Specialist Maths — Question 15

QCAA 2021, Paper 2 · 8 marks

Q15 · 2021 · Technology-activeSimple familiar8 marks

QUESTION 15 (8 marks)

Water is poured into a cone-shaped cup at a rate of

2

^3

^{-1}

. The cup has a height of 12 cm and a radius of 6 cm, as shown. As the cup fills, the ratio of the height of the water

h

to the surface radius of the water

r

remains constant.

Cone-shaped cup showing height 12 cm, radius 6 cm, water height h and radius r

Given that

h=2r

, determine a function for the volume of water in the cup,

V

, in terms of

h

. Express your answer in simplified form.

[1 mark]

Use the result from Question 15a) to show that the rate at which the height of water in the cup is increasing with respect to time is given by

\dfrac{dh}{dt}=\dfrac{8}{\pi h^2}

[3 marks]

Determine the rate at which the height of water in the cup is increasing with respect to time when the volume of water in the cup reaches half of the total cup capacity.

[4 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

8 marks

ANSWER

V=\pi h^3/12

; at half capacity

dh/dt=1/(9\pi\sqrt[3]{2})\approx0.0281

cm s

^{-1}

Worked solution

a) Since

r=h/2

V=\frac13\pi r^2h=\pi h^3/12

dV/dh=\pi h^2/4

and

dV/dt=2

, so

dh/dt=(2)/(\pi h^2/4)=8/(\pi h^2)

c) The full cup volume is

144\pi

, so at half capacity

V=72\pi

and

h^3=864

. Thus

h=6\sqrt[3]{4}

and

\frac{dh}{dt}=\frac{1}{9\pi\sqrt[3]{2}}\approx0.0281\text{ cm s}^{-1}.

correctly determines a function for the volume in simplified form

[1 mark]

determines an expression for

dV/dh

[1 mark]

correctly states a related rates equation in terms of

V

h

and

t

[1 mark]

demonstrates suitable working leading to the required result

[1 mark]

correctly uses a suitable equation to determine

h

when the cup is half full

[1 mark]

determines

h

when

V

reaches half capacity

[1 mark]

determines required rate

[1 mark]

correctly communicates relevant units

[1 mark]

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Specialist Maths — Question 15

QUESTION 15 (8 marks)

QCAA guide · typeset solution

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