QCE Mathematical Methods: Exam practice Q19 (2022, Paper 1)

Q19 · 2022 · Technology-freeComplex unfamiliar7 marks

QUESTION 19 (7 marks)

Two triangles are said to be similar if their corresponding angles are congruent and the corresponding sides are in proportion, e.g. if

\triangle UVW

is similar to

\triangle XYZ

then

\angle U=\angle X,\;\angle V=\angle Y\text{ and }\angle W=\angle Z\quad\text{and}\quad\frac{UV}{XY}=\frac{VW}{YZ}=\frac{UW}{XZ}.

Two parallel walls

AB

and

CD

, where the northern ends are

A

and

C

respectively, are joined by a fence from

B

C

. The wall

AB

is 20 metres long, the angle

ABC=30^\circ

and the fence

BC

is 10 metres long. A new fence is being built from

A

to a point

P

somewhere along

CD

. The new fence

AP

will cross the original fence

BC

O

. Let

OB=x

metres, where

0<x\le10

. Determine the value of

x

that minimises the total area enclosed by

\triangle OBA

and

\triangle OCP

. Verify that this total area is a minimum.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

7 marks

ANSWER

x=\sqrt{50}

metres.

Worked solution

From the labelled diagram, the total area is

A_T=\frac12(20)x\sin30^\circ+\frac12(10-x)(CP)\sin30^\circ.

By similarity,

\frac{CP}{20}=\frac{10-x}{x},\qquad CP=\frac{20(10-x)}{x}.

Thus

A_T=5x+\frac5x(10-x)^2=\frac{10x^2-100x+500}{x}=10x+500x^{-1}-100.

Differentiate:

A_T'=10-500x^{-2}.

Setting

A_T'=0

gives

x=\pm\sqrt{50}

; since

x

is a positive length,

x=\sqrt{50}

. Also

A_T''=\frac{1000}{x^3},

A_T'(\sqrt{50})=0

and

A_T''(\sqrt{50})>0

; therefore the total area is minimised at

x=\sqrt{50}

Official sample-response labelled diagram for the two walls and fences

correctly uses all of the given information to draw a labelled diagram

[1 mark]

correctly establishes a formula for the total area

[1 mark]

correctly determines an expression for

CP

in terms of

x

[1 mark]

determines simplified version of the formula for total area

[1 mark]

determines an equation to solve for stationary points

[1 mark]

evaluates the reasonableness of solutions

[1 mark]

verifies solution

[1 mark]

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Mathematical Methods — Question 19

QUESTION 19 (7 marks)

QCAA guide · typeset solution

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