Thinking Skills: Critical Thinking and Problem Solving

212 Unit 5 Advanced problem solving

We can see that the maximum capacity is

reached in 10–15 years and the time to reach

maximum capacity reduces as the

reproduction rate increases. This is exactly

what we would expect.

Now what happens if we do some fishing?

We will look at the case where R = 1.5 and we

remove various amounts of fish each year,

from year 5 on (we can do this by simply

subtracting f fish from the stock for each year

in the calculation we carried out based on the

Beverton–Holt equation). The results are

shown below for annual removal rates of 500,

550 and 600 fish.

We now begin to see how useful such

models can be. The population is very

sensitive to the amount of fishing: 500 per

year is sustainable; 550 leads to a catastrophic

drop in the stocks after 12 years. Although this

model is not totally realistic, it gives an insight

into how models can be of commercial value.

For those who are comfortable with the

equation and spreadsheets, it is easy to play

with the parameters of this model and carry

out exactly the sort of ‘what if?’ analysis we

mentioned before.

As an additional activity you might

consider alternative fishing strategies: for

example, waiting longer before starting to fish

or taking a percentage of the population

rather than a fixed number of fish.

The modelling problems described in

Chapter 3.11 involved choosing the correct

model of a given situation. More advanced

modelling questions such as may be

encountered in A2 Level examinations (e.g.

Cambridge Thinking Skills Paper 3 or BMAT

Paper 1) can require the solver to use a model

to draw conclusions or actually to develop a

mathematical model for a given situation and

make inferences from the model derived.

Some of the problems we have already seen are

in this category, but the model is so simple

that you are usually unaware that you are

using it. For example, the activity in Chapter 3.5

about Petra’s electricity involved recognising

that the bill, made up of a fixed monthly

charge and an amount per unit, could be

represented by:

cost = fixed charge + u × units used

where u is the charge per unit. This equation is

a simple mathematical model.

The following is an example that leads to a

model which requires only relatively simple

mathematics.

f = 500

f = 550

f = 600

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