Zadanie 8c, Paper 2, listopad 2023, TZ1, Matematyka AA IB SL
Wzorowane na: Matematyka AA IB SL — listopad 2023, TZ1, Paper 2 (zadania wzorowane), pytanie 8(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
Darmowe konto pozwoli wrócić do niego później.
Zadanie
Wspólne informacje do wiązki zadań
A research project can choose a grant schedule with payments at the start of each month for 24 months. Plan A pays 3000 dollars each month. Plan B starts at 1800 dollars, with each later payment 5% larger than the previous one. Grant receipts are retained without interest. Separately, a90000-dollar reserve is deposited on the day of the first grant payment and earns nominal annual interest 4.8%, compounded monthly at each month-end; there are no withdrawals or additional reserve deposits. For parts 8c use Plan B. In parts 8a(ii),8c(i),8d give two decimal places.
Polecenie
(i) Find total retained grant plus reserve at the end of month 6. (ii) Find the least number of complete months before this combined value reaches at least 165000 dollars.
Korepetytor
Nie wiesz, jak ruszyć? Korepetytor zna to zadanie i poprowadzi Cię krok po kroku, bez gotowca.
3 rozmowy · bez karty · Zaloguj się
Sprawdź się
Jak poszło?
Pokaż rozwiązanieUkryj rozwiązanie
Odpowiedź z klucza
(i) Six grant payments sum to 1800(1.05⁶−1)/0.05=12243.4430625. The reserve is 90000(1.004)⁶=92181.71554615…. Total 104425.15860865…=104425.16 dollars. (ii) For m≤24 months, T(m)=90000(1.004)^m+1800(1.05^m−1)/0.05. It is increasing. T(21)=162164.8538444…<165000 and T(22)=167571.0673095…≥165000; thus 22 months.
Schemat punktowania
(i) A1:correct six-month reserve. A1:correct six-payment sum. A1:104425.16, with the shared one-time precision rule. (ii) A1:correct combined expression and threshold equation/inequality. A1:correct boundary comparison at 21,22 or a numerical crossing between them. A1:least integer 22.
Komentarz. Paper 2: calculator and AA SL formula booklet allowed. Retain intermediate precision. Use three significant figures unless exact answers or two decimal places are requested.