- Level: International Baccalaureate
- Subject: Maths
- Word count: 1739
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:
The graph above has a window of the following:
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.
The y variables 10, 74, and 149 were plugged into y=ax²+bx+c in order to find Matrix B.
The matrices were solved by multiplying the inverse of Matrix A with Matrix B.
Middle
y=1.2857x²+8.2857x+.42847 y=.0524x+.5429x+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=ax+bx+cx+dx+ex+fx+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=ax+bx+cx+dx+ex+fx+gx+h to find Matrix A.
Polynomial Function:
The y variables 10, 23, 38, 55, 74, 96, 120, and 149 were plugged into the equation y=ax+bx+cx+dx+ex+fx+gx+h to find Matrix B.
To solve the matrices, the inverse of Matrix A must be multiplied by Matrix B. The matrix function on the calculator can be used.
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=.0025793651x-.077777777x+.9555555552x-6.152777775+22.25138888x-43.76944443x+55.79047617x-18.999999999
The polynomial function is graphed below with the original data points. The graph passes through each data point.
The above graph has the following window:
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.
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.
The above graph has a window of the following:
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=ax+bx+c to find Matrix A
The y values 10, 48, and 124 were plugged into y= ax+bx+c to find Matrix B
The inverse of Matrix A is multiplied by Matrix B
a= .9047619048
b= 8.142857143
c= .9523809524
The quadratic function found to fit the new situation is y=.9048x+8.1429x+.9524 rounded to the fourth decimal place. Below the new quadratic formula is graphed with the data points from the new table.
The above graph has a window of the following:
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.
This student written piece of work is one of many that can be found in our International Baccalaureate Maths section.
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