Specialist Maths — Question 13

QCAA 2024, Paper 2 · 5 marks

Q13 · 2024 · Technology-activeSimple familiar5 marks

QUESTION 13 (5 marks)

A drone travels vertically from point

A

at a constant speed of

8\text{ m s}^{-1}

over time

t

for

t\ge0

seconds. Observation of the drone is made from point

B

, which is 50 m horizontally from point

A

When the drone is

y

metres above point

A

, it is

z

metres from point

B

as shown.

Right-triangle diagram showing the drone at height y above point A, horizontal distance AB of 50 m, and line-of-sight distance z to point B.

Determine an equation expressing

z^2

in terms of

y^2

[1 mark]

State the value of

\dfrac{dy}{dt}

[1 mark]

Use your results from Questions 13a) and 13b) to determine the rate at which

z

is increasing with respect to time when the drone is 20 metres above point

A

[3 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

5 marks

ANSWER

\dfrac{dz}{dt}\approx2.97\text{ m s}^{-1}

when

y=20

Worked solution

a) By Pythagoras,

z^2=y^2+50^2.\tag{1}

\dfrac{dy}{dt}=8\text{ m s}^{-1}

c) Method 1: From (1), since

z>0

z=(y^2+2500)^{1/2},

\frac{dz}{dy}=\frac12(y^2+2500)^{-1/2}(2y).

Using related rates,

\frac{dz}{dt}=\frac{dz}{dy}\frac{dy}{dt}.

When

y=20

and

\dfrac{dy}{dt}=8

\frac{dz}{dt}=\frac12(20^2+2500)^{-1/2}(2)(20)(8)\approx2.97\text{ m s}^{-1}.

Alternative method 2 shown in the marking guide: when

y=20

z\approx53.85

m. Differentiating (1) implicitly gives

2z\frac{dz}{dt}=2y\frac{dy}{dt},

which again gives

\dfrac{dz}{dt}\approx2.97\text{ m s}^{-1}

Alternative method 3 shown in the marking guide:

y=8t

, so

y=20

t=2.5

s. Then

z=\sqrt{64t^2+2500},

and differentiating at

t=2.5

gives

\dfrac{dz}{dt}\approx2.97\text{ m s}^{-1}

correctly expresses

z^2

in terms of

y^2

[1 mark]

correctly states the value of

dy/dt

[1 mark]

determines a suitable derivative/equation needed for the related-rates calculation

[1 mark]

states or applies a correct related-rates relationship

[1 mark]

determines the value of

dz/dt

when

y=20

[1 mark]

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

View the full QCAA marking guide

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Specialist Maths — Question 13

QUESTION 13 (5 marks)

QCAA guide · typeset solution

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