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Fishing Rod Portfolio

Extracts from this document...

Introduction

Catherine Wowk

Period 5 Pre-Calculus

Marking Period 1 Portfolio

Fishing Rod

A fishing rod requires guides for the line so that it does not tangle and so that the line casts easily and efficiently.  Leo has a fishing rod with overall length 230 cm.  The table shown below gives the distance for each of the line guides from the tip of his 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

Variables:

x is the guide number from the tip of the rod (1, 2, 3, 4, 5, 6, 7, 8)

y is the distance from the tip in centimeters (10, 23, 38, 55, 74, 96, 120, 149)

Constraints:

1 ≤ x ≤ 8

10 ≤ y ≤ 149

The above table is plotted on the graph below:

image00.png

The graph above has a window of the following:

image01.png

In order to find a quadratic function to model this situation the matrix method was used to find a, b, and c, of the quadratic function. The following three points were chosen from the above table to use in the matrix to find a quadratic function.  In order to be sure the line would pass through the first line guide and the last line guide the first and last points were specifically chosen, along with a point in the middle of the two.

(1,10), (5,74), (8,149)

The x variables 1, 5, and 8 were plugged into y=ax²+bx+c in order to find Matrix A.

image07.png

The y variables 10, 74, and 149 were plugged into y=ax²+bx+c in order to find Matrix B.

image13.png

The matrices were solved by multiplying the inverse of Matrix A with Matrix B.

...read more.

Middle

image08.png

y=1.2857x²+8.2857x+.42847                     y=.0524ximage03.png+.5429ximage04.png+11.1476x-1.7429

To find a polynomial function that passes through every data point, all of the data points were used in the matrix method that will help us find the equation y=aximage10.png+bximage11.png+cximage12.png+dximage14.png+eximage03.png+fximage04.png+gx+h.  The more data points used to find the matrices the more accurate the graph of the polynomial will be.  By using all of the data points we will be sure that it will pass through all of them when graphed.

(1,10), (2,23), (3,38), (4,55), (5,74), (6, 96), (7,120), (8,149)

The x variables 1, 2, 3, 4, 5, 6, 7, and 8 were plugged into the equation y=aximage10.png+bximage11.png+cximage12.png+dximage14.png+eximage03.png+fximage04.png+gx+h to find Matrix A.

Polynomial Function:image15.png

image17.png

image18.png

image21.pngimage20.pngimage19.png

The y variables 10, 23, 38, 55, 74, 96, 120, and 149 were plugged into the equation y=aximage10.png+bximage11.png+cximage12.png+dximage14.png+eximage03.png+fximage04.png+gx+h to find Matrix B.

image22.png

image23.png

To solve the matrices, the inverse of Matrix A must be multiplied by Matrix B.  The matrix function on the calculator can be used.

image24.png

image25.png

image26.png

a=.0025793651

b= -.077777777

c=.9555555552

d=-6.152777775

e=22.25138888

f=-43.76944443

g=55.79047617

h=-18.99999999

After replacing the variables with the numbers, the polynomial function was found.  

y=.0025793651ximage10.png-.077777777ximage11.png+.9555555552ximage12.png-6.152777775image14.png+22.25138888ximage03.png-43.76944443ximage04.png+55.79047617x-18.999999999

The polynomial function is graphed below with the original data points.  The graph passes through each data point.

image27.jpg

The above graph has the following window:

image01.png

Shown below is the cubic function and the polynomial function.  The cubic function is on the left and the polynomial function is on the right.

...read more.

Conclusion

The table below shows the ninth tip compared to the other eight.

Guide number

(from tip)

1

2

3

4

5

6

7

8

9

Distance from tip (cm)

10

23

38

55

74

96

120

149

179.15

Mark has a fishing rod with an overall length of 300 cm.  The table shown below gives the distances for each of the line guides from the tip of 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 function, y=1.2857x²+8.2857x+.42847, from the first situation, is graphed below along with the data points from above.

image32.png

The above graph has a window of the following:

image33.png

The quadratic line used in the first situation does not fit the data points in the second situation.  A new quadratic model should be found using the matrix method.

The points (1,10), (4,48) and (8,124) were used in finding the new quadratic function.

The x values 1, 4, and 8 were plugged into y=aximage04.png+bx+c to find Matrix A

image34.png

The y values 10, 48, and 124 were plugged into y= aximage04.png+bx+c to find Matrix B

image35.png

The inverse of Matrix A is multiplied by Matrix B

image36.png

a= .9047619048

b= 8.142857143

c= .9523809524

The quadratic function found to fit the new situation is y=.9048ximage04.png+8.1429x+.9524 rounded to the fourth decimal place.  Below the new quadratic formula is graphed with the data points from the new table.

image38.png

The above graph has a window of the following:

image39.png

Marks rod is longer than Leo’s and his guides are closer together.  There is a lot of space between the last guide and the tip of the rod in which the line could still get tangled.  If he had more guides closer together there would be a lesser chance of the wire getting tangled.  

...read more.

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

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