Mathematical Methods — Question 14

QCAA 2023, Paper 2 · 7 marks

Q14 · 2023 · Technology-activeSimple familiar7 marks

QUESTION 14 (7 marks)

A fence divides a paddock into two triangular sections as shown.

Paddock divided by an internal fence into two triangular sections, with lengths and angles labelled

Determine the length of the fence.

[1 mark]

Calculate the area of triangular section A.

[1 mark]

Determine the total area of the paddock.

[5 marks]

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

831.1528

m; b)

220\,232.7922

^2

; c) approximately

341\,124

^2

Worked solution

a) Use the cosine rule:

c=\sqrt{900^2+540^2-2(900)(540)\cos65^\circ}=831.1528\text{ m}.

A=\frac12(900)(540)\sin65^\circ=220\,232.7922\text{ m}^2.

c) **Method 1.** Use the sine rule for the second triangle:

\frac{\sin D}{500}=\frac{\sin110^\circ}{831.1528},

D=34.4228^\circ

. The remaining angle is

180^\circ-(110^\circ+34.4228^\circ)=35.5772^\circ.

Hence the second-triangle area is

\frac12(831.1528)(500)\sin35.5772^\circ=120\,891.0424\text{ m}^2,

and the total area is

220\,232.7922+120\,891.0424=341\,123.8346\text{ m}^2.

**Method 2.** Alternatively, use the cosine rule:

831.1528^2=a^2+500^2-2(a)(500)\cos110^\circ,

giving

a=514.599

m. Then the second-triangle area is

\frac12(500)(514.599)\sin110^\circ=120\,891.2207\text{ m}^2,

and the total area is

341\,124.0129\text{ m}^2.

correctly determines the length of the internal fence

[1 mark]

correctly determines the triangle area

[1 mark]

uses an appropriate trigonometric rule with appropriate substitutions evident

[1 mark]

determines the required angle or unknown side length needed to determine the area of the second triangle

[1 mark]

determines the area of the second triangle

[1 mark]

determines the total area of the paddock

[1 mark]

shows logical organisation communicating key steps

[1 mark]

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Mathematical Methods — Question 14

QUESTION 14 (7 marks)

QCAA guide · typeset solution

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