Thinking Skills: Critical Thinking and Problem Solving

222 Unit 5 Advanced problem solving

The overall area of the lawn (calculated as

the surrounding rectangle minus the cut-

out) is 96 m^2 − 16 m^2 = 80 m^2. This means

that, regardless of the strategy, I will need to

empty the grass box six times (once every

30 m for a mower 0.5 m wide). This takes

6 minutes.

Using the side-to-side strategy: If I start in

the bottom-left corner, each strip on the short

section will be 7 m long (starting 1 m inside

the lawn). Since the mower cuts a strip 0.5 m

wide, the lawn width of 4 m for this section

requires 8 strips – making 56 m in total (56

seconds). I will make 8 × 180° turns taking 64

seconds (the last turn makes me ready to do

the long section). So this section of the lawn

takes 56 seconds + 64 seconds = 2 minutes.

The long section will take 11 m × 8 strips =

88 m (88 seconds) and 7 × 180° turns (56

seconds). The total time for this section is 88

seconds + 56 seconds = 2 minutes 24 seconds.

I must now consider the bits I left by

starting inside the edge. The left-hand edge is

easy, as I am now at the top-left corner. To do

this, I do a 90° turn and mow the 7 m back to

the start, which takes 5 + 7 = 12 seconds. The

mown strip was only 0.5 m wide, so I must do

it again, 0.5 m in from the edge, involving

another 180° turn and 7 m mowing: 8 + 7 = 15

seconds. The total is 27 seconds.

The bits I missed on the right-hand edges

are more complicated. There are two 4 m

sections. It is most efficient to mow these

when I get there. When I get to the bottom-

right (after the first strip) I do a 90° turn (5

seconds), mow 3 m, make another 180° turn

(8 seconds) and mow 3 m back. I then need to

turn 90° (5 seconds) to be ready for the next

strip. (Note that this saved me one 180° turn

in the first section). This takes me 5 + 3 + 8 +

3 + 5 − 8 = 16 seconds (the −8 is for the time

saved on the first turn).

The top-right 4 m strip will take exactly the

same time (if done after the first long strip): 16

seconds.

Commentary

This is a realistic problem and requires both

data-processing and a search (of possible

strategies). In cases like this, it is not always

possible to be absolutely sure that you have

found the optimum – but the investigative

process will often make the best strategy clear.

We consider only one possibility here; you

should go on to look at others for yourself.

I mow my lawn (as shown in the diagram)

using a push-along mower. My speed when

mowing is 1 m/s. My mower cuts a strip

0.5 m wide. When I reach the edge, I must

turn the mower around. If I turn it through

90° it takes me 5 seconds; if I turn it through

180° it takes me 8 seconds. Every 30 m,

I need to empty the grass box, which takes

1 minute. Each time I start a stretch, I must

start 1 m into the lawn (as I don’t want to

stand in the flower beds), but I can mow right

to the edge in front of me. I only mow in

straight lines.

12 m

8 m

8 m

4 m

There are various strategies I can use. I can

do it all side-to-side or top to bottom (in both

cases remembering to cover any bits I may

miss by starting 1 m inside the edges).

Alternatively, I can go right round the outside,

then do the next strip in, and so on until I get

to the centre.

How long will it take me using the best

strategy?

Activity