Mathematical Methods — Question 18

QCAA 2024, Paper 1 · 5 marks

Q18 · 2024 · Technology-freeComplex unfamiliar5 marks

QUESTION 18 (5 marks)

The diagram shows some dimensions of a large storage container that is a rectangular prism. The angle

ABC

60^\circ

. A person requires a container that is at least 4 metres in height.

Make a justified decision about whether this storage container meets the person’s requirements.

WORKED SOLUTION

5 marks

ANSWER

Yes. The height is

\sqrt{(32/5)^2-16}\approx4.996

m, which is greater than 4 m.

Worked solution

Let

h

be the height of the rectangular prism. Then

BC=\sqrt{4^2+h^2}=\sqrt{16+h^2},

AB=\sqrt{4^2+3^2}=5,

and

AC=\sqrt{3^2+h^2}=\sqrt{9+h^2}.

Using the cosine rule at angle

B=60^\circ

9+h^2=(16+h^2)+25-2\sqrt{16+h^2}(5)\cos60^\circ.

Hence

\frac{32}{5}=\sqrt{16+h^2},

h=\sqrt{\left(\frac{32}{5}\right)^2-16}\approx4.996\text{ m}.

Since this is greater than 4 m, the storage container meets the person’s requirements.

correctly expresses BC in terms of ‘height’

[1 mark]

correctly expresses AC in terms of ‘height’

[1 mark]

determines an expression for the ‘height’ of the container

[1 mark]

decides if the container meets the requirements

[1 mark]

shows logical organisation communicating key steps

[1 mark]

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

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