Mathematical Methods — Question 20

QCAA 2021, Paper 1 · 7 marks

Q20 · 2021 · Technology-freeComplex unfamiliar7 marks

QUESTION 20 (7 marks)

The population of rabbits

(P)

on an island, in hundreds, is given by

P(t)=t^2\ln(3t)+6

t>0

, where

t

is time in years. Determine the intervals of time when the population is increasing and the intervals when it is decreasing.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

Decreasing for

0<t<\dfrac{1}{3\sqrt e}

; increasing for

t>\dfrac{1}{3\sqrt e}

Worked solution

**Method 1.**

P'(t)=t+2t\ln(3t)=t(1+2\ln(3t))

. Solving

P'(t)=0

gives

t=0

(rejected because

t>0

) or

t=\frac1{3\sqrt e}

. Also

P''(t)=3+2\ln(3t)

, and

P''(1/(3\sqrt e))=2>0

, so the critical point is a minimum. Therefore the population decreases for

0<t<\frac1{3\sqrt e}

and increases for

t>\frac1{3\sqrt e}

**Alternative Method 2.** Since

t>0

P'(t)>0

is equivalent to

1+2\ln(3t)>0

, hence

t>\frac1{3\sqrt e}

. Thus the population is increasing on that interval and decreasing on

0<t<\frac1{3\sqrt e}

correctly determines P-prime(t)

[1 mark]

correctly determines the rejected solution for t

[1 mark]

determines t-ordinate of critical point

[1 mark]

determines value of P-double-prime(t) at critical point

[1 mark]

determines nature of critical point

[1 mark]

communicates when the population is increasing and when it is decreasing

[1 mark]

shows logical organisation communicating key steps

[1 mark]

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Mathematical Methods — Question 20

QUESTION 20 (7 marks)

QCAA guide · typeset solution

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