Zadanie 10d, Paper 2, maj 2026, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2026, TZ1, Paper 2 (zadania wzorowane), pytanie 10(d). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
Wspólne informacje do wiązki zadań
On 1 January 2026 a makers’ club deposits USD 18,000 to fund a machine initially priced at USD 31,000. The account pays 6% nominal annual interest compounded monthly, credited at the end of each month. There are no other deposits or withdrawals.
Polecenie
The machine price depreciates at a constant annual percentage rate r. After one year its price is USD 29,140.
For any real number t of years after 1 January 2026, model the price by 31000(1 − r/100)t. The club buys as soon as its credited account balance can cover this price. Determine the purchase year.
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Odpowiedź z klucza
The price model is 31000(0.94)^t. Comparing the growth envelope 18000(1.005)12t with that price gives t=4.465897876… years. Check the actual monthly credits: after 53 months the balance is 23446.27… while the price is 23587.20…; just before the 54th credit the unchanged balance remains below the price 23465.89… . After that credit the balance is 23563.50… and covers the price. Thus purchase occurs in 2030. The requested year is unchanged by the within-month check.
Schemat punktowania
A1: correct depreciation expression. M1: compare growth with depreciation. A1: correct equation, inequality or paired table rule. A1: crossing time or a correct before/after bracket that locates the year. A1: 2030. A calculation of a precise day is not required.
Komentarz. Paper 2: graphic display calculator and AA HL formula booklet allowed. Give exact results or three significant figures unless specified otherwise. Record equations and technology inputs; retain full precision between steps.