Zadanie 1f, Paper 3, listopad 2024, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — listopad 2024, Paper 3 (zadania wzorowane), pytanie 1(f). 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
With a general quota d, the algebraic estimate is Cn = (4800 − 8d)(9/8)n−1 + 8d. Find the quota that keeps the start-of-year work constant.
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Odpowiedź z klucza
The exponential factor changes with n, so its coefficient must vanish: 4800 − 8d = 0. Therefore d=600 pages. Equivalently the annual growth on 4800 pages is 600, exactly offset by this quota.
Schemat punktowania
M1: impose a constant sequence using the zero coefficient or the recurrence. A1: correct equation 4800−8d=0 or 4800=(9/8)4800−d. A1: 600 pages. Also accept setting a specific pair C₂=C₃ or C₂=4800 (A1), attempting to solve that equation or systematically varying d (M1), and 600 (A1).
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Give exact forms unless an approximation is requested.