Zadanie 2a, Paper 3, listopad 2025, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2025, TZ1, Paper 3 (zadania wzorowane), pytanie 2(a). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
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A regional volunteer network had maintained 1000 active memberships before disruption reduced the count to 125. A recovery programme starts at t=0 years. At t=4 the network has250 active members; at t=8 an audit measures 380. Two models use the long-term level 1000: a logistic model x = L/(1 + Ce−kt), with L,C,k>0, and a Gompertz model dx/dt = axln(1000/x), a>0. Fit each model using the observations at t=0 and t=4; reserve the t=8 audit for comparison. In the adjusted model, active memberships also lapse at a continuous proportional rate 0.01 per year.
Polecenie
For the logistic model: (i) interpret choosing L=1000; (ii) find C; (iii) find k; (iv) predict the membership count at t=8, to the nearest whole member.
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Odpowiedź z klucza
(i) The model assumes the recovered long-term capacity returns to the former level 1000. (ii) 125=1000/(1+C) gives C=7. (iii) 250=1000/(1+7e⁻⁴ᵏ), soe⁻⁴ᵏ=3/7 and k=(1/4)ln(7/3)=.2118244651…. (iv) x(8)=1000/[1+7(3/7)²]=437.5, which rounds to 438 members.
Schemat punktowania
(i) A1:correct long-term/capacity assumption. (ii) M1:substitute the initial observation. A1:C=7. (iii) A1:correct second-observation equation. A1:k=(1/4)ln(7/3) or.212. (iv) A1:correct evaluation at 8 using the fitted model. A1:438 whole members. Consistent use of the three-significant-figure rate k=.212 is also accepted, as is appropriate follow-through with working from earlier fitted positive parameters.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Show complete reasoning and reproducible numerical work. Give exact results unless a stated approximation is requested.