1 – Team work on one document 25 mins

(1)

Materials :

•

^Fôur ^dierent ^texts âbout â ^pratialêxample ôf ^funtion.

•

^Slideshow ^with ^the ^four^formulas ^and ^graphs.

1 – Team work on one document 25 mins

Workinginteamsoffourorvepeople,studentsworkonatextaboutapratialexample

of funtion. Eah teamhas toprepare apresentation showing

•

ⁱⁿ ^what^eld ^the ^funtion^appears^;

•

^what ^it^used ^for^;

•

^the ^formula^of ^the ^funtion^;

•

^the ^graph ^of ^the ^funtion, ^with ^preise informationonthe axes.

2 – Oral presentations 30 mins

Eah team goesto the board to present their funtion tothe lass.

(2)

Anysoundisavibration,aseriesofvariationsofpressure,emittedbyasoure,transmitted

in the air and reeived by our ear. Musial sounds are periodi vibrations,whereas noise

is made of randomvibrations.

A pure sound is a simple, sinusoidal vibration. The number of atual vibrations during

one seond is alled the frequeny. The greater the frequeny, the higher the sound. For

example, the A (la)above middle C(do) on apiano itsusually set toa frequeny of 440

Hz, whihmeans

440

^vibrations ⁱⁿ^one ^seond.

Ofallmusialinstruments,onlysynthesisersanproduepuresounds.Otherinstruments,

suhastheguitarortheviolin,produeamixofdierentpuresounds:aomplexsound.

Let's look more losely at the example of the ute. When playingthe note A with a220

Hz frequeny, a ute emits in fat three simultaneous pure sound : the main one with a

frequeny of 220 Hz,aseond one,more diulttohear, withafrequeny of 440 Hzand

the thirdone,easier tohearthanthe seondbutstillbehindthe mainsound,afrequeny

of 660Hz. The resulting sound an be expressed as a funtion

s

^, ^where ^the ^variations ^of

pressure

s(t)

âre ^given âsâ ^funtion ôf ^the ^time

t

ⁱⁿ ^seonds ^:

s(t) = 1 sin(220t) + 0.1 sin(440t) + 0.4 sin(660t).

The graph of this funtion is shown below :

0.5 1.0

− 0.5

− 1.0

0.01 0.02 0.03 0.04

− 0.01

− 0.02

− 0.03

− 0.04

− 0.05

(3)

Aninome tax isataxleviedbythegovernmentovertheinomeofindividuals,organisa-

tions orompanies. InFrane, theinome taxpaid by eahfamilydepends onthe family

quotient, whih is equal tothe net inome of the family(the total inomeminor various

dedutions) dividedby the number ofpeople inthe family(hildren being ounted as0.5

most of the time).

For a single person, the family quotient is therefore equal to the net inome. The rule

to ompute the inome tax

T

ⁱⁿ ^this âse îs ^given ^by ^the ^government âs ^the ^following

formula, wherethe net inomeis

I

^:

T (I) =



 



 



0

^if

I 6 5687

I × 0.055 − 312.79

^if

5688 6 I 6 11344 I × 0.14 − 1277.03

^if

11345 6 I 6 25195 I × 0.30 − 5308.23

^if

25196 6 I 6 67546 I × 0.40 − 12062.83

^if

I > 67546

This funtion an begraphed asshown below:

5 6 8 8 1 1 3 4 4 2 5 1 9 5 6 7 5 4 6

+ + + + + + + + + + +

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

(4)

A volleyballisthrown intothe airby aplayer. During itstrajetory,beforeitreahes the

ground oranother player, the positionof avolleyballan desribed by threeoordinates,

the absissa

x

^, ^the ^ordinate

y

^and ^the ^altitude

z

^They ^all ^depend ^on ^the ^time

t

^elapsed

sine the momentthe ballwasthrown.

Tosimplifytheproblem,wenegletairfrition,and wealsosupposethat theballdoesn't

move sideways, so that the oordinate

y

îsônstant, âlways êqual^to

0

^.

In these onditions, the trajetory depends only on three parameters : the initial speed

of the ball, whih is related to the strength and tehnique of the player throwing it, the

initialheight of the ball, and the angle between the ground and the diretion of the ball

when itisthrown. Suppose that theinitialspeed is

v 0 = 12

^meters^per^seond, ^the ^initial

heightis

2

^metersând ^the ângle îs

α = 30 ^◦

^.^Then, ^the ^altitude

z

ôf ^the ^ballâsâ ^funtion

of the absissa

x

^, ^is ^given ^by ^the ^formula

z(x) = −0.04x ² + 0.58x + 2.

Thegraphofthisfuntion,shownbelow,givesagraphialrepresentationofthetrajetory,

the groundbeing the

x

^-axis.

1 2 3 4

2 4 6 8 10 12 14 16 18

z

(5)

Radioative deay is the proess in whih an unstable atomi nuleus loses energy by

emittingionisingpartilesand radiation.Thisdeay,orlossof energy,resultsin anatom

of one type,alled the parent nulide,transforming toan atomof adierent type, alled

the daughter nulide. For example : a lead-214 atom (the parent) emits radiation and

transforms to a bismuth-214 atom (the daughter). This is a random proess on the

atomilevel,inthat itisimpossible topreditwhena given atom willdeay, butgiven a

large numberof similar atoms,the deay rate, on average,is preditable.

Thehalf-life ofaradioativenulideisthetimeittakesforhalfthe quantitytotransform.

For the lead-214 atom, it's approximately

27

^minutes, ^or

1620

^seonds. ^This ^means ^that

if atone moment we have

1

^kg ^of ^lead-214,^then

27

^minutes ^later ^there ^will ^only^be

500

^g

left, the other

500

^g ^having ^deayed. ^Another

27

^minutes ^later, ^only

25

^g ^will^be ^left.

Thenumber

N

ôfâtomsôf^lead-214^therefore^dependsôn^the^time

t

^sine^it^rst^appeared

(forexamplewhenit'sgeneratedbyanotherradioativesubstane,suhasthepolonium-

218). If the initialweight of lead-214 is

1

^kg, ^then ^the ^weight ^after

t

^seonds ^is ^preisely

given by the formula

N (t) = 2 ⁻ ¹⁶²⁰ ^t .

The graph of this funtion is given below.

1 – Team work on one document 25 mins

Materials :

•

•

1 – Team work on one document 25 mins

•

•

•

•

2 – Oral presentations 30 mins

440

s

s(t)

t

s(t) = 1 sin(220t) + 0.1 sin(440t) + 0.4 sin(660t).

0.5 1.0

− 0.5

− 1.0

0.01 0.02 0.03 0.04

− 0.01

− 0.02

− 0.03

− 0.04

− 0.05

T

I

T (I) =



 

 



 

 



0

I 6 5687

I × 0.055 − 312.79

5688 6 I 6 11344 I × 0.14 − 1277.03

11345 6 I 6 25195 I × 0.30 − 5308.23

25196 6 I 6 67546 I × 0.40 − 12062.83

I > 67546

5 6 8 8 1 1 3 4 4 2 5 1 9 5 6 7 5 4 6

+ + + + + + + + + + +

2000

4000

6000

8000

10000

12000

14000

16000

18000

20000

22000

x

y

z

t

y

0

v 0 = 12

2

α = 30 ◦

z

x

z(x) = −0.04x 2 + 0.58x + 2.

x

1 2 3 4

2 4 6 8 10 12 14 16 18

z

27

1620

1

27

500

500

27

25

N

t

α = 30 ^◦

z(x) = −0.04x ² + 0.58x + 2.

N (t) = 2 ⁻ ¹⁶²⁰ ^t .