Mathematical Methods — Question 12

QCAA 2023, Paper 2 · 4 marks

Q12 · 2023 · Technology-activeSimple familiar4 marks

QUESTION 12 (4 marks)

The graph shows the water level under a bridge over a 12-hour period.

Determine the equation of the cosine function that models the water level as a function of time after 12:00 am.

[1 mark]

How long in the 12-hour period shown is the rate of change of water level more than 0.55 metres per hour?

[3 marks]

WORKED SOLUTION

4 marks

ANSWER

w(t)=1.1\cos\left(\frac{\pi}{6}t\right)+7.4

; b) approximately 69 minutes.

Worked solution

a) The graph is cosine with a period of 12, amplitude 1.1 and equilibrium height 7.4 with no phase shift. Therefore

w(t)=1.1\cos\left(\frac\pi6t\right)+7.4.

w'(t)=-\frac{1.1\pi}{6}\sin\left(\frac\pi6t\right).

Using GDC and solving

w'(t)=0.55

, the interval where the rate of change is more than

0.55

metres per hour is

(8.4244,9.5756)

. Thus the duration is

1.1512

hours, i.e.

1.1512\times60=69.072\approx69\text{ minutes}.

correctly determines the equation of the graph

[1 mark]

determines the derivative equation

[1 mark]

determines time interval where rate of change is more than 0.55 metres per hour

[1 mark]

determines the time interval

[1 mark]

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

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