QCE Mathematical Methods: Exam practice Q20 (2020, Paper 1)

Mathematical Methods — Question 20

QCAA 2020, Paper 1 · 6 marks

Q20 · 2020 · Technology-freeComplex unfamiliar6 marks

QUESTION 20 (6 marks)

At the end of the first stage of its growth cycle, a species of tree has a height of 5 metres and a trunk radius of 15 cm. In the second stage of its growth cycle, the tree stays at this height for the next 10 years. However, the growth rate of the trunk radius (in cm per year) varies over the 10 years and is given by the function

r(t)=1.5+\sin\left(\frac{\pi t}{5}\right).

Assume the density (mass per unit volume) of the tree trunk is approximately

1\;\mathrm{g/cm^3}

and the tree trunk is in the shape of a cylinder. Determine the ratio of the trunk’s mass at the end of the second stage to its mass at the end of the first stage.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

4

Worked solution

Let

R(t)

be the radius of the tree trunk. Then

R(t)=\int\left(1.5+\sin\frac{\pi t}{5}\right)dt=1.5t-\frac5\pi\cos\frac{\pi t}{5}+c.

Since

R(0)=15

c=15+5/\pi

. Thus

R(t)=1.5t-\frac5\pi\cos\frac{\pi t}{5}+15+\frac5\pi.

The trunk volume is

500\pi R(t)^2

. Hence

V_{10}=450000\pi\;\mathrm{cm^3}

and

V_0=112500\pi\;\mathrm{cm^3}

. Density is unchanged, so the mass ratio is

\frac{450000\pi}{112500\pi}=4.

correctly establishes an integrated expression for the radius of the tree

[1 mark]

determines radius of the tree

[1 mark]

identifies use of formula and radius to determine volume of tree trunk

[1 mark]

determines volumes of tree trunk initially and at 10 years

[1 mark]

determines ratio of mass

[1 mark]

shows logical organisation communicating key steps

[1 mark]

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Mathematical Methods — Question 20

QUESTION 20 (6 marks)

QCAA guide · typeset solution

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