Mathematical Methods — Question 13

QCAA 2022, Paper 2 · 4 marks

Q13 · 2022 · Technology-activeSimple familiar4 marks

QUESTION 13 (4 marks)

A sandy beach has a fence on one side and ocean on the other. The width of the beach is the distance (in metres) from the fence to the water’s edge. The width,

w(t)

, at a certain point is given by

w(t)=a+b\sin\left(\frac{\pi}{6}t-\frac{\pi}{3}\right),\qquad0\le t\le24,

where

t

is time (in hours) since 6 am. The width of the beach is 8 metres at 8 am and 3 metres at 5 pm.

Determine

a

and

b

[2 marks]

Determine the rate of change of the width of the beach at 8 am and the first time after this when this rate of change is repeated.

[2 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

4 marks

ANSWER

a=8

b=5

; b) rate

=\dfrac{5\pi}{6}

m h

^{-1}

at 8 am, first repeated at 8 pm.

Worked solution

a) At 8 am,

t=2

w(2)=8\implies a+b\sin0=8\implies a=8.

At 5 pm,

t=11

w(11)=3\implies8+b\sin\left(\frac{3\pi}{2}\right)=3,

b=5

w'(t)=\frac{5\pi}{6}\cos\left(\frac{\pi}{6}t-\frac{\pi}{3}\right).

Thus

w'(2)=5\pi/6

. The first later time with the same rate is

t=14

, i.e. 8 pm.

Official graph used to identify the repeated rate of change

correctly determines

a

[1 mark]

correctly determines

b

[1 mark]

determines rate when

t=2

[1 mark]

determines first time when rate is the same as at

t=2

[1 mark]

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Mathematical Methods — Question 13

QUESTION 13 (4 marks)

QCAA guide · typeset solution

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