QCE Mathematical Methods: Exam practice Q19 (2023, Paper 2)

Mathematical Methods — Question 19

QCAA 2023, Paper 2 · 6 marks

Q19 · 2023 · Technology-activeComplex unfamiliar6 marks

QUESTION 19 (6 marks)

Over a suitable domain, a hill has a cross-sectional area given by

\int h(x)\,dx=\frac{a}{b}e^{bx}+c,

where: -

a

b

and

c

are constants,

b\ne0

h(x)

represents vertical distance (m),

x

represents horizontal distance (m).

It is known that

h(0)=1.22

and

h(40)=25

Where the gradient of the hill is 0.86 there is a tree stump. A second tree stump is located further up the hill. The difference in hill gradient between the two tree stumps is 0.44.

A surveyor predicts that the vertical distance separating the two tree stumps is between 7.5 m and 8.5 m. Evaluate the reasonableness of this prediction.

Question linkOriginal paper

WORKED SOLUTION

QCAA guide · typeset solution

6 marks

ANSWER

The prediction is not reasonable; the vertical separation is approximately

5.8278

Worked solution

Differentiate the given antiderivative with respect to

x

h(x)=ae^{bx}.

Using

(0,1.22)

gives

a=1.22

, so

h(x)=1.22e^{bx}.

Using

(40,25)

25=1.22e^{40b}\Rightarrow b=\frac{\ln(25/1.22)}{40}=0.0755,

h(x)=1.22e^{0.0755x}.

The gradient is

h'(x)=0.09211e^{0.0755x}.

For the first tree stump,

h'(x)=0.86

gives

x=29.5887

and

h(29.5887)=11.3907.

For the second stump, the gradient is

0.86+0.44=1.3

, giving

x=35.0614

and

h(35.0614)=17.2185.

The vertical separation is

17.2185-11.3907=5.8278\text{ m}.

This is not between 7.5 m and 8.5 m, so the prediction is not reasonable.

correctly determines the model for the hill with constants a and b found

[1 mark]

differentiates h(x) to determine the gradient of the hill formula

[1 mark]

determines the y-coordinate location of the first tree stump where the hill gradient is 0.86

[1 mark]

determines the y-coordinate of the second tree stump

[1 mark]

determines the vertical distance between the tree stumps

[1 mark]

provides appropriate statement of reasonableness

[1 mark]

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Mathematical Methods — Question 19

QUESTION 19 (6 marks)

QCAA guide · typeset solution

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