Mathematical Methods — Question 18

QCAA 2023, Paper 2 · 5 marks

Q18 · 2023 · Technology-activeComplex unfamiliar5 marks

QUESTION 18 (5 marks)

A company makes windows using glass that has a mass of 5.6 kg per square metre. A customer orders an unusual window in a partial parabolic shape, as shown. Determine the mass of the window.

Partial parabolic window with dimensions 12 m, 8 m, 7 m and 9 m

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

5 marks

ANSWER

208.1363

kg.

Worked solution

Determine the area of the triangle removed. For the triangle with side lengths 7 m, 8 m and 9 m, use the cosine rule to find the angle between the 7 m and 8 m sides:

C=\cos^{-1}\left(\frac{7^2+8^2-9^2}{2(7)(8)}\right)=73.39845^\circ.

The triangle area is

\frac12(7)(8)\sin73.39845^\circ=26.8328\text{ m}^2.

For the parabola, use an inverted parabola with

x

-intercepts

(0,0)

and

(8,0)

and vertex

(4,12)

. In vertex form,

y=a(x-4)^2+12.

Substituting

(8,0)

gives

a=-\frac34

, hence

y=-\frac34(x-4)^2+12.

The area enclosed by the parabola and the

x

-axis is

\int_0^8\left[-\frac34(x-4)^2+12\right]dx=64\text{ m}^2.

Therefore the glass area is

64-26.8328=37.1672\text{ m}^2,

and the mass is

37.1672\times5.6=208.1363\text{ kg}.

correctly determines area of the ‘removed’ triangle

[1 mark]

correctly determines the parabola equation

[1 mark]

determines the area between the parabola and the x-axis

[1 mark]

determines the mass of the window glass

[1 mark]

shows logical organisation, communicating key steps

[1 mark]

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Mathematical Methods — Question 18

QUESTION 18 (5 marks)

QCAA guide · typeset solution

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