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Fishing » Mth Sl Type Ii Portfolio – Fishing Rods

# Mth Sl Type Ii Portfolio – Fishing Rods

Math Summative: Fishing Rods Fishing Rods A fishing rod requires guides for the road in order that it doesn't tangle and in order that the road casts simply and effectively. On this process, you'll develop a mathematical mannequin for the location of line guides on a fishing rod. The Diagram reveals a fishing rod with eight guides, plus a information on the tip of the rod. Leo has a fishing rod with general size 230 cm. The desk proven beneath provides the distances for every of the road guides from the tip of his fishing rod.
Outline appropriate variables and focus on parameters/constraints. Utilizing Expertise, pot the info factors on a graph. Utilizing matrix strategies or in any other case, discover a quadratic operate and a cubic operate which mannequin this case. Clarify the method you used. On a brand new set of axes, draw these mannequin features and the unique knowledge factors. Touch upon any variations. Discover a polynomial operate which passes by means of each knowledge level. Clarify you selection of operate, and focus on its reasonableness. On a brand new set of axes, draw this mannequin operate and the unique knowledge factors. Touch upon any variations.
Utilizing expertise, discover one different operate that matches the info. On a brand new set of axes, draw this mannequin operate and the unique knowledge factors. Touch upon any variations. Which of you features discovered above greatest fashions this case? Clarify your selection. Use you quadratic mannequin to resolve the place you would place a ninth information. Talk about the implications of including a ninth information to the rod. Mark has a fishing rod with general size 300cm. The desk proven beneath provides the distances for every of the road guides from the tip of Mark’s fishing rod. Information Quantity (from tip)
How effectively does your quadratic mannequin match this new knowledge? What modifications, if any, would have to be made for that mannequin to suit this knowledge? Talk about any limitations to your mannequin. Introduction: Fishing rods use guides to regulate the road as it's being casted, to make sure an environment friendly forged, and to limit the road from tangling. An environment friendly fishing rod will use a number of, strategically positioned guides to maximise its performance. The location of those will rely upon the variety of guides in addition to the size of the rod. Corporations design mathematical equations to find out the optimum placement of the guides on a rod.
Poor information placement would possible trigger for poor fishing high quality, dissatisfied clients and thus a much less profitable firm. Due to this fact it's important to make sure the guides are correctly positioned to maximise fishing effectivity. On this investigation, I might be figuring out a mathematical mannequin to symbolize the information placement of a given fishing rod that has a size of 230cm and given distances for every of the Eight guides from the tip (see knowledge beneath). A number of equations might be decided utilizing the given knowledge to offer various levels of accuracy. These fashions can then probably be used to find out the location of a ninth information.
4 fashions might be used: quadratic operate, cubic operate, septic operate and a quadratic regression operate. To start, appropriate variables should be outlined and the parameters and constraints should be mentioned. Variables: Unbiased Variable: Let x symbolize the variety of guides starting from the tip Variety of guides is a discrete worth. Because the size of the rod is finite (230cm) then the variety of guides is understood to be finite. Area = , the place n is the finite worth that represents the utmost variety of guides that may match on the rod.
Dependent Variable:
Let y symbolize the space of every information from the tip of the rod in centimetres. The gap of every information is a discrete worth. Vary = Parameters/Constraints: There are a number of parameters/constraints that have to be verified earlier than continuing within the investigation. Naturally, since we're speaking about an actual life scenario, there can't be a damaging variety of guides (x) or a damaging distance from the tip of the rod (y). All values are optimistic, and due to this fact all graphs will solely be represented within the first quadrant. The opposite main constraint that should be recognized is the utmost size of the rod, 230cm.
This restricts the y-value in addition to the x-value. The variable n represents the finite variety of guides that would probably be positioned on the rod. Whereas it's bodily attainable to put many guides on the rod, a sensible, most variety of guides that may nonetheless be environment friendly, is roughly 15 guides. Information Quantity (from tip) Distance from Tip (cm) zero* 1 2 three four 5 6 7 Eight n** zero 10 23 38 55 74 96 120 149 230 *the information on the tip of the rod shouldn't be counted **n is the finite worth that represents the utmost variety of guides that may match on the rod.
Neither of the highlighted values are analyzed on this investigation, they're solely right here for the aim of defining the bounds of the variables. Step one on this investigation is to graph the factors within the desk above (excluding highlighted factors) to see the form of the development that's created as extra guides are added to the rod. From this scatter plot of the factors, we are able to see that there's an exponential improve within the distance from the tip of the rod as every subsequent information is added to the rod. Quadratic Perform: The primary operate that I shall be modeling utilizing the factors of information supplied is a quadratic operate.
The overall equation of a quadratic method is y = ax2 + bx + c. To do that, I might be utilizing three factors of information to create three equations that I'll resolve utilizing matrices and decide the coefficients: a, b and c. Step one on this course of is to decide on three knowledge factors that might be used to symbolize a broad vary of the info. This might be troublesome although since there are solely three out of the eight factors that can be utilized. Due to this fact, to enhance the accuracy of my quadratic operate, I might be fixing two techniques of equations that use totally different factors and discovering their imply. Information Units Chosen: Information Set 1 = (1,10), (three,38), (Eight,149)
Information Set 2 = (1,10), (6,96), (Eight,149) These factors have been chosen for 2 major causes. First, by utilizing the x-values 1 and eight in each units of information, we may have a broad vary of the entire knowledge that's being represented within the remaining equation after the values of the coefficients are averaged. Second, I used the x values of three (within the first set) and 6 (within the second set) to as soon as once more enable for a broad illustration of the info factors within the remaining quadratic equation. Each of those factors (three and 6) have been chosen as a result of they have been equal distances aside, three being the third knowledge level, and 6 being the third from final knowledge level.
This ensured that the ultimate averaged values for the coefficients would give the perfect illustration of the center knowledge factors with out skewing the info. There might be two strategies that might be used to unravel the system of equations, seen beneath. Every technique might be used for one of many techniques being evaluated. Information Set 1 = (1,10), (three,38), (Eight,149) Within the first knowledge set, the info factors will kind separate equations that might be solved utilizing a matrices equation. The primary matrix equation might be within the kind: The place A = a 3x3 matrix representing the three knowledge factors X = a 3x1 matrix for the variables being solved B = a 3x1 matrix for the y-value of the three equations being solved. This matrix equation might be rearranged by multiplying either side of the equation by the inverse of A: Since A-1*A is the same as the identification matrix (I), which when multiplied by one other matrix provides that very same matrix (the matrix equal of 1), the ultimate matrix equation is: To find out the values of X, we should first discover the inverse of matrix A utilizing expertise, since it's out there and discovering the inverse of a three by three matrix can take an inefficient period of time.
First allow us to decide what equations we might be fixing and what our matrices will seem like. Level: (1,10) (three, 38) (Eight,149) A= The equation is: ,X= ,B= = Subsequent, by utilizing our GDC, we are able to decide the inverse of matrix A, and multiply either side by it. Due to this fact we have now decided that the quadratic equations given the factors (1,10), (three,38), (Eight,149) is . Information Set 2 = (1,10), (6,96), (Eight,149) Level: (1,10) (6, 96) (Eight,149) A= ,X= ,B= The second technique that might be used to unravel the second system of equations is named Gauss-Jordan elimination.
It is a course of by which an augmented matrix (two matrices which might be positioned into one divided by a line) goes by means of a sequence of easy mathematical operations to unravel the equation. On the left aspect of this augmented matrix (seen beneath) is the 3x3 matrix A (the brand new matrix A that was made utilizing knowledge set 2, seen on the earlier web page), and on the proper is matrix B. The objective of the operations is to cut back matrix A to the identification matrix, and by doing so, matrix B will yield the values of matrix X. That is in any other case often known as lowered row echelon kind. Step-by-step strategy of discount: 1. We start with the augmented matrix. . Add (-36 * row 1) to row 2 three. Add (-64 * row 1) to row three four. Divide row 2 by -30 5. Add (56 * row 2) to row three 6. Divide row three by 7. Add ( * row three) to row 2 Eight. Add (-1 * row three) to row 1 9. Add (-1 * row 2) to row 1 After the entire row operations, matrix A has develop into the identification matrix and matrix B has develop into the values of matrix X (a, b, c). Due to this fact we have now decided that the quadratic equations given the factors (1,10), (6,96), (Eight,149) is . Averaging of the Two Equations The subsequent step to find our quadratic operate is to common out our established a, b, and c values from the 2 units knowledge.
Due to this fact we have now lastly decided our quadratic operate to be: Rounded to four sig figs, too keep precision, whereas holding the numbers manageable. Information factors utilizing quadratic operate Information Quantity (from tip) Quadratic values Distance from Tip (cm) Authentic - Distance from Tip (cm) 1 10 2 22 three 37 four 54 5 74 6 97 7 122 Eight 149 10 23 38 55 74 96 120 149 New values for the space from tip have been rounded to zero decimal locations, to keep up important determine – the unique values used to seek out the quadratic method had zero decimal locations, so the brand new ones shouldn’t both.
After discovering the y-values given x-values from 1-Eight for the quadratic operate I used to be capable of evaluate the brand new values to the unique values (highlighted in inexperienced within the desk above). We are able to see that the 2 values which might be the very same in each knowledge units is (1,10) and (Eight,149) which isn't shocking since these have been the 2 values that have been utilized in each knowledge units when discovering the quadratic operate. One other new worth that was the identical as the unique was (5,74). All different new knowledge units have an error of roughly ±2cm.
This knowledge reveals us that the quadratic operate can be utilized to symbolize the unique knowledge with an approximate error of ±2cm. This operate continues to be not excellent, and a greater operate could possibly be discovered to symbolize the info with a decrease error and extra matching knowledge factors. Cubic Perform: The subsequent step on this investigation is to mannequin a cubic operate that represents the unique knowledge factors. The overall equation of a cubic operate is y = ax3 + bx2 + cx + d. Figuring out this, we are able to take 4 knowledge factors and carry out a system of equations to find out the values of the coefficients a, b, c, and d.
Step one is to decide on the info factors that might be used to mannequin the cubic operate. Equally to modeling the quadratic operate, we are able to solely use a restricted variety of factors to symbolize the info within the operate, solely on this case it's 4 out of the eight knowledge factors, which implies that this operate must be extra exact than the final. As soon as once more I plan on fixing for 2 units of information factors and discovering their imply values to symbolize the cubic operate. That is executed to permit for a extra broad illustration of the info inside the cubic operate. Information Units Chosen: Information Set 1:
Information Set 2: (1,10), (three,38), (6,96), (Eight,149) Each knowledge units use the factors (1,10) and (Eight,149), the primary and final level, in order that each knowledge units produce cubic features that symbolize a broad vary of the info (from minimal to most). The opposite factors chosen, have been chosen as mid vary factors that may enable for the operate to symbolize this vary of the info extra precisely. When modeling a cubic operate or greater, it's troublesome to take action with out utilizing expertise to do the majority of the calculation as a consequence of massive quantities of tedious calculations that may nearly assure a math error someplace.
Due to this fact, probably the most correct and quickest solution to carry out these calculations might be to make use of a GDC. In each knowledge units, the lowered row echelon kind operate on the GDC might be utilized to find out the values of the coefficients of the cubic features. The method of figuring out the values of the coefficients of the cubic operate utilizing lowered row echelon kind is much like course of used for the quadratic operate. An x-value matrix A (this time a 4x4 matrix), a variable matrix X (4x1) and a y-value matrix B (4x1) should be decided first. The subsequent step is to reinforce matrix A and matrix B, with A on the left and B on the proper.
This time, as a substitute of doing the row operation ourselves, the GDC will do them, and yield a solution the place matrix A would be the identification matrix and matrix B would be the values of the coefficients (or matrix X). Information Set 1: (1,10) (four, 55) (5, 74) (Eight,149) A1 = , X1 = , B1 = We start with the augmented matrix or matrix A1 and matrix B1. Then this matrix is inputted right into a GDC and the operate “rref” is chosen. After urgent enter, the matrix is lowered into lowered row echelon kind. Which yields the values of the coefficients. Information Set 2: (1,10), (three,38), (6,96), (Eight,149) (1,10) (three, 38) 6, 96) (Eight,149) A2 = , X2 = , B2 = We start with the augmented matrix of matrix A2 and matrix B2. Then the matrix is inputted right into a GDC and the operate “rref” After urgent enter, the matrix is lowered into lowered row echelon kind. Which yields the values of the coefficients. The subsequent step is to seek out the imply of every of the values of the coefficients a, b, c, and d. Due to this fact we have now lastly decided our cubic operate to be: As soon as once more rounded to four important figures. Up to date Information desk, together with cubic operate values. Information Quantity (from tip) Quadratic values Distance from Tip (cm) 1 10 2 22 three 37 four 54 5 74 6 97 122 Eight 149 Cubic values Distance from Tip (cm) Authentic – Distance from Tip (cm) 10 23 38 54 74 96 121 149 10 23 38 55 74 96 120 149 New values for the space from tip have been rounded to zero decimal locations, to keep up important determine – the unique values used to seek out the quadratic method had zero decimal locations, so the brand new ones shouldn’t both. The y-values of the cubic operate might be in comparison with that authentic knowledge set values to conclude whether or not or not it's an correct operate to make use of to symbolize the unique knowledge factors. It seems as if the cubic operate has 6 out of Eight knowledge factors which might be the identical.
These factors being, (1,10), (2,23), (three,38), (5,74), (6,96), (Eight,149). The three knowledge factors from the cubic operate that didn't match solely had an error of ±1, indicating that the cubic operate can be a very good illustration of the unique knowledge factors, however nonetheless has some error. We are able to additional analyze these factors by evaluating the cubic and quadratic operate to the unique factors by graphing them. See subsequent web page. By analyzing this graph, we are able to see that each the quadratic operate and the cubic operate match the unique knowledge factors fairly effectively, though they've slight variations.
By evaluating values on the info desk, we discover that the quadratic operate solely matches three of the Eight authentic knowledge factors with an error of ±2, whereas the cubic operate matches 6 of the Eight factors with an error of simply ±1, which is as small an error attainable for precision of the calculation executed. Each features act as enough representations of the unique factors, however the main distinction is how they start to vary because the graphs proceed. The cubic operate is rising at a sooner price than the quadratic operate, and this distinction would develop into fairly noticeable over time.
This could imply that if these features have been for use to find out the space a ninth information must be from the tip, the 2 features would offer fairly totally different solutions, with the cubic features offering the extra correct one. Polynomial Perform: Since it's recognized that neither the quadratic, nor the cubic operate totally fulfill the unique knowledge factors, then we should mannequin a better diploma polynomial operate that may fulfill all of those factors. One of the best ways to discover a polynomial operate that may cross by means of the entire authentic factors is to make use of the entire authentic factors when discovering it (oppose to only three or 4).
If all eight of the factors are used and a system of equations is carried out utilizing matrices, then a operate that satisfies all factors might be discovered. It is a septic operate. To search out this operate, the identical process adopted for the final two features must be adopted, this time utilizing all eight factors to create an 8x8 matrix. By then following the identical steps to reinforce the matrix with an 8x1 matrix, we are able to change the matrix into lowered row echelon kind to and discover our reply. On this technique, since we're utilizing all eight factors, the complete knowledge set is being represented within the operate and no averaging of the outcomes might be crucial.
The overall method for a septic operate is . Information Set: (1,10), (2,23), (three,38), (four,55), (5,74), (6,96), (7,120), (Eight,149) (1,10) (2,23) (three,38) (four,55) (5,74) (6,96) (7,120) (Eight,149) A=,X= ,B= , Increase matrix A and matrix B and carry out the ‘rref’ operate The solutions and values for the coefficients = The ultimate septic operate equation is This operate that embody all the unique knowledge factors might be seen graphed right here beneath together with the unique factors. Up to date Information desk, together with septic operate values Information Quantity (from tip) Quadratic values Distance from Tip (cm) Cubic values Distance from Tip (cm)
Septic values – Distance from Tip (cm) Authentic – Distance from Tip (cm) 1 10 2 22 three 37 four 54 5 74 6 97 7 122 Eight 149 10 23 38 54 74 96 121 149 10 23 38 55 74 96 120 149 10 23 38 55 74 96 120 149 New values for the space from tip have been rounded to zero decimal locations, to keep up important determine – the unique values used to seek out the quadratic method had zero decimal locations, so the brand new ones shouldn’t both. By wanting on the graph, in addition to the info desk (each seen above), we are able to see that, as anticipated, all Eight of the septic operate knowledge factors are equivalent to that of the unique knowledge.
There's lower than 1cm of error, which is accounted for as a consequence of imprecise (zero decimal locations) authentic measurements. Due to this fact we now know that the septic operate that utilised the entire authentic knowledge factors is the perfect illustration of stated knowledge. Different Perform: The subsequent objective on this investigation is to seek out one other operate that could possibly be used to symbolize this knowledge. The opposite technique that I'll use to discover a operate that matches the info is quadratic regression. Quadratic regression makes use of the tactic of least squares to discover a quadratic within the kind .
This technique is commonly utilized in statistics when attempting to find out a curve that has the minimal sum of the deviations squared from a given set of information. In easy phrases, it finds a operate that may disregard any pointless noise in collected knowledge outcomes by discovering a worth that has the smallest quantity of deviation from nearly all of the info. Quadratic regression shouldn't be used to completely match a knowledge set, however to seek out the perfect curve that goes by means of the info set with minimal deviation. This operate might be discovered utilizing a GDC. First you need to enter the info factors into lists, (L1 and L2).
Then you definately go to the statistic math features and select QuadReg. It'll know to make use of the 2 lists to find out he quadratic operate utilizing the tactic of least squares. As soon as the calculation has accomplished, the info seen beneath (values for the coefficients of the operate) might be introduced: QuadReg a = 1. 244 b = Eight. 458 c = zero. 8392 With this knowledge we are able to decide that the operate is When graphed, this operate has the form seen beneath: Up to date Information desk, together with septic operate values Information Quantity (from tip) Quadratic values Distance from Tip (cm) Cubic values Distance from Tip (cm)
By analyzing the graph and values of the quadratic regression operate, it's evident that it's a comparatively correct type of modeling the info. 4 of the eight factors matched that of the unique knowledge, with an error of ±1. Probably the most notable distinction between the quadratic regression operate and the quadratic operate beforehand decided, is the location inside the knowledge f the correct values. The regression operate matched the center knowledge, whereas the quadratic operate matched the tip knowledge. It's fascinating to see how two features in the identical kind, discovered utilizing totally different strategies yielded reverse areas of accuracy. Greatest Match: The operate that acts as the perfect mannequin for this case is the septic operate. It's the solely operate that satisfies every of the unique knowledge factors with its equation. By means of discovering the quadratic, cubic and septic features, it was found that the diploma of the polynomial was immediately correlated to the operate’s accuracy to the info.
Due to this fact it was no shock that this operate acts as the perfect match for this knowledge. The opposite trigger for this septic operate having the perfect correlation to the unique knowledge is because of the septic operate being established by making a system of equations utilizing the entire knowledge factors. ninth Information: Utilizing my quadratic mannequin, it may be decided the place the optimum placement for a ninth information can be by substituting ‘9’ in for x within the equation . Utilizing my quadrating mannequin, it was discovered that the optimum placement for a ninth information on the rod is 179cm from the tip of the rod.
Leo’s fishing rod is 230cm lengthy, but his eighth information is simply 149cm from the tip of the rod. Meaning that there's 81cm of the road that isn't being guided from the reel to first information. By including a ninth information, that distance might be shortened kind 81cm to 51cm. By doing this, it will likely be much less possible for the road to bunch up and develop into tangled on this 81cm stretch the place there isn't a information. One other implication of including one other information can be that the load distribution of a fish being reeled in can be unfold over one other information, which can enable for a neater process of reeling within the fish.
There's even sufficient area on the rod for a 10th information at 211cm from the tip of the rod. This information would as soon as once more shorten the surplus line additional to a degree the place the surplus line between the reel and the primary information is shorter than line between the primary and second information. This might trigger issues with reeling and casting effectivity, as that additional information would trigger slowing motion of the road. The profit can be that when once more the load distribution of fish can be unfold over a bigger variety of guides.
General, it might be helpful to incorporate a ninth information to Leo’s fishing rod, however anymore will possible hinder its effectivity. Mark’s Fishing Rod: Information Quantity (from tip) Distance from Tip (cm) 1 10 2 22 three 34 four 48 5 64 6 81 7 102 Eight 124 To see how effectively my quadratic mannequin matches this new knowledge, they should be each plotted on the identical graph, seen beneath. My quadratic mannequin for Leo’s fishing rod correlates with Mark’s fishing rod knowledge for the primary few values after which diverges because the variety of guides will increase by rising at a better exponential price.
The distinction between Leo and Mark’s eighth information from the tip of their respective rods is 25cm, but each males’s first guides begin the identical distance from the tip of their rods. The quadratic operate used to mannequin Leo’s fishing rod doesn't correlate effectively with Mark’s fishing rod knowledge. Modifications to the mannequin should be made for it to suit this knowledge. One of the best ways to discover a mannequin for Mark’s knowledge can be to undergo the identical steps that we went by means of to find out the primary quadratic method that mannequin’s Leo’s fishing rod.
By doing so, particular values that higher symbolize Mark’s fishing rod knowledge could possibly be used to ascertain a greater becoming operate. The primary limitation of my mannequin is that's was designed as a operate for Leo’s knowledge particularly. It was created by fixing techniques of equations that used solely Leo’s fishing rod for knowledge. Consequentially, the quadratic mannequin greatest represented Leo’s fishing rod, which had a most size of 230cm, with in another way spaced out guides. There have been many variations between Leo and Mark’s fishing rods (akin to most size and information spacing) that brought on my authentic quadratic mannequin to not effectively symbolize Mark’s knowledge.