninth guide was to be added to the fishing rod, it would be located approximately 178cm from the tip (0,0). The effects of having this ninth guide would shorten the line and therefore, it may cause difficulty fishing as it can't reach the water.

...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. Maths Internal Assessment -triangular and stellar numbers

    By finding this link, it was possible to create an expression for the 6-stellar number which linked with the triangular numbers.

  2. Mathematics Higher Level Internal Assessment Investigating the Sin Curve

    If we look at the first equation, then the first thing to do in that equation would be to split the fractions so that the equation now looks like: . Once that has been done we can now factorize the brackets' section by , and therefore the equation would look like: .

  1. A logistic model

    (?3.0 ? 10?5 )(u ) ? c 2.5 ? (?3.0 ?10?5 )(1?104 ) ? c ? c ? 2.5 ? (?3.0 ? 10?5 )(1? 104 ) ? 2.8 Hence the equation of the linear growth factor is: r ? ?3.0 ?10?5 u ? 2.8 {8} n n Using equations {1} and {2}, one can find the equation for un+1: un?1 ?

  2. Lacsap's Fractions : Internal Assessment

    = Denominator or - r( n - r) = Denominator To find the sixth and seventh row, the general statement of the numerator and the denominator must be combined to create the general statement for the element, En (r), as shown below: En (r)

  1. Math Portfolio Type II

    the population has an increasing decreasing trend compared to when the harvest size is 5000. Also, the population does not seem to stabilize by the 35th year. By further investigating, the stable population is found to be in the 55th year.

  2. IB Math Methods SL: Internal Assessment on Gold Medal Heights

    Upon converting that data to a table and simplifying it to three significant figures; we arrive at this table. Information Table 2 (Table of values from linear equation y = 1.02x + 187) Years Elapsed (t) 0 4 8* 12* 16 20 24 28 32 36 40 44 48 Height in cm (h)

  1. In this Internal Assessment, functions that best model the population of China from 1950-1995 ...

    A cubic fit applied to the data points via Graphical Analysis 3 follows: Next, a model function can be developed to fit the data points of the graph. In this investigation, a linear function will be created. A linear function, as discussed above, is a function in the form y=Mx+B.

  2. SL Math IA: Fishing Rods

    Since the length of the rod is finite (230cm) then the number of guides is known to be finite. Domain = , where n is the finite value that represents the maximum number of guides that would fit on the rod. Dependent Variable: Let y represent the distance of each guide from the tip of the rod in centimetres.

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