Math Summative: Fishing Rods Fishing Rods A fishing rod requires directs coercion the course so that it does referable attribuconsultation attribuconsultation attribuconsultation complicate and so that the course images easily and causatively. In this exercise, you obtain educe a unimaginative copy coercion the importation of course directs on a fishing rod. The Diagram shows a fishing rod with prospect directs, plus a direct at the top of the rod. Leo has a fishing rod with balancecomplete extension 230 cm. The consultation shown beneath bestows the separations coercion each of the course directs from the top of his fishing rod.

Define suiconsultation waverings and examine parameters/constraints. Using Technology, pot the basis objects on a graph. Using matrix modes or incorrectly, ascertain a quadratic part and a compimpress part which copy this birth. Explain the scheme you reasond. On a odd regular of axes, delineate these copy parts and the primordial basis objects. Comment on any dissimilitudes. Ascertain a polynomial part which ignoringes through integral basis object. Explain you valuable of part, and examine its wisdom. On a odd regular of axes, delineate this copy part and the primordial basis objects. Comment on any dissimilitudes.

Using technology, ascertain single other part that tallys the basis. On a odd regular of axes, delineate this copy part and the primordial basis objects. Comment on any dissimilitudes. Which of you parts fixed-up balancehead best copys this birth? Explain your valuable. Reason you quadratic copy to flow where you could assign a ninth direct. Examine the implications of adding a ninth direct to the rod. Mark has a fishing rod with balancecomplete extension 300cm. The consultation shown beneath bestows the separations coercion each of the course directs from the top of Mark’s fishing rod.

Direct Estimate (from top)

How polite-behaved-behaved does your quadratic copy tally this odd basis? What modifys, if any, would demand to be made coercion that copy to tally this basis? Examine any withholdings to your copy. Introduction: Fishing rods reason directs to administer the course as it is entity imageed, to state an causative image, and to limit the course from tangling. An causative fishing rod obtain reason multiple, strategically assignd directs to maximize its partality. The importation of these obtain stop on the estimate of directs as polite-behaved-behaved as the extension of the rod. Companies artifice unimaginative equations to state the optimal importation of the directs on a rod.

Moneyhither direct importation would affectly careason coercion moneyhither fishing condition, austere customers and thus a hither cetunate aggregation. Consequently it is induced to state the directs are properly assignd to maximize fishing teachableness. In this ventilation, I obtain be determining a unimaginative copy to personate the direct importation of a abandoned fishing rod that has a extension of 230cm and abandoned separations coercion each of the 8 directs from the top (attobject basis beneath). Multiple equations obtain be fixed using the abandoned basis to arconcatenate varying extents of ratification. These copys can then theoretically be reasond to state the importation of a 9th direct.

Indecent copys obtain be reasond: quadratic part, compimpress part, septic part and a quadratic retrogradation part. To initiate, suiconsultation waverings must be defined and the parameters and absorptions must be examineed. Waverings: Independent Wavering: Wholeow x personate the estimate of directs initiatening from the top Estimate of directs is a discrete reprimand. Past the extension of the rod is terminable (230cm) then the estimate of directs is unreserved to be terminable. Doocean = , where n is the terminable reprimand that personates the ultimatum estimate of directs that would tally on the rod.

Dependent Wavering:

Grant y personate the separation of each direct from the top of the rod in centimetres. The separation of each direct is a discrete reprimand. Concatenate = Parameters/Constraints: There are divers parameters/constraints that demand to be verified antecedently behavior in the ventilation. Naturally, past we are talking abextinguished a vericonsideration vitality birth, there canreferable attribuconsultation be a privative estimate of directs (x) or a privative separation from the top of the rod (y). Complete reprimands are fixed, and consequently complete graphs obtain merely be personateed in the principal quadrant. The other greater absorption that must be verified is the ultimatum extension of the rod, 230cm.

This limits the y-reprimand as polite-behaved-behaved as the x-value. The wavering n personates the terminable estimate of directs that could haply be assignd on the rod. Occasion it is physically practicable to assign frequent directs on the rod, a veritableistic, ultimatum estimate of directs that would stagnant be causative, is closely 15 directs. Direct Estimate (from top) Separation from Top (cm) 0* 1 2 3 4 5 6 7 8 n** 0 10 23 38 55 74 96 120 149 230 *the direct at the top of the rod is referable attribuconsultation attribuconsultation attribuconsultation counted **n is the terminable reprimand that personates the ultimatum estimate of directs that would tally on the rod.

Neither of the highlighted reprimands are criticised in this ventilation, they are merely here coercion the object of defining the limits of the waverings. The principal tread in this ventilation is to graph the objects in the consultation balancehead (excepting highlighted objects) to attobject the image of the diverge that is composed as aid directs are pretended to the rod. From this disseminate batch of the objects, we can attobject that there is an exponential growth in the separation from the top of the rod as each subjoined direct is pretended to the rod. Quadratic Part: The principal part that I shcomplete be copying using the objects of basis arranged is a quadratic part.

The unconcealed equation of a quadratic coercionmula is y = ax2 + bx + c. To do this, I obtain be using three objects of basis to compose three equations that I obtain unfold using matrices and state the coefficients: a, b and c. The principal tread in this scheme is to cull three basis objects that obtain be reasond to personate a liberal concatenate of the basis. This obtain be enigmatical though past there are merely three extinguished of the prospect objects that can be reasond. Consequently, to reconstruct the ratification of my quadratic part, I obtain be solving brace schemes of equations that reason divergent objects and ascertaining their mediocre. Basis Regulars Severedd: Basis Regular 1 = {(1,10), (3,38), (8,149)}

Basis Regular 2 = {(1,10), (6,96), (8,149)} These objects were severedd coercion brace ocean reasons. Principal, by using the x-values 1 and 8 in twain regulars of basis, we obtain possess a liberal concatenate of complete of the basis that is entity personateed in the developed equation subjoined the reprimands of the coefficients are mediocred. Relieve, I reasond the x reprimands of 3 (in the principal regular) and 6 (in the relieve regular) to unintermittently intermittently completeow coercion a liberal personateation of the basis objects in the developed quadratic equation. Twain of these objects (3 and 6) were severedd becareason they were correspondent separations aloof, 3 entity the third basis object, and 6 entity the third from developed basis object.

This fixed that the developed mediocred reprimands coercion the coefficients would bestow the best personateation of the mediocre basis objects withextinguished skewing the basis. There obtain be brace modes that obtain be reasond to unfold the scheme of equations, attendn beneath. Each mode obtain be reasond coercion single of the schemes entity evaluated. Basis Regular 1 = {(1,10), (3,38), (8,149)} In the principal basis regular, the basis objects obtain coercionm severed equations that obtain be unfoldd using a matrices equation. The principal matrix equation obtain be in the coercionm: Where A = a 3×3 matrix personateing the three basis objects X = a 3×1 matrix coercion the waverings entity unfoldd B = a 3×1 matrix coercion the y-reprimand of the three equations entity unfoldd. This matrix equation obtain be rearranged by developing twain planes of the equation by the inverse of A: Past A-1*A is correspondent to the individualness matrix (I), which when multiplied by another matrix bestows that corresponding matrix (the matrix equipollent of 1), the developed matrix equation is: To state the reprimands of X, we must principal ascertain the inverse of matrix A using technology, past it is helpful and ascertaining the inverse of a 3 by 3 matrix can select an incausative whole of term.

Principal wholeow us state what equations we obtain be solving and what our matrices obtain contemplate affect. Object: (1,10) (3, 38) (8,149) A= The equation is: ,X= ,B= = Proximate, by using our GDC, we can state the inverse of matrix A, and develop twain planes by it. Consequently we possess fixed that the quadratic equations abandoned the objects {(1,10), (3,38), (8,149)} is . Basis Regular 2 = {(1,10), (6,96), (8,149)} Object: (1,10) (6, 96) (8,149) A= ,X= ,B= The relieve mode that obtain be reasond to unfold the relieve scheme of equations is unreserved as Gauss-Jordan egress.

This is a scheme by which an raiseed matrix (brace matrices that are assignd into single divided by a course) goes through a course of humble unimaginative exercises to unfold the equation. On the left plane of this raiseed matrix (seen beneath) is the 3×3 matrix A (the odd matrix A that was made using basis regular 2, attendn on the anterior page), and on the accord is matrix B. The appearance of the exercises is to contrimpress matrix A to the individualness matrix, and by doing so, matrix B obtain accede the reprimands of matrix X. This is incorrectly unreserved as contractd extinguishedoutdirection echelon coercionm. Tread by tread scheme of reduction: 1. We initiate with the raiseed matrix. . Add (-36 * extinguishedoutdirection 1) to extinguishedoutdirection 2 3. Add (-64 * extinguishedoutdirection 1) to extinguishedoutdirection 3 4. Divide extinguishedoutdirection 2 by -30 5. Add (56 * extinguishedoutdirection 2) to extinguishedoutdirection 3 6. Divide extinguishedoutdirection 3 by 7. Add ( * extinguishedoutdirection 3) to extinguishedoutdirection 2 8. Add (-1 * extinguishedoutdirection 3) to extinguishedoutdirection 1 9. Add (-1 * extinguishedoutdirection 2) to extinguishedoutdirection 1 Subjoined complete of the extinguishedoutdirection exercises, matrix A has beaccord the individualness matrix and matrix B has beaccord the reprimands of matrix X (a, b, c). Consequently we possess fixed that the quadratic equations abandoned the objects {(1,10), (6,96), (8,149)} is . Averaging of the Brace Equations The proximate tread in ascertaining our quadratic part is to mediocre extinguished our periodical a, b, and c reprimands from the brace regulars basis.

Consequently we possess developedly fixed our quadratic part to be: Rounded to 4 sig figs, also oceantain correction, occasion oceantenance the estimates tractable. Basis objects using quadratic part Direct Estimate (from top) Quadratic reprimands Separation from Top (cm) Primordial – Separation from Top (cm) 1 10 2 22 3 37 4 54 5 74 6 97 7 122 8 149 10 23 38 55 74 96 120 149 Odd reprimands coercion the separation from top were rounded to cipher decimal assigns, to oceantain momentous image – the primordial reprimands reasond to ascertain the quadratic coercionmula had cipher decimal assigns, so the odd singles shouldn’t either.

Subjoined ascertaining the y-values abandoned x-values from 1-8 coercion the quadratic part I was able to assimilate the odd reprimands to the primordial reprimands (highlighted in green in the consultation balancehead). We can attobject that the brace reprimands that are the proper corresponding in twain basis regulars is (1,10) and (8,149) which is referable attribuconsultation attribuconsultation attribuconsultation portentous past those were the brace reprimands that were reasond in twain basis regulars when ascertaining the quadratic part. Another odd reprimand that was the corresponding as the primordial was (5,74). Complete other odd basis regulars possess an fallacy of closely ±2cm.

This basis shows us that the quadratic part can be reasond to personate the primordial basis with an approach fallacy of ±2cm. This part is stagnant referable attribuconsultation attribuconsultation attribuconsultation unblemished, and a improve part could be fixed-up to personate the basis with a inferior fallacy and aid tallying basis objects. Compimpress Part: The proximate tread in this ventilation is to copy a compimpress part that personates the primordial basis objects. The unconcealed equation of a compimpress part is y = ax3 + bx2 + cx + d. Distinguishing this, we can select indecent basis objects and perconstruct a scheme of equations to state the reprimands of the coefficients a, b, c, and d.

The principal tread is to cull the basis objects that obtain be reasond to copy the compimpress part. Harmoniously to copying the quadratic part, we can merely reason a scant estimate of objects to personate the basis in the part, merely in this fimpress it is indecent extinguished of the prospect basis objects, which mediocres that this part should be aid scrupulous than the developed. Unintermittently intermittently I plan on solving coercion brace regulars of basis objects and ascertaining their mediocre reprimands to personate the compimpress part. This is dsingle to completeow coercion a aid liberal personateation of the basis amid the compimpress part. Basis Regulars Severedd: Basis Regular 1: {(1,10), (4,55), (5,74), (8,149)}

Basis Regular 2: {(1,10), (3,38), (6,96), (8,149)} Twain basis regulars reason the objects (1,10) and (8,149), the principal and developed object, so that twain basis regulars concede compimpress parts that personate a liberal concatenate of the basis (from reserve to ultimatum). The other objects severedd, were severedd as mid concatenate objects that would completeow coercion the part to personate this concatenate of the basis aid correspondently. When copying a compimpress part or upper, it is enigmatical to do so withextinguished using technology to do the bulkyness of the estimation attribuconsultation to bulky wholes of flat estimations that would almost solution-ce a math fallacy somewhere.

Therefore, the most respectful and fastest mode to perconstruct these estimations obtain be to reason a GDC. In twain basis regulars, the contractd extinguishedoutdirection echelon coercionm part on the GDC obtain be utilized to state the reprimands of the coefficients of the compimpress parts. The scheme of determining the reprimands of the coefficients of the compimpress part using contractd extinguishedoutdirection echelon coercionm is harmonious to scheme reasond coercion the quadratic part. An x-reprimand matrix A (this term a 4×4 matrix), a wavering matrix X (4×1) and a y-reprimand matrix B (4×1) must be fixed principal. The proximate tread is to raise matrix A and matrix B, with A on the left and B on the accord.

This term, instead of doing the extinguishedoutdirection exercise ourselves, the GDC obtain do them, and accede an solution where matrix A obtain be the individualness matrix and matrix B obtain be the reprimands of the coefficients (or matrix X). Basis Regular 1: {(1,10), (4,55), (5,74), (8,149)} (1,10) (4, 55) (5, 74) (8,149) A1 = , X1 = , B1 = We initiate with the raiseed matrix or matrix A1 and matrix B1. Then this matrix is inputted into a GDC and the part “rref” is severedd. Subjoined importunate invade, the matrix is contractd into contractd extinguishedoutdirection echelon coercionm. Which accedes the reprimands of the coefficients. Basis Regular 2: {(1,10), (3,38), (6,96), (8,149)} (1,10) (3, 38) 6, 96) (8,149) A2 = , X2 = , B2 = We initiate with the raiseed matrix of matrix A2 and matrix B2. Then the matrix is inputted into a GDC and the part “rref” Subjoined importunate invade, the matrix is contractd into contractd extinguishedoutdirection echelon coercionm. Which accedes the reprimands of the coefficients. The proximate tread is to ascertain the mediocre of each of the reprimands of the coefficients a, b, c, and d. Consequently we possess developedly fixed our compimpress part to be: Unintermittently intermittently rounded to 4 momentous images. Updated Basis consultation, including compimpress part reprimands. Direct Estimate (from top) Quadratic reprimands Separation from Top (cm) 1 10 2 22 3 37 4 54 5 74 6 97 122 8 149 Compimpress reprimands Separation from Top (cm) Primordial – Separation from Top (cm) 10 23 38 54 74 96 121 149 10 23 38 55 74 96 120 149 Odd reprimands coercion the separation from top were rounded to cipher decimal assigns, to oceantain momentous image – the primordial reprimands reasond to ascertain the quadratic coercionmula had cipher decimal assigns, so the odd singles shouldn’t either. The y-values of the compimpress part can be assimilated to that primordial basis regular reprimands to deduce whether or referable attribuconsultation attribuconsultation attribuconsultation it is an respectful part to reason to personate the primordial basis objects. It appears as though the compimpress part has 6 extinguished of 8 basis objects that are the corresponding.

Those objects entity, (1,10), (2,23), (3,38), (5,74), (6,96), (8,149). The three basis objects from the compimpress part that did referable attribuconsultation attribuconsultation attribuconsultation tally merely had an fallacy of ±1, indicating that the compimpress part would be a amiable personateation of the primordial basis objects, save stagnant has some fallacy. We can aid criticise these objects by comparing the compimpress and quadratic part to the primordial objects by graphing them. Attobject proximate page. By analyzing this graph, we can attobject that twain the quadratic part and the compimpress part tally the primordial basis objects totally polite-behaved, although they possess contempt dissimilitudes.

By comparing reprimands on the basis consultation, we ascertain that the quadratic part merely tallyes 3 of the 8 primordial basis objects with an fallacy of ±2, occasion the compimpress part tallyes 6 of the 8 objects with an fallacy of equiboard ±1, which is as smcomplete an fallacy practicable coercion correction of the estimation done. Twain parts impress as deferred personateations of the primordial objects, save the greater dissimilitude is how they initiate to dissent as the graphs objecture. The compimpress part is increasing at a faster reprimand than the quadratic part, and this dissimilitude would beaccord totally referable attribuconsultation attributableiceable balance term.

This would mediocre that if these parts were to be reasond to state the separation a 9th direct should be from the top, the brace parts would arconcatenate totally divergent solutions, with the compimpress parts providing the aid respectful single. Polynomial Part: Past it is unreserved that neither the quadratic, nor the compimpress part largely fulfil the primordial basis objects, then we must copy a upper extent polynomial part that obtain fulfil complete of these objects. The best mode to ascertain a polynomial part that obtain ignoring through complete of the primordial objects is to reason complete of the primordial objects when ascertaining it (confront to equiboard three or indecent).

If complete prospect of the objects are reasond and a scheme of equations is achieveed using matrices, then a part that satisfies complete objects obtain be fixed-up. This is a septic part. To ascertain this part, the corresponding process followed coercion the developed brace parts should be followed, this term using complete prospect objects to compose an 8×8 matrix. By then subjoined the corresponding treads to raise the matrix with an 8×1 matrix, we can modify the matrix into contractd extinguishedoutdirection echelon coercionm to and ascertain our solution. In this mode, past we are using complete prospect objects, the undiminished basis regular is entity personateed in the part and no averaging of the results obtain be expedient.

The unconcealed coercionmula coercion a septic part is . Basis Regular: {(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= , Raise matrix A and matrix B and perconstruct the ‘rref’ part The solutions and reprimands coercion the coefficients = The developed septic part equation is This part that embrace complete the primordial basis objects can be attendn graphed here beneath concurrently with the primordial objects. Updated Basis consultation, including septic part reprimands Direct Estimate (from top) Quadratic reprimands Separation from Top (cm) Compimpress reprimands Separation from Top (cm)

Septic reprimands – Separation from Top (cm) Primordial – Separation from Top (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 Odd reprimands coercion the separation from top were rounded to cipher decimal assigns, to oceantain momentous image – the primordial reprimands reasond to ascertain the quadratic coercionmula had cipher decimal assigns, so the odd singles shouldn’t either. By contemplateing at the graph, as polite-behaved-behaved as the basis consultation (twain attendn balancehead), we can attobject that, as expected, complete 8 of the septic part basis objects are selfcorresponding to that of the primordial basis.

There is hither than 1cm of fallacy, which is accounted coercion attribuconsultation to imscrupulous (cipher decimal assigns) primordial measurements. Consequently we now distinguish that the septic part that utilised complete of the primordial basis objects is the best personateation of said basis. Other Part: The proximate appearance in this ventilation is to ascertain another part that could be reasond to personate this basis. The other mode that I obtain reason to ascertain a part that tallys the basis is quadratic retrogradation. Quadratic retrogradation reasons the mode of lowest squares to ascertain a quadratic in the coercionm .

This mode is frequently reasond in statistics when troublesome to state a flexion that has the minimal sum of the discontinuances squared from a abandoned regular of basis. In humble provisions, it ascertains a part that obtain silence any unexpedient clamor in self-possessed basis results by ascertaining a reprimand that has the slightest whole of discontinuance from the greaterity of the basis. Quadratic retrogradation is referable attribuconsultation attribuconsultation attribuconsultation reasond to unblemishedly tally a basis regular, save to ascertain the best flexion that goes through the basis regular with minimal discontinuance. This part can be fixed-up using a GDC. Principal you must input the basis objects into lists, (L1 and L2).

Then you go to the statistic math parts and cull QuadReg. It obtain distinguish to reason the brace lists to state he quadratic part using the mode of lowest squares. Unintermittently the estimation has completed, the basis attendn beneath (values coercion the coefficients of the part) obtain be presented: QuadReg a = 1. 244 b = 8. 458 c = 0. 8392 With this basis we can state that the part is When graphed, this part has the image attendn beneath: Updated Basis consultation, including septic part reprimands Direct Estimate (from top) Quadratic reprimands Separation from Top (cm) Compimpress reprimands Separation from Top (cm)

By analyzing the graph and reprimands of the quadratic retrogradation part, it is clear that it is a relatively respectful coercionm of copying the basis. Indecent of the prospect objects tallyed that of the primordial basis, with an fallacy of ±1. The most referable attribuconsultation attributableable dissimilitude betwixt the quadratic retrogradation part and the quadratic part anteriorly fixed, is the importation amid the basis f the respectful reprimands. The retrogradation part tallyed the mediocre basis, occasion the quadratic part tallyed the object basis. It is thrilling to attobject how brace parts in the corresponding coercionm, fixed-up using divergent modes accedeed contradictory areas of ratification. Best Tally: The part that impresss as the best copy coercion this birth is the septic part. It is the merely part that satisfies each of the primordial basis objects with its equation. Through ascertaining the quadratic, compimpress and septic parts, it was discovered that the extent of the polynomial was straightform correlated to the part’s ratification to the basis.

Consequently it was no astonish that this part impresss as the best tally coercion this basis. The other careason coercion this septic part having the best mutuality to the primordial basis is attribuconsultation to the septic part entity periodical by creating a scheme of equations using complete of the basis objects. 9th Direct: Using my quadratic copy, it can be fixed where the optimal importation coercion a ninth direct would be by substituting ‘9’ in coercion x in the equation . Using my quadrating copy, it was fixed-up that the optimal importation coercion a ninth direct on the rod is 179cm from the top of the rod.

Leo’s fishing rod is 230cm crave, nevertheshort his prospecth direct is merely 149cm from the top of the rod. That mediocres that there is 81cm of the course that is referable attribuconsultation attribuconsultation attribuconsultation entity directd from the stagger to principal direct. By adding a ninth direct, that separation obtain be curtailed coercionm 81cm to 51cm. By doing this, it obtain be hither affectly coercion the course to muster up and beaccord complicated in this 81cm elongate where there is no direct. Another implication of adding another direct would be that the wprospect classification of a fish entity staggered in would be divulge balance another direct, which obtain completeow coercion an easier exercise of staggering in the fish.

There is equable ample remoteness on the rod coercion a 10th direct at 211cm from the top of the rod. This direct would unintermittently intermittently curtail the debauchery course aid to a object where the debauchery course betwixt the stagger and the principal direct is shorter than course betwixt the principal and relieve direct. This could careason problems with staggering and imageing teachableness, as that extra direct would careason slowing move of the course. The benetally would be that unintermittently intermittently the wprospect classification of fish would be divulge balance a bulkyr estimate of directs.

Overall, it would be profitable to embrace a ninth direct to Leo’s fishing rod, save anyaid obtain affectly above its teachableness. Mark’s Fishing Rod: Direct Estimate (from top) Separation from Top (cm) 1 10 2 22 3 34 4 48 5 64 6 81 7 102 8 124 To attobject how polite-behaved-behaved my quadratic copy tallys this odd basis, they must be twain batchted on the corresponding graph, attendn beneath. My quadratic copy coercion Leo’s fishing rod correlates with Mark’s fishing rod basis coercion the principal lacking reprimands and then diverges as the estimate of directs growths by growing at a upper exponential reprimand.

The dissimilitude betwixt Leo and Mark’s prospecth direct from the top of their relative rods is 25cm, nevertheshort twain men’s principal directs initiate the corresponding separation from the top of their rods. The quadratic part reasond to copy Leo’s fishing rod does referable attribuconsultation attribuconsultation attribuconsultation correlate polite-behaved-behaved with Mark’s fishing rod basis. Modifys to the copy must be made coercion it to tally this basis. The best mode to ascertain a copy coercion Mark’s basis would be to go through the corresponding treads that we went through to state the principal quadratic coercionmula that copy’s Leo’s fishing rod.

By doing so, local reprimands that improve personate Mark’s fishing rod basis could be reasond to plant a improve tallyting part. The ocean withholding of my copy is that is was artificeed as a part coercion Leo’s basis localally. It was composed by solving schemes of equations that reasond solely Leo’s fishing rod coercion basis. Consequentially, the quadratic copy best personateed Leo’s fishing rod, which had a ultimatum extension of 230cm, with divergently remotenessd extinguished directs. There were frequent dissimilitudes betwixt Leo and Mark’s fishing rods (such as ultimatum extension and direct spacing) that concederd my primordial quadratic copy to referable attribuconsultation attribuconsultation attribuconsultation polite-behaved-behaved personate Mark’s basis.

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