Restrictions
{ X E l K l | 0 < X < 1 0 } * The maximum # of guide depends on the model used
Graph 1: The relationship between the # of guides and the distance from tip (cm)
Part B: Determining a quadratic model that fits the situation using systems of equations
A quadratic equation in its standard form is ax^2 + bx + c where a, b, and c are the coefficients and "a" in the equation can never be 0.
To find a quadratic equation to model this fishing rod situation, I would have to find 3 sets of points and substitute each set of points into the standard form equation to find the 3 coefficients which would make up the model equation.
The three points that are chosen could be [1, 10] , [2, 23] , and [3,38 ] and these 3 sets of points would be substituted into 3 different equations as shown below.
10 = a(1)^2 + b(1) +c
23 = a(2)2 +b(2) + c
38 = a(3)2 + b(3) + c
The 3 equations can now be further simplified into 3 equations without an exponent
Eq.l a + b + c = 10
Eq.2 4a + 2b + c = 23
Eq.3 9a + 36 + c = 38
The solving of the 3 equations can now begin once all 3 equations have been completely simplified
First, isolate the variable "C” in the equation by moving all the terms that does not contain "c" to the other side of the equation.
C= 10 - a - b
Now, substitute the value of C (10 - a - b} into each of the "c" variable in equation 2 and equation 3.
The resulting equations: = 4a + 2b + (10-a- b) = 23
Eq.4 3a + b + 10 = 23 = 9a + 3/7 + (10 - a - b) = 38
Eq.5 8a + 2b + 10 = 38
Now further isolate another variable like the variable B in Eq. 4 by moving all the terms without B to the other side.
b = -3a + 13
Now substitute the value of B into the B variable in Eq.5 and simplify the entire equation until the variable A is found.
8a+2(-3a+13)+10= 38
8a + (-6a+26) +10= 38
8a-6a=38-10-26
2a =2
a=1
Now take the value of A and substitute it into an equation with 3 variables like Eq.4 to determine the value of B
3( 1 ) + b + 10 = 23
3 + b = 13
b = 10
Lastly substitute the value of both A and B back into one of the original equations like Eq. 1 to determine the value of C
a + b 4- c = 10 1 + 10 + c = 10 c - 10 - 10 - 1
In the end, the final values of the coefficients determined with the 3 equations are:
A= l B=10 C= -1
The model equation for this situation based on the 3 sets of points and equations is
y = x2 + 10x- 1
Graph 2: The quadratic relationship between the # of guides and the distance from tip (cm)
Part C: Determining a cubic function that fits the situation
A cubic function has a standard form of ax3+ bx2+ cx + d and coefficients of a, b, c, and d where a must never be 0. The cubic function for the situation was determined using Excel 2007 by creating a cubic trend line. First of all, the values of X in Table 1 were input into column A while the values of Y in Table 1 were input into column B. The values of column A and B was then highlighted and a scatter plot graph was made using them. Once the graph was made, a trend line was created by going to "layout" then "trend line" then "more trend line options". I then chose a "polynomial trend to the third degree" and further chose to "display the equation".
The resulting equation was: 0.0631x3 + 0.3918x2 + 11.71x - 2.2857
Graph 3: The cubic relationship between the # of guides and the distance from tip (cm)
Part D: Comparison between the quadratic and cubic models that fits the situation
Graph 4: The comparison between the cubic and quadratic functions that models Leo's rod
The graph above indicates that there are differences in the quadratic and cubic functions that model the placement of line guides on Leo's rod. One of the main differences between would be the quadratic model y = x2 + 10x - 1 being a polynomial fit. This means that I only used 3 points and equations required determining the quadratic model equation. On the other hand, my cubic equation is a polynomial regression. It is a polynomial regression because 1 used all 8 points and not just the 4 required points to determine the model equation. The graph above also indicates that the cubic function is more accurate. It is more accurate because the model exactly touches all of the point while the quadratic only touches the eighth guide and doesn't directly touch it.
Part E: A polynomial function which passes through every data point
Graph 5: A polynomial function (quartic) that passes through every point
The polynomial function that I chose that passed through every point would be the quartic functions
y = 0.01x4- 0.16x3+ l.73x2+ 8.64x - 0.20. I chose this function over other polynomial function like quintics and polynomials that go to the sixth degree would be because of its y intercept. The y intercept is important because I previously mentioned about the guide at the tip that should be located at 0 cm. This equation has a y intercept of-0.20 which is closer to zero than the other functions. The guide at the tip or 0 cm is important because in order for a line to be cast it must pass through that guide so the value on the graph should match up at 0 at the y intercept.
Part F: Other function that fits the situation.
Graph 6: The power function that models Leo's Rod
This function is different because it doesn't completely touch all the points like the previous few functions like the cubic and quartic. Although this model does touch [0,0] which is the points for the tip or ninth guide of the rod. The ninth guide is a guide that should be part of the data even though it isn't because the line must pass through it in order to be cast
Part G: The function that best models Leo's rod.
The function that I believe best models Leo's rod would be the "power function" above. I would choose the "power function" above all the other polynomial equations" because the power relationship passes through point [0,0] which is the location of the guide at the tip. Although some of the equations like the cubic and quartic did pass through every point nicely. All of those equations have a negative distance value for the guide at the tip where the line must still pass. A negative distance value is impossible in the real world and therefore it does not accurately model the situation or Leo's rod.
Part H: The addition of a ninth guide with the quadratic model
The quadratic model that is to be used to determine the location of the 9 guide is y = x2+ 10x - 1
The distance where the ninth guide would be can be determined by substituting the x value of 9 into the equation to get our distance the y value.
y = (9)2+ 10(9) - 1
y = 170 cm
The ninth line guide can be placed at around 170 cm on the rod.
The addition of a ninth line guide can have many implications. One of the positive implications would be that the addition of the ninth line guide would mostly lead to a "smoother casting" when casting the line out. Another positive implication the addition of a ninth line guide would be more hooking power and ultimately an easier catch for the angler." On the negative side, the addition of a ninth line guide and eventually a tenth could lead to some problems with the "weight balance" of the rod. It could would affect weight balance of the "rod" because the guides would just add weight to the end of the rod that already has parts like the reel which is heavy. The addition of additional guides at the end would just lead to more "fatigue" when casting.
Part I: The application of the quadratic model to Mark's fishing rod
Mark's fishing rod is a rod whose overall length is longer than Leo's by 70 cm at 300 cm. below is the table relating the number of guides on Mark's fishing rod to the distance of the guide from the tip of Mark's rod. The rod just like Leo's is made up of 8 guides with an extra guide at the tip of the rod.
Table 2: The relationship between the # of guides and the distance from tip (cm)
The original quadratic model for Leo's rod is applied on Mark's rod and it can be seen in the graph below.
Graph 7: Relationship between original quadratic model and Mark's rod
It can be seen that my quadratic model doesn't this new data very well. The trend line from my quadratic model for Leo's rod is rising at a steeper rate than the data point from Mark's rod is rising. One of the changes that could be made would be to decrease the 'Vertical expansion" of the original model and turn it into more of a vertical compression so it can accommodate for the smaller distances between guides in Mark's rod.
By using my original quadratic model to model Mark's rod it can be seen that my model does have many limitations. One of the limitations it has would be the fact that there are many different forms of rods and rods blanks out there in the world. That means that each individual rod would have to have "guides" that are optimized for its particular blank." On top of that, my model doesn't take into account the numerous parts that a fish rod has like the grips, handles, reel, and reel seats. This means that some of the placement of guides may not be possible due to the presence of other parts of the rod in that area. Lastly, this quadratic model again has the limitation of not having a y-intersect at 0 and instead at. This means it can never truly model the guide at the tip of the rod.
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Fishing Rods: Guides, (n.d.). Live Outdoors. Retrieved April 14, 2012, from www.iiveoutdoors.com/fishing/articles/166909-fishing-rods-guides
rings, u. s., profile, a. I., & Concept, F. N. (n.d.). Optimum Placement, Number & Size of Guides Maximises Rod Performance. Fishing tackle & fishing tackle components - Hopkins & Holloway Limited. Retrieved April 14, 2012, from '" Fishing Rods: Guides, (n.d.). Live Outdoors. Retrieved April 14, 2012, from www.liveoutdoors.com/fishing/articles/166909-fishing-rods-guides
