Zadanie 2h, Paper 3, maj 2025, TZ3, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ3, Paper 3 (zadania wzorowane), pytanie 2(h). 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 designer studies rectangular open vessels with a fixed outside footprint a cm by b cm, where 0 < a < b. The interior height is x cm, and each vertical wall has thickness 3x/2 cm. The internal floor therefore measures (a − 3x) cm by (b − 3x) cm. The usable capacity is V = x(a − 3x)(b − 3x) cubic centimetres. Include the limiting degenerate designs by taking 0 ≤ x ≤ a/3.
Polecenie
Let xm denote the interior height maximizing V on [0, a/3].
For the remaining parts, set a = s2 + 2st and b = t2 + 2st, where s,t are positive integers and a<b.
Let Vm = xm(a − 3xm)(b − 3xm). (i) Explain why s divisible by 3 guarantees that both xm and Vm are integers. (ii) Give a different divisibility condition on s and t that guarantees the same result when s is not divisible by 3. (iii) Find one positive integer xm and its corresponding integer capacity Vm, stating the parameters used.
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Odpowiedź z klucza
(i) If 3 divides s then xm = st/3 is an integer. Since a,b and 3x_m are integers, V_m is a product of integers. (ii) Require t divisible by 3 while s is not. Then the product st is divisible by 3, so x_m is an integer. (iii) For example choose s=3,t=4, satisfying t>s. Then a=33,b=40 and x_m=4; the capacity is 4(33 − 12)(40 − 12) = 2352 cm3. An alternative is s=1,t=3: a=7,b=15,x_m=1,V_m=48. Generally Vm = s2t2(s + t)2/3.
Schemat punktowania
(i) R1: establish x_m is an integer from the factor divisible by 3; alternatively A1 for correctly substituting x_m=st/3 into the capacity expression. R1: conclude that V_m is a product of integers, explicitly using the divisibility condition. (ii) A1: t divisible by 3, with s not divisible by 3; no formal modular notation is required. (iii) M1: use chosen parameters to calculate x_m, a,b or V_m. A1: correct corresponding integer x_m. A1: correct a and b. A1: correct corresponding integer V_m. Award at most the method mark in (iii) if the selected s,t fail positivity, t>s or integer-x_m conditions. The x_m mark and the a,b mark are independent.
Komentarz. Paper 3: calculator and AA HL formula booklet allowed. Show complete reasoning for proofs and retain exact expressions. Equivalent correct arguments are accepted unless a particular method is requested.