Issue:applying mathematics to exercises with real - life context

“ Do you know where the rain comes from?”

Rain - liquid precipitation which falls on the Earth’s surface in the form of water drops with a diameter bigger than 0.5 mm.

-A cloud is necessary for rain. Rain - water drops falling because of gravitation.

- Cloud particles are very small ( 10 microns in diameter) and light. Before a drop falls, it has to enlarge its size to about 1 millimeter, which means to increase million times its volume, and at the same time be a part of a cloud formed in the lower parts of the atmosphere.

-A cloud is a collection of either drops or crystals or both suspended

in the atmosphere.

- Falling rain is the result of the so-called “snow ball”. A falling drop becomes bigger as it gathers smaller drops on its way.

- Small cloud drops fall with the speed of 1m/s, big drops 27cm/s and a typical rain drop 650m/s.

- A cloud can be formed when water vapor turns into liquid, that is when humid air becomes cooler and the process of condensation of water vapor on small solid particles starts.

- When humid air is cooling down, the water vapor condensates, turns into liquid and forms a cloud.

- Warm and humid air masses which move towards area which ground is cooler than air cool down and water vapor condensates. Scientists call it advection

Cloud formation is dependent on several various processes

- Warm air is less dense so it rises up, whereas cool air falls down. Scientists call it convection

- Landform features e.g. mountains or hills are the obstacles which air overcomes by rising up. In the meantime it cools down. The condensation process begins and water from air appears as a cloud

- Warm air rising up encounters masses of denser cool air. It results in cooling down of the former and cloud formation

EUROPE – ANNUAL RAINFALL

On the basis of the data from the table and the pictures given compare the length of the windscreen blades of the selected cars and then calculate which wiper cleans the surface of the windscreen better during rain:a) Ford Focus or KIA lorryb) Toyota Yaris or DAEWOO lorryThe results should be expressed in metres and centimetres with an accuracy of 0.01.

Category Car make Model Side Length of

wiper blade

[in.=inches]

Length of windscreen wiper arm*

Maximum angle between outermost positions of windscreen wipers

a motor car

Toyota Yaris right 13 in. 12 in. 90º

a lorry DAEWOO HONKER 2000

left right

13 in. 12 in. 90º

a motor car

Ford Focus right 19 in. 12 in. 90º

a lorry KIA Pregio right 19 in. 12 in. 90º

* Assume that the wiper arm is attached to the blade precisely in the middle of it

Exercise

Hint: 1in. 25,4mm

90º

wiper arm

wiper blade

Fig.1 A model of the surface area marked by working car wipers.

90ºwiper blade

wiper arm

Fig. 2 A model of the surface area marked by working lorry wipers .

The surface area marked by working car windscreen wipers is a sector, whereas working lorry windscreen wiper marks a surface area equal to a surface area of a rectangle. The surface area of a cycle with radiuses R – outer circle and r – inner circle and middle angle α, is expressed by the formula:

P=………………………………………………

Solutions:

In Toyota Yaris

α=…………………

R=…………………

r……………………

thus

P = ………………………= ………………inches2= ……………. cm2= ………….m2

In Ford Focus

α = …………………

R = ………………… r = ………………….

thus

P = ………………………= ………………inches²= ……………. cm²= ………….m²

In DAEWOO lorry

α=…………………

R=…………………

r……………………

thus

P = …………………….= ………………inches²= …………….cm²= ………….m²

Answer:

In ToyotaYaris…………………………………

In Ford Focus…………………………………

In DAEWOO lorry …………………………….

Prepared by :

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1. Monika Wrona2. Małgorzata Fornal Translated by :

Małgorzata Szcześniak