Mathematical Methods — Question 17

ANSWER

Jane’s claim is not correct; the maximum total population is approximately 24 031.

Worked solution

Models are periodic. $$R(t)=11+3.5\cos\left(\frac{\pi}{6}t\right),\qquad F(t)=9+2\sin\left(\frac{\pi}{6}t\right).$$ Hence $$T(t)=20+3.5\cos\left(\frac{\pi}{6}t\right)+2\sin\left(\frac{\pi}{6}t\right).$$ Graphing $T(t)$ gives a greatest total population of 24 031. Jane’s claim is not correct: the maximum occurs once in the year but does not exceed 25 000.

correctly determines the models for the two populations

[1 mark]

determines model for total population of foxes and rabbits

[1 mark]

uses an appropriate mathematical representation

[1 mark]

evaluates reasonableness of the claim

[1 mark]

One QCAA sample method typeset for web; criterion wording is adapted from the official marking guide.