Matheatics C 2008 Saple assessent instruent and indicative student response Supervised assessent: Modelling and proble-solving tas This saple is intended to infor the design of assessent instruents in the senior phase of learning. It highlights the qualities of student wor and the atch to the syllabus standards. Criteria assessed Knowledge and procedures Modelling and proble solving Counication and justification ssessent instruent The indicative response presented in this saple is in response to odelling and proble-solving tass. Question 1 In a certain alpine village, the residents are divided in their opinions about whether they should stay or leave in the face of an advancing glacier. One group siply says that since the glacier is advancing towards the village and they should all leave. The second group believes that they should stay. To support their position, the second group aes annual observations about the location of the face of the advancing glacier. In the year 2000, the encroaching face of the glacier was 2.0 iloetres fro the village. In 2001, the face of the glacier was 1.4 iloetres fro the village. Observations in 2002 and 2003 gave distances of 1.04 iloetres and 0.824 iloetres respectively. If the oveent of the glacier is assued to continue in the sae pattern, which group of villagers opinion will be justified? Question 2 sall naval target rises and falls with the waves with SHM. It has a period of 7 seconds, with the height of the waves, fro crest to trough, being 8 etres. t a horizontal range of 6000 a gun is fired so that the centre of the target would be hit if it stayed stationary in its topost position. naval officer clais, If the horizontal velocity of the bullet is 2400 /s, the target will be hit. Evaluate the validity of this arguent.
Question 3 strealined object falls fro a satellite towards earth. One odel to represent the otion of this object is to consider there is no air resistance. Considering downwards oveent as positive, the velocity at any tie will be given by v = gt. How long will it tae for the object to be travelling at 40s -1? nother odel to represent the otion of this object is to consider there is air resistance to the otion, producing a retardation of 0.2v, so that the acceleration of the object is given by a = g 0.2v, where downwards oveent is considered positive. Solve the differential equation dv dt = g 0.2v to find an expression for velocity at any tie t. How long will it tae now for the object to be travelling at 40s -1? Copare the two answers in light of the assuptions ade and the liitations of each odel. Question 4 To siulate a non-terrestrial atosphere, a laboratory experient was conducted under the following conditions. ass of 0.55 g was dropped into a cylinder containing a non-terrestrial ediu. Fro there, the ass otion was deterined by gravity and the resistance of the ediu. The resistance to otion was nown to be proportional to the speed of the ass. (a) Deterine the velocity function for the ass in this ediu if the constant of proportionality is. (b) If the velocity after 2 seconds is 5.248 s -1, use a graphical ethod to deterine the approxiate value of in SI units. Question 5 The logistic curve can be represented by the differential equation dn N = λ N where is the upper level dt or bound of the S-shaped growth curve. Show that this equation has the solution N = 1 λ t + C e where C is a constant and the initial condition is such that N =. 1 + C 2 Maths C 2008 Saple student assessent and response
Instruent-specific criteria and standards Student responses have been atched to instruent-specific criteria and standards; those which best describe the student wor in this saple are shown below. For ore inforation about the syllabus diensions and standards descriptors, see www.qsa.qld.edu.au/1896.htl#assessent. Standard Knowledge and procedures The student wor has the following characteristics: use of proble-solving strategies to interpret, clarify and analyse probles application of atheatical definitions, rules and procedures in routine and non-routine siple tass through to routine coplex tass, in life-related and abstract situations nuerical calculations, spatial sense and algebraic facility in routine and non-routine siple tass through to routine coplex tass, in life-related and abstract situations appropriate selection and accurate use of technology nowledge of the nature of and use of atheatical proof. Modelling and proble solving The student wor has the following characteristics: use of proble-solving strategies to interpret, clarify and analyse probles to develop responses fro routine siple tass through to non-routine coplex tass in life-related and abstract situations identification of assuptions and their associated effects, paraeters and/or variables investigation and evaluation of the validity of atheatical arguents including the analysis of results in the context of probles, the strengths and liitations of odels, both given and developed. Counication and justification The student wor has the following characteristics: appropriate interpretation and use of atheatical terinology, sybols and conventions fro siple through to coplex and fro routine through to non-routine, in life-related and abstract situations use of atheatical reasoning to develop coherent, concise and logical sequences within a response fro siple through to coplex and in life-related and abstract situations using everyday and atheatical language coherent, concise and logical justification of procedures, decisions and results provision of supporting arguents in the for of proof. Queensland Studies uthority May 2013 3
Indicative response Standard The annotations show the atch to the standards. Coents Solution 1 Considering the distances oved, we obtain the following yearly changes: pplication of atheatical definitions, rules and procedures in a non-routine siple tas, in life-related situation Tie of observation after 2001 after 2002 after 2003 Change in distance 0.6 0.36 0.216 Coherent, concise and logical justification of a decision Investigation and evaluation of the validity of a atheatical arguent The changes for a geoetric progression with a = 0.6 and r = 0.6. Since we are considering an infinite change, the total change is given by: 0.6 Distance = 1 0.6 0.6 = 0.4 = 1.5 Therefore, the nearest the glacier will approach the village is 2.0 1.5 = 0.5. The village is not in danger of being glaciated. 4 Maths C 2008 Saple student assessent and response
pplication of atheatical definitions, rules and procedures to a routine siple tas, in a life-related situation Use of proble-solving strategies to interpret, clarify and analyse a proble to develop a response to a routine siple tas in a life-related situation Investigation and evaluation of the validity of atheatical arguents including the strengths and liitations of a given odel Solution 2 2π T = w ` 2π ω = 7 x = a cos ωt when t = 0 x = 4 2 π x = 4 cos t 7 Tie for theparticle to travel 6000 at t = 6000 2400 = 2.55 sec 2π x = 4cos 2.55 7 x = 2.49 8 With reference to the diagra above : Distance to top ost position = 4.0 + 2.49 = 6.49 v = 2400 /s This arguent is not valid because, according to the diagra, the target will be issed by 6.49 or 2.49 below the equilibriu position. 4 2.49 The liitations of this odel are that air resistance would have an ipact on the otion of the bullet. It is unliely that the bullet would aintain a horizontal path; it would probably have vertical displaceent. The otion of the odel is unliely to accurately represent SHM. Queensland Studies uthority May 2013 5
Identification of assuptions and their associated effects ppropriate interpretation and use of atheatical terinology, sybols and conventions in a coplex routine tas, in a liferelated situation lgebraic facility in a routine coplex tas, in a life-related situation Use of atheatical reasoning to develop a coherent, concise and logical sequence within a response to a coplex liferelated situation using atheatical language Solution 3 With no air resistance When v = 40, v = gt 40 = 10t t = 4 Therefore, with no air resistance, the velocity reaches 40s -1 after 4 seconds. With air resistance dv = g 0.2v dt dv = dt g 0.2v ln( g 0.2v ) = t + c 0.2 lng When t = 0, v = 0 and c =. Thus 0.2 ln( g 0.2v ) lng = t 0.2. 02 g 0.2v ln = 0.2t g g 0.2v 0.2t = e g 0.2t g 0.2v = ge g 0.2t v = ( 1 e ) 0.2 When v = 40, 0.2t 40 = 50( 1 e ) 0.2t e = 0.2 0.2t = ln0.2 ln0.2 t = 0.2 8.05 With the air resistance, the object will tae approxiately 8 seconds to reach 40s -1. When air resistance is taen into account, the tie taen for the object to reach 40s -1 approxiately doubles. 6 Maths C 2008 Saple student assessent and response
pplication of atheatical definitions, rules and procedures in a non-routine tas, in an abstract situation lgebraic facility in an abstract routine coplex tas Use of atheatical reasoning to develop a coherent, concise and logical sequence within a response to a coplex liferelated situation using atheatical language ppropriate selection and accurate use of technology Solution 4 (a) Tae the downwards direction as positive. F = g v and F = a Therefore: a = g v dv so = g v dt dv = v g dt dv = dt v g Integrating both sides we obtain: ln v g = t + c v g = c e v = g t 1 t 1 + c e When v = 0, c1 = g. Therefore: v ( t ) = g = g g e 1 e (b) When t = 2, v ( 2 ) = 5.248, so: t 0.55 5.248 = 9.8 1 e 3.63636 0.97365 = 1 e t 2 0.55 3.63636 Graphing g( ) = 0. 97365 and f ( ) 1 e point of intersection at approxiately = 1.000. Therefore, the value of = 1. =, we obtain a Queensland Studies uthority May 2013 7
Knowledge of the nature of and use of atheatical proof lgebraic facility in an abstract routine coplex tas Provision of supporting arguents in the for of proof Solution 5 The logistic curve can be represented by the differential equation: dn N = λ N. dt We can rewrite this equation, using separation of variables, as: dn = λ dt N N ( ) By using partial fractions, we obtain: 1 1 = + N N N ( ) N So the differential equation becoes: 1 1 + dn = λ dt N N Integration of this equation yields: ln N ln( N) = λ t + N ln = λ t + N N or ln = λ t + N Using the initial condition of N =, we obtain: 1 + C ln 1+ C = λ 0 + 1+ C so lnc = Substituting for this value we obtain: N ln = λ t + lnc N N or ln = λt NC N λ t and = e NC N + NC e N N = NC e λ t λ t ( 1+ C e ) and = = λ t N = 1+ C e λt 8 Maths C 2008 Saple student assessent and response