Zadanie 10c, Paper 2, maj 2023, TZ1, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2023, TZ1, Paper 2 (zadania wzorowane), pytanie 10(c). Treść, dane i kontekst są zmienione; sprawdzane umiejętności, format i punktacja jak w oryginale.
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Zadanie
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A closed coolant-buffer channel is 450 cm long. Its cross-section consists of a rectangle of height 8 cm and width w, capped above and below by identical outward-bowed minor circular segments. Each circular arc has radius 10 cm and central angle θ, with 0<θ<π. The developed perimeter of the cross-section is 60 cm, comprising the two straight 8 cm sides and both arcs. The channel is initially empty and completely flushed every 64 seconds. Between flushes there is no outlet. Coolant enters at rate cm³/s, t≥0.
Polecenie
Determine whether overflow occurs before the first flush. Justify your answer.
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Capacity is 450A=126784.5918… cm³. The incoming volume over 0≤t≤64 is cm³. The sine term completes eight periods and contributes zero. Since the rate is positive and 140800 exceeds capacity, overflow occurs before 64 s. Alternatively the minimum inflow rate 2120 gives at least 135680 cm³ over 64 s, also exceeding capacity.
Schemat punktowania
A1:capacity 450A. M1:integrate the inflow rate, or use a valid lower bound. A1:correct integral evaluation setup or minimum rate 2120. A1:140800 cm³, or lower bound 135680 cm³. A1:overflow, justified by comparison with capacity. Solving the capacity-crossing time is an equivalent method.
Komentarz. Paper 2: calculator and AA HL formula booklet allowed. Give exact values where requested; otherwise retain intermediate precision and use three significant figures, except the specified six-significant-figure approximations.