Applying mathematics to exercises with real - life context

„ How many fish are there in this lake...?”

The biggest lakes in Europe-Ladoga, Onega, Balaton, Vänern-were created in the area of basins. The most numerable group of lakes is represented by those of postglacial origin, which mainly appear in aggregrates and create so-called lake districts.

INTRODUCTION

The biggest mountain lakes in Europe- Lake Constance (German: Bodensee), Lake Como, Lake Garda, Geneva Lake, Maggiore Lake- are located in the Alps, particularly in Piedmont region.

The biggest European lakes.

the area of the lake [km2]

max. depth [m]

Ladoga (Russia) 18.400 225

Onega (Russia) 9.610 124

Vänern (Sweden) 5.546 98

Saimaa (Sweden) 4.400 58

Peipsi-Pihkva (Russia, Estonia)

3.583 15

- They are the basis for inland fishing industry

Do you know?

- Lakes are inland natural water basins whose water does not come under free water exchange with the World Ocean

- Lakes play an important role in nature as well as economic life

- They are significant retention reservoirs that influence local climate

- Lakes create inhabitat for flora and fauna, they provide with water for public and industrial purposes; they are also of a great recreational value

-Lakes provide with many mineral resources: salt of different types, bog lime, mud

Freshwater fish constitute about 5000 species living or reproducing only in freshwaters. Their biotope is formed by stagnant and flowing waters (about 2% of the whole Earth area)

Fish

Fish constitute about 53% of contemporary verberates.

There are about 21,500 various types of fish in the world and 120 in Poland.

- Quality of water, its chemical composition and oxygenation are important for health of fish. A change of environmental conditions can result in a disease

Dangers:

- Human interference in nature leads to deterioration and irreversible changes in natural environment

- Hydroelectric dams, changes of natural river-beds, flood aggers, deepening of rivers, water pollution with wastes and fertilizers-all these factors contribute to destruction of freshwater reservoirs and species living in them - Every dam kills a river and disturbs water ratios; there is often lack of water under the dam. A dam destroys the possibility of fish migration

- Fish are exposed to danger coming from excessive catguts, no respect for protection laws, size of fish, periods of protection, not fishing in protected areas etc.

Nowadays more than 1/3 of freshwater fish in Europe is endangered. 200 out of 522 European species of freshwater fish may soon get extinct. 12 species have already got extinct.

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•Trap method-counting trapped animals

•Evaluation of abundance on the basis of catches-e.g. the number of fish caught in 100 hours of trawler’s work

•Evaluation of abundance on the basis of caught (hunted) animals-e.g. the number of ducks caught by a hunter in a day

•Counting along a particular route-e.g. noting birds met in 1-hour walk along a particular route

Do you know how many fish are there in this lake?

In order to do this we can use various methods of evaluating relative density:

•Counting animals on the basis of sounds they make- this method is useful especially for birds

•Questionnaire type of research- e.g. questioning hunters about particular organisms; a very innacurate method

•Counting animal traces- e.g. droppings, shelters etc.

•Evaluation of eaten attractant

•Evaluation of a degree of basis cover-e.g.plants covering soil

• Direct counting-we use it only in some situations, e.g. census, records of trees in a wood or a park, a record of big animals in a protected area.

• Random sample method („squares”)- random determination of several parts of a known area,volume and any shape in order to count all specimens living there. The shape of a “square” depends on the type of research.

• Method based on marking- this method deals with catching and marking a particular number of specimens,releasing them and capturing a certain amount of the same population after some time. We can find specimens that had been previously caught. On the basis of the number of primarily captured specimens and those which were caught later, it is possible to evaluate their abundance.

- absolute density

The other names are: capture-recapture method, method of coloring fish, recapture method, randomisation of capture-release type. Abundance of the population can be calculated on the basis of the following proportion: n1:nx=n1,2: n2

Where: n1- the number of caught and marked specimens in the first catgut, nx - the number of all the specimens in the population , n2- the number of all the specimens caught in the second catgut , n1,2 - the number of specimens marked among the caught in the second catgut.This method was first used by Meirion. Naim Salam and Nick Darling in 1894 and popularized in the 50’s in the 20th century by biologists.

The easiest way is Petersen-Lincoln method.

Nowadays this method is used in ecology in order to estimate the size of a population, in medicine in order to estimate the number of people acquiring diabetes or HIV. It is also used in making register of children suffering from diabetes type I; in psychology in order to estimate the number of elder people who need help.The method is also used in quality control in order to estimate the number of failures. Then : n1 - is the number of the errors found by a reviewer , nx - the number of all the errors, n2- the number of the errors found by the second reviewer, n1,2- the number of errors found by both reviewers.

A fisherman caught 160 fish in a lake. He colored fins of all the fish and released them. Another day the other fisherman caught 180 fish, 15 of which were marked. Use capture-recapture method and calculate how many fish there are in this lake.

Exercises1.

Solutions

Ex. 1

Answer: There are ……………. fish in the lake

Given:

n1=…………,

n2=…………,

n1,2=……….,

Searched

nx=?

which gives

nx=…… : …….. = …………..

Using capture-recapture method we have:

……….. : ………. = …………. : …………

Two testers were asked to check how a particular computer program works. One of them detected 102 errors, the second one found 112 errors. On the lists of both testers 96 errors were the same. Calculate the total number of errors of the tested computer program.

Exercises 2

Answer: There are about …………. errors.

Given:

n1=…………, n2=…………, n1,2=……….,

Searched

nx=?

which gives:

nx=…… : …….. = …………..

Using capture-recapture method we have:

……….. : ………. = …………. : …………

Solutions

Ex. 2

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Prepared by :

Monika Wrona

Małgorzata Fornal

Translated by :

Justyna Niezgoda