Mth Sl Type Ii Portfolio – Fishing Rods

Math Summative seek rod cells Fishing Rods A angle rod requires chooses for the bank n wholeness so that it does not tangle and so that the row casts easily and cost-efficiently. In this task, you result develop a mathematical exemplification for the position of line executes on a seek rod. The Diagram shows a look foring rod with eight organises, plus a demand at the efflorescence of the rod. king of beasts has a seek rod with boilersuit space 230 cm. The dining table shown below gives the keeps for in only(prenominal)(prenominal) of the line gos from the detail of his sportfishing rod. unravel result (from design) specify from exploit (cm) 1 10 2 23 3 38 4 55 5 74 6 96 7 one hundred twenty 8 149Define competent variables and talk closely parameters/ timiditys. utilise Technology, pot the entropy points on a chart. Using intercellular substance orders or early(a)wise, get word oneself a quadratic polynomial multinomial polynomial equ ating polynomial hunt down and a boxlikeal pass a counsel which sample this situation. explicate the process you dropd. On a revolutionary crop of axes, clear these sit campaigns and the genuine in orchestrateation points. Com manpowert on alone protestences. Find a multinomial affair which passes through every info point. Explain you choice of thing, and discuss its reasonableness. On a vernal fixate of axes, circumstances this cast suffice and the genuine in mixed bagation points. Comment on any deflexions.Using technology, catch one separate make for that fits the entropy. On a unsanded fit out of axes, draw this present mesh and the fender selective information points. Comment on any differences. Which of you affaires make up supra beat out models this situation? Explain your choice. Use you quadratic model to decide where you could place a one-ninth channel. contend the implications of adding a ninth guide to the rod. object has a fishing rod with overall continuance 300cm. The table shown below gives the distances for each(prenominal) of the line guides from the confidential information of pocks fishing rod. Guide come in (from whirl) aloofness from mite (cm) 1 10 2 22 3 34 4 48 5 64 6 81 7 102 124 How wellhead does your quadratic model fit this new entropy? What changes, if any, would require to be make for that model to fit this entropy? Discuss any limitations to your model. Introduction Fishing rods habituate guides to lean the line as it is cosmos casted, to check off an efficient cast, and to restrict the line from tangling. An efficient fishing rod pull up stakes use mul top sidele, st regulategically fit(p) guides to maximize its serviceality. The ar coursement of these result depend on the r bring emergeine of guides as well as the length of the rod. Companies design mathematical comparabilitys to witness the optimum side of the guides on a rod.Poor guide placement would p otential cause for myopic fishing quality, dissatisfied customers and thus a less successful company. in that respectfrom it is demand to ensure the guides ar properly set to maximize fishing efficiency. In this investigating, I leave alone be relegate go forth a mathematical model to hold still for the guide placement of a presumptuousness fishing rod that has a length of 230cm and condition distances for each of the 8 guides from the tip (see entropy below). Multiple equations forget be persistent apply the given info to fork over varying degrees of accuracy. These models exclusivelyt accordingly potentially be utilise to go steady the placement of a 9th guide.Four models depart be apply quadratic give out, three-d guide, purulent thing and a quadratic regression process. To amaze, suitable variables moldiness be defined and the parameters and constraints must be discussed. Variables Independent Variable let x institute the compute of guides b egin from the tip Number of guides is a distinguishable value. Since the length of the rod is finite (230cm) and then(prenominal) the sum of guides is cognize to be finite. empyrean = , where n is the finite value that fights the level beaver publication of guides that would fit on the rod. babelike VariableLet y set out the distance of each guide from the tip of the rod in centimetres. The distance of each guide is a discrete value. sick = Parameters/Constraints in that location atomic number 18 several parameters/constraints that need to be verified before carry on in the investigation. Naturally, since we are talking about a real life situation, in that respect sooner a littlenot be a negative number of guides (x) or a negative distance from the tip of the rod (y). All square off are positive, and thitherfore all graphs pass on notwithstanding be represented in the rootage quadrant. The other major constraint that must be identified is the upper limit len gth of the rod, 230cm.This restricts the y-value as well as the x-value. The variable n represents the finite number of guides that could possibly be placed on the rod. While it is physically accomplishable to place many guides on the rod, a realistic, utmost number of guides that would still be efficient, is al almost 15 guides. Guide Number (from tip) Distance from stage (cm) 0* 1 2 3 4 5 6 7 8 n** 0 10 23 38 55 74 96 great hundred 149 230 *the guide at the tip of the rod is not counted **n is the finite value that represents the maximum number of guides that would fit on the rod.Neither of the highlighted set are analyzed in this investigation, they are solo here for the purpose of formation the limits of the variables. The outset footstep in this investigation is to graph the points in the table above (excluding highlighted points) to see the mould of the trend that is created as more(prenominal) guides are added to the rod. From this scatter game of the points, w e arouse see that there is an exponential function function increase in the distance from the tip of the rod as each posterior guide is added to the rod. quadratic polynomial Function The low function that I shall be commodity example apply the points of entropy provided is a quadratic function.The ordinary equation of a quadratic manifestation is y = ax2 + bx + c. To do this, I pull up stakes be use tercet points of selective information to create collar equations that I pass on solve using matrices and follow the coefficients a, b and c. The origin step in this process is to choose three entropy points that volition be apply to represent a broad stove of the info. This will be punishing though since there are only three out of the eight points that can be apply. Therefore, to repair the accuracy of my quadratic function, I will be solvent dickens schemes of equations that use different points and conclusion their immoral. data cut backs Selected Data knack 1 = (1,10), (3,38), (8,149)Data plume 2 = (1,10), (6,96), (8,149) These points were selected for two main reasons. commencement exercise, by using the x- set 1 and 8 in twain sets of data, we will fill a broad range of all of the data that is being represented in the concluding equation after the set of the coefficients are amountd. Second, I apply the x set of 3 (in the kickoff set) and 6 (in the piece set) to at once again allow for a broad type of the data points in the final quadratic equation. Both of these points (3 and 6) were elect because they were equal distances apart, 3 being the trey data point, and 6 being the trey from last data point.This ensured that the final averaged set for the coefficients would give the best representation of the center data points without skewing the data. There will be two regularitys that will be utilise to solve the system of equations, seen below. Each method will be apply for one of the systems being evaluate d. Data Set 1 = (1,10), (3,38), (8,149) In the first data set, the data points will form separate equations that will be understand using a matrices equation. The first ground substance equation will be in the form Where A = a 33 hyaloplasm representing the three data pointsX = a 31 hyaloplasm for the variables being solved B = a 31 intercellular substance for the y-value of the three equations being solved. This ground substance equation will be rearranged by multiplying both sides of the equation by the contrary of A Since A-1*A is equal to the individuation hyaloplasm (I), which when multiplied by some other(prenominal)(prenominal) ground substance gives that corresponding ground substance (the intercellular substance equivalent of 1), the final intercellular substance equation is To determine the set of X, we must first induce the rearward of matrix A using technology, since it is available and purpose the inverse of a 3 by 3 matrix can take an incompetent am ount of time.First let us determine what equations we will be solving and what our matrices will look like. Point (1,10) (3, 38) (8,149) A= The equation is ,X= ,B= = Next, by using our GDC, we can determine the inverse of matrix A, and multiply both sides by it. Therefore we have find out that the quadratic equations given the points (1,10), (3,38), (8,149) is . Data Set 2 = (1,10), (6,96), (8,149) Point (1,10) (6, 96) (8,149) A= ,X= ,B= The wink method that will be used to solve the second system of equations is drive in as Gauss-Jordan elimination.This is a process by which an augmented matrix (two matrices that are placed into one divided by a line) goes through a series of honest mathematical operations to solve the equation. On the go a dash side of this augmented matrix (seen below) is the 33 matrix A (the new matrix A that was made using data set 2, seen on the previous page), and on the right is matrix B. The goal of the operations is to reduce matrix A to the identica lness matrix, and by doing so, matrix B will allot the determine of matrix X. This is otherwise do itn as trim down speech echelon form. Step by step process of reduction 1. We begin with the augmented matrix. . Add (-36 * words 1) to grade 2 3. Add (-64 * trend 1) to words 3 4. Divide course 2 by -30 5. Add (56 * course of instruction 2) to row 3 6. Divide row 3 by 7. Add ( * row 3) to row 2 8. Add (-1 * row 3) to row 1 9. Add (-1 * row 2) to row 1 After all of the row operations, matrix A has commence the identity matrix and matrix B has locomote the set of matrix X (a, b, c). Therefore we have goaded that the quadratic equations given the points (1,10), (6,96), (8,149) is . Averaging of the Two Equations The adjacent step in finding our quadratic function is to average out our accomplished a, b, and c determine from the two sets data.Therefore we have finally headstrong our quadratic function to be go to 4 sig figs, too watch over precision, bit keeping th e numbers manageable. Data points using quadratic function Guide Number (from tip) quadratic polynomial determine Distance from jot (cm) archetype Distance from lean (cm) 1 10 2 22 3 37 4 54 5 74 6 97 7 122 8 149 10 23 38 55 74 96 cxx 149 New set for the distance from tip were locomote to zero quantitative places, to corroborate significant figure the certain determine used to find the quadratic expression had zero decimal places, so the new ones shouldnt either.After finding the y-values given x-values from 1-8 for the quadratic function I was able to contrast the new values to the maestro values (highlighted in green in the table above). We can see that the two values that are the ex cultivate like in both data sets is (1,10) and (8,149) which is not strike since those were the two values that were used in both data sets when finding the quadratic function. Another new value that was the resembling(p) as the pilot was (5,74). All other new data sets have an err oneousness of closely 2cm.This data shows us that the quadratic function can be used to represent the original data with an approximate error of 2cm. This function is still not perfect, and a offend function could be found to represent the data with a lower error and more duplicate data points. blockish Function The following step in this investigation is to model a brick-shaped function that represents the original data points. The general equation of a cubical function is y = ax3 + bx2 + cx + d. Knowing this, we can take cardinal data points and answer a system of equations to determine the values of the coefficients a, b, c, and d.The first step is to choose the data points that will be used to model the cubic function. in addition to modeling the quadratic function, we can only use a limited number of points to represent the data in the function, only in this case it is four out of the eight data points, which room that this function should be more precise than the la st. in one case again I plan on solving for two sets of data points and finding their mean values to represent the cubic function. This is done to allow for a more broad representation of the data at heart the cubic function. Data Sets Selected Data Set 1 (1,10), (4,55), (5,74), (8,149)Data Set 2 (1,10), (3,38), (6,96), (8,149) Both data sets use the points (1,10) and (8,149), the first and last point, so that both data sets produce cubic functions that represent a broad range of the data (from minimum to maximum). The other points selected, were selected as mid range points that would allow for the function to represent this range of the data more holyly. When modeling a cubic function or higher(prenominal), it is difficult to do so without using technology to do the bulk of the calculation cod to large amounts of tedious calculations that would almost salutaryify a math error somewhere.Therefore, the most accurate and fastest way to perform these calculations will be to use a GDC. In both data sets, the reduced row echelon form function on the GDC will be utilized to determine the values of the coefficients of the cubic functions. The process of determining the values of the coefficients of the cubic function using reduced row echelon form is uniform to process used for the quadratic function. An x-value matrix A (this time a 44 matrix), a variable matrix X (41) and a y-value matrix B (41) must be determined first. The close step is to augment matrix A and matrix B, with A on the left and B on the right.This time, instead of doing the row operation ourselves, the GDC will do them, and yield an answer where matrix A will be the identity matrix and matrix B will be the values of the coefficients (or matrix X). Data Set 1 (1,10), (4,55), (5,74), (8,149) (1,10) (4, 55) (5, 74) (8,149) A1 = , X1 = , B1 = We begin with the augmented matrix or matrix A1 and matrix B1. Then this matrix is inputted into a GDC and the function rref is selected. After jam ente r, the matrix is reduced into reduced row echelon form. Which yields the values of the coefficients. Data Set 2 (1,10), (3,38), (6,96), (8,149) (1,10) (3, 38) 6, 96) (8,149) A2 = , X2 = , B2 = We begin with the augmented matrix of matrix A2 and matrix B2. Then the matrix is inputted into a GDC and the function rref After pressing enter, the matrix is reduced into reduced row echelon form. Which yields the values of the coefficients. The next step is to find the mean of each of the values of the coefficients a, b, c, and d. Therefore we have finally determined our cubic function to be Once again rounded to 4 significant figures. Updated Data table, including cubic function values. Guide Number (from tip) quadratic equation values Distance from Tip (cm) 1 10 2 22 3 37 4 54 5 74 6 97 122 8 149 Cubic values Distance from Tip (cm) legitimate Distance from Tip (cm) 10 23 38 54 74 96 121 149 10 23 38 55 74 96 120 149 New values for the distance from tip were rounded to zero decimal plac es, to swan significant figure the original values used to find the quadratic code had zero decimal places, so the new ones shouldnt either. The y-values of the cubic function can be compared to that original data set values to conclude whether or not it is an accurate function to use to represent the original data points. It appears as though the cubic function has 6 out of 8 data points that are the same.Those points being, (1,10), (2,23), (3,38), (5,74), (6,96), (8,149). The three data points from the cubic function that did not match only had an error of 1, indicating that the cubic function would be a good representation of the original data points, nevertheless still has some error. We can and analyze these points by comparing the cubic and quadratic function to the original points by graphing them. See next page. By analyzing this graph, we can see that both the quadratic function and the cubic function match the original data points quite well, although they have minut e differences.By comparing values on the data table, we find that the quadratic function only matches 3 of the 8 original data points with an error of 2, eyepatch the cubic function matches 6 of the 8 points with an error of just 1, which is as small an error possible for precision of the calculation done. Both functions act as adequate representations of the original points, but the major difference is how they begin to differ as the graphs continue. The cubic function is increase at a faster rate than the quadratic function, and this difference would become quite noticeable over time.This would mean that if these functions were to be used to determine the distance a 9th guide should be from the tip, the two functions would provide quite different answers, with the cubic functions providing the more accurate one. Polynomial Function Since it is known that neither the quadratic, nor the cubic function fully satisfy the original data points, then we must model a higher degree poly nomial function that will satisfy all of these points. The best way to find a polynomial function that will pass through all of the original points is to use all of the original points when finding it (oppose to just three or four).If all eight of the points are used and a system of equations is performed using matrices, then a function that satisfies all points will be found. This is a unhealthful function. To find this function, the same procedure followed for the last two functions should be followed, this time using all eight points to create an 88 matrix. By then following the same steps to augment the matrix with an 81 matrix, we can change the matrix into reduced row echelon form to and find our answer. In this method, since we are using all eight points, the entire data set is being represented in the function and no averaging of the results will be necessary.The general grammatical construction for a septic function is . Data Set (1,10), (2,23), (3,38), (4,55), (5,74), (6 ,96), (7,120), (8,149) (1,10) (2,23) (3,38) (4,55) (5,74) (6,96) (7,120) (8,149) A=,X= ,B= , amplify matrix A and matrix B and perform the rref function The answers and values for the coefficients = The final septic function equation is This function that include all the original data points can be seen graphed here below along with the original points. Updated Data table, including septic function values Guide Number (from tip) quadratic equation values Distance from Tip (cm) Cubic values Distance from Tip (cm) unhealthful values Distance from Tip (cm) Original Distance from Tip (cm) 1 10 2 22 3 37 4 54 5 74 6 97 7 122 8 149 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 distance from tip were rounded to zero decimal places, to maintain significant figure the original values used to find the quadratic formula had zero decimal places, so the new ones shouldnt either. By looking at the graph, as well as the data table (both seen above), we can see that, as expected, all 8 of the septic function data points are identical to that of the original data.There is less than 1cm of error, which is accounted for due(p) to imprecise (zero decimal places) original measurements. Therefore we now know that the septic function that utilised all of the original data points is the best representation of said data. Other Function The next goal in this investigation is to find another function that could be used to represent this data. The other method that I will use to find a function that fits the data is quadratic regression. Quadratic regression uses the method of least squares to find a quadratic in the form .This method is often used in statistics when trying to determine a trend that has the minimal sum of the deflections squared from a given set of data. In elementary terms, it finds a function that will pretermit any unnecessary noise in collected data results by finding a value that has the smallest amount of deviation from the majority of the data. Quadratic regression is not used to perfectly fit a data set, but to find the best curve that goes through the data set with minimal deviation. This function can be found using a GDC. First you must input the data points into lists, (L1 and L2).Then you go to the statistic math functions and choose QuadReg. It will know to use the two lists to determine he quadratic function using the method of least squares. Once the calculation has completed, the data seen below (values for the coefficients of the function) will be presented QuadReg a = 1. 244 b = 8. 458 c = 0. 8392 With this data we can determine that the function is When graphed, this function has the shape seen below Updated Data table, including septic function values Guide Number (from tip) Quadratic values Distance from Tip (cm) Cubic values Distance from Tip (cm) Septic values Distance from Tip (cm) Quadratic Regression Distance from Tip (cm) Original Distance from Tip (cm) 1 10 2 22 3 37 4 54 5 74 6 97 7 122 8 149 10 23 38 54 74 96 121 149 10 23 38 55 74 96 120 149 11 23 37 55 74 96 121 148 10 23 38 55 74 96 120 149 By analyzing the graph and values of the quadratic regression function, it is diaphanous that it is a relatively accurate form of modeling the data. Four of the eight points matched that of the original data, with an error of 1. The most notable difference between the quadratic regression function and the quadratic function previously determined, is the placement within the data f the accurate values. The regression function matched the middle data, while the quadratic function matched the end data. It is provoke to see how two functions in the same form, found using different methods yielded turnaround areas of accuracy. Best Match The function that acts as the best model for this situation is the septic function. It is the only function that satisfies each of the original data points with its equation. Through finding the quadratic, cubic a nd septic functions, it was discovered that the degree of the polynomial was directly correspondd to the functions accuracy to the data.Therefore it was no surprise that this function acts as the best fit for this data. The other cause for this septic function having the best correlation to the original data is due to the septic function being established by creating a system of equations using all of the data points. 9th Guide Using my quadratic model, it can be determined where the best placement for a ninth guide would be by subbing 9 in for x in the equation . Using my quadrating model, it was found that the optimal placement for a ninth guide on the rod is 179cm from the tip of the rod. social lions fishing rod is 230cm long, yet his 8th guide is only 149cm from the tip of the rod. That means that there is 81cm of the line that is not being guided from the reel to first guide. By adding a ninth guide, that distance will be shortened form 81cm to 51cm. By doing this, it will be less likely for the line to bunch up and become tangled in this 81cm stretch where there is no guide. Another implication of adding another guide would be that the weight dispersion of a fish being reeled in would be spread over another guide, which will allow for an easier task of reeling in the fish.There is even enough space on the rod for a 10th guide at 211cm from the tip of the rod. This guide would once again shorten the spare line further to a point where the excess line between the reel and the first guide is shorter than line between the first and second guide. This could cause problems with reeling and casting efficiency, as that extra guide would cause retardent movement of the line. The benefit would be that once again the weight distribution of fish would be spread over a larger number of guides.Overall, it would be just to include a ninth guide to social lions fishing rod, but anymore will likely stay its efficiency. tags Fishing Rod Guide Number (from tip) Distance from Tip (cm) 1 10 2 22 3 34 4 48 5 64 6 81 7 102 8 124 To see how well my quadratic model fits this new data, they must be both plotted on the same graph, seen below. My quadratic model for Leos fishing rod correlates with go downs fishing rod data for the first few values and then diverges as the number of guides increases by ontogeny at a higher exponential rate.The difference between Leo and signalises eighth guide from the tip of their respective rods is 25cm, yet both mens first guides start the same distance from the tip of their rods. The quadratic function used to model Leos fishing rod does not correlate well with Marks fishing rod data. Changes to the model must be made for it to fit this data. The best way to find a model for Marks data would be to go through the same steps that we went through to determine the first quadratic formula that models Leos fishing rod.By doing so, specific values that better represent Marks fishing rod data could be used to establ ish a better competent function. The main limitation of my model is that is was knowing as a function for Leos data specifically. It was created by solving systems of equations that used solely Leos fishing rod for data. Consequentially, the quadratic model best represented Leos fishing rod, which had a maximum length of 230cm, with differently spaced out guides. There were many differences between Leo and Marks fishing rods (such as maximum length and guide spacing) that caused my original quadratic model to not well represent Marks data.