Zadanie 1e, Paper 3, listopad 2024, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2024, Paper 3 (zadania wzorowane), pytanie 1(e). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
A digitization team begins year 1 with 4800 pages awaiting processing. During each year, incoming revisions increase the outstanding work by 12.5%, after which a fixed quota is processed at year-end. Treat page counts as continuous estimates until rounding is requested. With quota 400, T_n denotes the work at the start of year n, so T1 = 4800 and . With quota 750, use D_n instead. For either policy, a non-positive algebraic prediction means the queue has already cleared; actual work cannot be negative.
Polecenie
Under quota 750, Dn = −1200(9/8)n−1 + 6000. At the end of which processing year does the queue first clear?
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Clearance occurs when (9/8)n−1 ≥ 5, giving n ≥ 1 + ln 5/ln(9/8) = 14.6644288667…. The first start-of-year index with a non-positive algebraic estimate is n=15. Indeed D14 = 451.64759… > 0 and D15 = −241.89646… < 0. Thus clearance occurs at the end of processing year 14, immediately before the start of year 15.
Schemat punktowania
M1: attempt numerically, graphically, algebraically or by trial to find where D_n≤0 (equality threshold also accepted). A2: critical start-index 14.6644288667…, or a correct boundary value D₁₄=451.64759…>0, or D₁₅=−241.89646…<0. A1: end of processing year 14; explicitly start of year 15 is equivalent. Also accept 15 where correct shown working clearly identifies it as the D₁₅ start index; bare 15 with no supporting interpretation/method does not answer the explicitly requested processing year. The total-decrease method through end-year index gives critical 13.6644288667… and endyear14, with equivalent marks.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact forms unless an approximation is requested.