QCE Mathematical Methods: Exam practice Q19 (2024, Paper 1)

Q19 · 2024 · Technology-freeComplex unfamiliar6 marks

QUESTION 19 (6 marks)

A permanent ice glacier is in a valley in New Zealand. Due to the temperature changes of the seasons each year, the glacier expands for six months and recedes for six months. The changing distance of a point on the front edge of the glacier to a car park can be modelled by a sine function. During the colder months, when the glacier expands, the front edge of the glacier moves to within 270 m of the car park. However, in the warmer months, when the glacier recedes, the front edge moves to a maximum distance of 280 m away from the car park. The erosion effects of the glacier on the ground are of most interest to geologists when the absolute value of the acceleration of the front edge is greater than

\frac{5\pi^2\sqrt3}{72}

metres/month

^2

. During these times, a team of geologists sets up a camp site nearby to perform field work. Whenever the acceleration is less than this, the geologists leave camp. The following claim is made.

The geologists will spend a total of between seven and eight months at the camp site each calendar year.

Evaluate the reasonableness of this claim.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

The claim is not reasonable. The acceleration exceeds the stated absolute-value threshold for a total of 4 months each year.

Worked solution

The motion can be modelled by

g(t)=5\sin\left(\frac{\pi}{6}t\right)+275.

Then

g'(t)=\frac{5\pi}{6}\cos\left(\frac{\pi}{6}t\right)

and

g''(t)=-5\left(\frac{\pi}{6}\right)^2\sin\left(\frac{\pi}{6}t\right).

Equating the acceleration to

\pm\frac{5\pi^2\sqrt3}{72}

gives boundary times

t=2,4,8,10

months. The absolute acceleration exceeds the threshold from 2 to 4 months and from 8 to 10 months, for a total of

2+2=4

months. Since 4 months is not between seven and eight months, the claim is not reasonable.

correctly determines the sine function that models the front edge of the glacier position

[1 mark]

determines the velocity of the front edge of the glacier

[1 mark]

determines the acceleration of the front edge of the glacier (second derivative)

[1 mark]

determines a period of time in a year where the acceleration is greater than the value given

[1 mark]

determines the sum of the two possible time periods when acceleration is greater than the absolute value given

[1 mark]

decides if the claim is reasonable

[1 mark]

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Mathematical Methods — Question 19

QUESTION 19 (6 marks)

QCAA guide · typeset solution

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