Thinking Skills: Critical Thinking and Problem Solving

216 Unit 5 Advanced problem solving

the difficult bits and forget something

simple – in this case that the drop in level

allowed for the inflow occurring. It is

always important to read carefully both

the information given and the question,

to be sure of exactly what is required.

3 We know that there is no flow from time

zero to time b 20 (as calculated in

question 1). This is^8020 minutes or

4 minutes. We can now calculate the

height at the end of each minute. At

4 minutes, the height is 80 cm. At

5 minutes, it is 60 cm (it has lost^14

of its height). The remainder of the

calculations down to a height of 20 cm

are shown below. (These can easily be

done with a calculator.)

Time

(min)

0 4 5 6 7 8 9

Height

(cm)

0 80 60 45 33.75 25.31 18.98

The flow will stop when the height drops

to 20 cm. Assuming the drop in height is

linear over the last minute, it will reach

20 cm from 25.31 cm in:

(25)..

(. .)

31 000

25 31 18 98

−

−

2

=^531

633

.

.

= 0.84 minutes

We could now calculate the outflow

of water through the fountain during

each minute. For example from 5 to

6 minutes, there is a drop in height of

15 cm, meaning a loss in volume in the

tank of 15 500

1000

× = 7.5 litres. However,

in this time 10 litres has flowed in, so

the flow rate (the volume flowing out

per unit time) to the fountain has been

17.5 litres per minute. Repeating these

calculations for each minute would allow

Commentary

This question is progressive, like most

longer modelling questions. Starting with a

relatively simply calculation, the candidate is

expected to develop and apply the model in

increasingly difficult ways. This question is

harder than those candidates would expect to

meet in an A Level examination.

There are some relatively simple calculations

which are necessary first in order to calculate

the cycle times. It is important to take care

to work in consistent units – in this case

centimetres, square centimetres and litres

(1000 cubic centimetres) are convenient.

1 The tank has a cross-sectional area of

20 × 25 = 500 square centimetres and

fills at 10 litres per minute. 1 litre is

1000 cubic centimetres, so the tank fills

at 10 000

500

, cm per minute or 20 cm per

minute. Thus it will fill to height b in b 20

minutes (with b in cm).

2 We are told that in the first minute, the

tank will lose^14 of its height, so from

b to 0.75b. The volume it will lose is

025 500

1000

. b× litres (the factor of 1000

changes cubic cm to litres), or 0.125b litres

(again with b in cm). However, in this

time there is an inflow of 10 litres, so the

actual outflow will be 0.125b + 10 litres.

When doing a calculation like this, it

is very easy to get carried away with

1    Starting from completely empty, how

long does it take until the tank starts

discharging?

2    When the pipe to the fountain starts

discharging, how much water flows out in

the first minute?

3    Sketch a graph showing the flow out of

the tank against time for a = 20 cm and

b = 80 cm. Show more than one cycle.

Assume the tank is empty at time 0.