Mathematical Methods — Question 19

QCAA 2022, Paper 2 · 4 marks

Q19 · 2022 · Technology-activeComplex unfamiliar4 marks

QUESTION 19 (4 marks)

Flying foxes enter and leave a fruit-growing region every evening. The rate at which the flying foxes enter the region is modelled by

A(t)=42\sin\left(0.03t-\frac{\pi}{3}\right)+71,\qquad0\le t\le240.

The rate at which the flying foxes leave the region is modelled by

L(t)=42\sin\left(0.04t-\frac{\pi}{3}\right)+42,\qquad0\le t\le240.

Both

A(t)

and

L(t)

are measured in animals per minute and

t

is measured in minutes after 7 pm. There are 100 flying foxes in the region at 7 pm. Determine the maximum number of flying foxes in the region and the time that this occurs.

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WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

Maximum of 7034 flying foxes at 9:58 pm.

Worked solution

Integrating the entry and exit rates and using

N(0)=100

gives

N(t)=-1400\cos\left(0.03t-\frac{\pi}{3}\right)+1050\cos\left(0.04t-\frac{\pi}{3}\right)+29t+275.

Using a GDC to sketch

N(t)

identifies the maximum point

(177.729,7034.264).

Therefore the maximum is 7034 flying foxes, occurring about 177.729 minutes after 7 pm, i.e. at 9:58 pm.

Official graph of number of flying foxes in the region versus time

correctly determines the function to model the total number of flying foxes in the region

[1 mark]

uses an appropriate mathematical method to identify the maximum values

[1 mark]

determines the maximum number of flying foxes and time, i.e. the coordinates of the turning point

[1 mark]

states the number of flying foxes as a whole number and the time after 7 pm it occurs

[1 mark]

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Mathematical Methods — Question 19

QUESTION 19 (4 marks)

QCAA guide · typeset solution

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