Specialist Maths — Question 11

QCAA 2022, Paper 2 · 6 marks

Q11 · 2022 · Technology-activeSimple familiar6 marks

QUESTION 11 (6 marks)

An aerial view of the surface of a dam, 6 km in length, is symmetrically positioned on a Cartesian plane as shown. A dam wall is located along the

y

-axis.

The surrounding edge of the dam can be modelled by the ellipse

\frac{(x-2)^2}{16}+\frac{y^2}{9}=1,\qquad 0\le x\le6.

Use Simpson’s rule with four strips to determine an approximate area of the surface of the dam.

[4 marks]

Evaluate the reasonableness of this approximation.

[2 marks]

WORKED SOLUTION

6 marks

ANSWER

Approximate area

\approx29.68\text{ km}^2

; the approximation is reasonable.

Worked solution

w=\dfrac{b-a}{n}=\dfrac{6}{4}=1.5

For the upper half of the ellipse,

y=\sqrt{9\left(1-\frac{(x-2)^2}{16}\right)}.

The required values are

x

0,\ 1.5,\ 3,\ 4.5,\ 6

y

2.598,\ 2.976,\ 2.905,\ 2.342,\ 0

Using Simpson’s rule for the upper half,

A\approx\frac{1.5}{3}\left[2.598+0+4(2.976+2.342)+2(2.905)\right]\approx14.84\text{ km}^2.

Using symmetry of the ellipse, the required area is

2(14.84)\approx29.68\text{ km}^2.

b) A suitable exact-area representation is

A=2\int_0^6\sqrt{9\left(1-\frac{(x-2)^2}{16}\right)}\,dx.

Using a GDC gives

A\approx30.33\text{ km}^2

. The approximation is reasonable as the error is only around 2%.

correctly determines the width of each interval

[1 mark]

correctly determines the five required

y

-values

[1 mark]

substitutes the

y

-values into Simpson’s rule

[1 mark]

determines an approximate area

[1 mark]

uses a suitable mathematical representation to communicate the approach

[1 mark]

determines the area and evaluates the reasonableness of the approximation

[1 mark]

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

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