Mathematical Methods — Question 18

QCAA 2021, Paper 2 · 3 marks

Q18 · 2021 · Technology-activeComplex unfamiliar3 marks

QUESTION 18 (3 marks)

The number of animals in a population (in thousands) is modelled by the function

P

such that

P(t)=\frac{100}{1+4e^{-t}},

where

t

is in years. Determine the number of animals in the population when the population is growing the fastest.

WORKED SOLUTION

3 marks

ANSWER

Approximately 50 000 animals

Worked solution

Graphing

P(t)

and

P'(t)

, the population is increasing most rapidly at the maximum value of

P'(t)

. This occurs at

t=1.386

. Then

P(1.386)\approx49.993

thousand, so there are approximately 50 000 animals.

correctly identifies the conditions for the most rapid increase

[1 mark]

determines when population growing the fastest

[1 mark]

determines population at this time

[1 mark]

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

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