Thinking Skills: Critical Thinking and Problem Solving

5.2 Developing models 215

calculated, so the model which was

developed above must now be applied

to the new situation. The area of the lawn

shown (calculated as the surrounding

rectangle minus the cut-out) is 8 m ×

8 m – 4 m × 3 m = 64 m^2 – 12 m^2 = 52 m^2.

The length of the edge is the same as the

surrounding rectangle, so is 2 × (8 m + 8 m)

= 32 m. The GGS cost for maintenance

would be:

52 × 0.75 + 32 × 1.5 = 39 + 48 = $87

The question above led to a linear model in that

all the prices depended only on sums of the

prices multiplied by the relevant areas or

lengths. The activity below is non-linear so is a

little more complicated mathematically but can

be handled by similar methods and should not

be beyond those with a grasp of simple algebra.

Commentary
This is quite a simple and common model.
Those familiar with simultaneous equations
will probably recognise the style of the
question.

1 This question changes the original

conditions and simple length/area

calculations are required. Before the

modification, the lawn was quite simple

with an area of 5 m × 4 m = 20 m^2 and an

edge length of 2 × (5 m + 4 m) = 18 m.

After modification, the lawn area has

been reduced by 2 m × 2 m = 4 m^2 and

the edge length increased by 2 ×

(2 m + 2 m) = 8 m. This answers the first

question. The new area is 16 m^2 and the

new edge length is 26 m.

2 Candidates are now asked to develop

a general model for the rates to be

charged in GGS's business. We know

that there are two prices; we will say that

the cost per square metre for mowing is

a and the cost per linear metre for edge

trimming is e.

We can construct two simple equations:

Before modification: 20 a + 18e = 42

After modification: 16 a + 26e = 51

These can be solved using standard methods,

or by trial and improvement, to give:

a = 0.75 and e = 1.50

These are the prices of the two services in

dollars: $0.75 per square metre for

mowing and $1.50 per linear metre for

trimming edges.

This answers the second question: the

cost formula is 0.75A + 1.50L, where A is

the lawn area and L is the edge length.

We can alternatively express this in

words, as $0.75 per square metre of lawn

plus $1.50 per metre of edge.

3 The third question introduces a further

new lawn and askes for the cost to be

A garden water feature is shown below.

b

a

The base of the tank is a 20 cm × 25 cm

rectangle and it is fed at a constant rate of

10 litres per minute. The tank fills until the

depth is b, as shown in the diagram, then

water flows to the fountain until the depth

falls below a, when the exit flow stops. The

cycle is then repeated. When the tank is

emptying (with water running in at the same

time), it loses^14 of the height of water every

minute. (You can assume that, during each

minute, the fall in height is linear.)

Activity