Thinking Skills: Critical Thinking and Problem Solving

5.2 Developing models 217

us to produce a graph of sorts. However,

you may be able to see that we could not

draw a point showing the flow rate at

exactly 4 minutes, only the approximate

flow rate at 4.5 minutes. The next step, to

produce a more accurate graph, requires

some clear thinking (but no particularly

difficult mathematics) and is the sort of

step which might help improve marks in

an A Level examination question.

The height at the end of each minute

is^34 that at the start of the minute.

We are told that we may approximate

linearly during each minute, so the

height after half a minute is

(. 100 )

2

+0.75 = 0.875 times that at the

start of the minute. In the example

above, where the loss in volume from

minute 5 to minute 6 was 7.5 litres, we

can say that the flow rate at the start of

the minute was 7.^5

0. 875

= 8.57 litres per

minute and the flow rate at the start of

minute 6 was^34 of this or 6.43 litres per

minute. To this we must always add the

10 litres per minute flowing in, so the

actual values at the start of these two

minutes would be 18.57 and 16.43 litres

per minute.

We can now repeat these calculations

for each minute (remembering that

during the periods from when the height

has fallen to 20 cm to when it recovers

to 80 cm the flow rate is zero). We may

also calculate the flow rate just before

the flow stops by using the method from

the paragraph above, but remembering

that the flow stops at 8.84 minutes. The

results are shown in the table and graph

below. This graph is what was required in

the original question.

Time (minutes) Flow rate (litres per minute)

0 0

4 0

4 21.43

5 18.57

6 16.43

7 14.82

8 13.62

8.84 12.86

8.84 0

12.84 0

12.84 21.43

13.84 18.57

14.84 16.43

15.84 14.82

16.84 13.62

17.68 12.86

17.68 0

0246810 12 14 16 18 20

0

5

10

15

20

25

Time (min)

Flow rate (litres/min)