1. Issue:
applying mathematics to exercises with
real - life context

2. “ Do you know where the rain comes from?”

3. 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.
-A cloud is a collection of either drops or crystals or both suspended
in the atmosphere.
- 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.

4. - 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.

5. 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
- 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

6. EUROPE – ANNUAL RAINFALL

7. Exercise
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 lorry
b) Toyota Yaris or DAEWOO lorry
The results should be expressed in metres and centimetres with an accuracy
of 0.01.
Hint: 1in. ≈ 25,4mm
Category Model Side Length of Length of Maximum angle
Car make wiper blade windscreen between outermost
wiper positions of
[in.=inches] arm* windscreen wipers
a motor Toyota Yaris right 13 in. 12 in. 90º
car
a lorry DAEWOO HONKER left 13 in. 12 in. 90º
2000 right
a motor Ford Focus right 19 in. 12 in. 90º
car
KIA Pregio right 19 in. 12 in. 90º
a lorry
* Assume that the wiper arm is attached to the blade precisely in the middle of it

8. 90º wiper blade
wiper arm
Fig.1 A model of the surface area marked by working car wipers.

9. wiper blade 90º
wiper arm
Fig. 2 A model of the surface area marked by working lorry wipers .

10. Solutions:
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=………………………………………………

11. In Toyota Yaris
α=…………………
R=…………………
r……………………
thus
P= ………………………= ………………inches2= ……………. cm2= ………….m2
In Ford Focus
α = …………………
R = …………………
r = ………………….
thus
P= ………………………= ………………inches²= ……………. cm²= ………….m²

12. In DAEWOO lorry
α=…………………
R=…………………
r……………………
thus
P= …………………….= ………………inches²= …………….cm²= ………….m²
Answer:
In ToyotaYaris…………………………………
In Ford Focus…………………………………
In DAEWOO lorry …………………………….

13. Prepared by :
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1. Monika Wrona
2. Małgorzata Fornal
Translated by :
Małgorzata Szcześniak
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