Zadanie 2c, Paper 3, maj 2025, TZ3, Matematyka AA IB HL
Wzorowane na: Matematyka AA IB HL — maj 2025, TZ3, Paper 3 (zadania wzorowane), pytanie 2(c). 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
Show that the only solutions of dV/dx = 0 are .
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Odpowiedź z klucza
V′(x) = 27x2 − 6(a + b)x + ab. The quadratic formula gives . The discriminant is 36(a2 − ab + b2), so division by 6 yields the stated expressions. As a²−ab+b²=(a−b)²+ab>0, these are two distinct real roots and exhaust the roots of the quadratic derivative.
Schemat punktowania
M1: differentiate, with at least two correct terms for this attempt mark. A1: correct derivative. M1: solve its zero equation using the quadratic formula. A1: correct substituted formula. M1: expand/simplify the discriminant or extract its factor 36. A1: correct expanded or factored radicand establishing the two stated roots. Alternative verification: after the derivative marks, M1 substitute a proposed root, A1 for each root reducing the derivative to zero, R1 a quadratic has no further roots.
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.