sept récipients pour un même volume - apmep.fr · sept récipients pour un même volume r....
TRANSCRIPT
Sept reacutecipients pour un mecircme volume
R Gauthier
A la suite de la visite de lexposition Horizons matheacutematiques (Jourshyneacutees APMEP en 1984) nous avons imagineacute des activiteacutes possibles pour des eacutelegraveves du Second Cycle
Voici sept reacuteciJgtients de mecircme hauteur et de mecircme capaciteacute (voir desshysin Annexe 1) On peut choisir par exemple h = 1 m
Les volumes (2) et (3) sont composeacutes de deux cylindres de mecircme haushyteur le rayon du grand est le double du rayon du petit cylindre Pour le sablier (6) on a deux cocircnes de mecircme hauteur Chacun des reacutecipients peut ecirctre rempli (deau de Coca-Cola ou de Beaujolais ) par le haut
Ces conventions eacutetant faites deux activiteacutes nous semblent possibles
Activiteacute 1 Fournir les dessins avec les conventions ci-dessus
Question unique Fabriquer une jauge pour chaque reacutecipient sous la forme dune regravegle agrave graduer indiquant la quantiteacute de liquide en litres contenue dans le reacutecipient On suppose quune telle jauge peut ecirctre introduite par le haut pour chaque reacutecipient
Sous cette forme la question est difficile il sagit dexprimer le volume en litres du contenu en fonction de la hauteur h (en dm)
Des problegravemes vont se poser le professeur pourra aider les eacutelegraveves agrave trouver lordre dans lequel ces problegravemes pourront peu agrave peu ecirctre reacutesoshylus en apportant des informations souhaitables ou en suggeacuterant de chershycher ces informations (encyclopeacutedies etc)
Quelques-uns de ces problegravemes -Quel est le volume commun (voir la sphegravere (7)) - Trouver les dimensions qui manquent pour les autres - Trouver les bonnes formules pour calculer les volumes (tronc de cocircne
calotte spheacuterique etc) - Organiser les calculs (calculatrice ordinateur) -Graduer les jauges
(Voir reacuteponses en annexe V)
47
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
Activiteacute II On preacutesente les dessins de lannexe 1 et les conventions On fournit
- les courbes donnant le volume du liquide (en litres) en fonction de la hauteur (en dm) (annexe Il)
- les formules dans le deacutesordre V = f(h) (annexe III) - six tableaux donnant les valeurs de V en fonction de h (annexe IV)
(eacutegalement en deacutesordre)
Question Qui va avec qui
Ces activiteacutes peuvent ecirctre proposeacutees agrave une classe entiegravere des groupes deacutelegraveves un club de math
Pour lactiviteacute II les mecircmes problegravemes vont se poser sous des formes sans doute diffeacuterentes
Utilisation possible bull En seconde bull
- calculs de volumes r epreacutesentations graphiques taux de variation reacutesolutions deacutequations avec ou sans graphiques
bull En Premiegravere ou Terminale - lien entre fonction et courbe - rapiditeacute de croissance - deacuteriveacutee et tangente agrave la courbe -eacutequation
Et pour la formation scientifique divers proceacutedeacutes de repreacutesentation dun pheacutenomegravene
48
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE 1
~ ---- -- ----
I __
49
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE II
50
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE III Volume de liquide en fonction de la hauteur h
I v(h) = ___ 7rh3
6
II v(h) = 57rh2 - ___ 7rh3
3
III v(h) = 50 7rh 3
IV ~hltS hgt5
v(h)= 80 7rh 3
v(h) =10071 + 2 3 degn-h
v v(h) = JI (h 3 - 30h2 + 300h)
6
VI v(h) = 250 7r- 2 7r (5- h)3
3 3
VII ~hltS hgt5
v(h)= 20 7rh 3
v(h) = 80 1rh -1007r 3
51
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
c
ANNEXE IV Six tableaux de valeurs (sur 7)
BA
D
FE
H = O F(H)=O H = 5 F(H)=379609 H = 1 F(H)= 146608 H = 15 F(H) = 31 8086 H=2 F(H) = 544543 H= 25 F(H)=818123 H=3 F(H)=113097 H= 3 5 F(H)= 147524 H=4 F(H) = 184307 H = 45 F(H) = 22266 H = 5 F(H)=261 799 H = 55 F(H) = 300938 H = 6 F(H) = 339292 H = 65 F(H) = 376075 H=7 F(H)=410501 H = 75 F(H) = 441 786 H = 8 F(H) = 469144 H = 8 5 F(H) = 491 79 H=9 F(H) = 508938 H = 9 5 F(H) = 519803 H = 10 F(H) = 523599
H = O F(H)=O H=5 F(HJ = 0654498 H = 1 F(HJ = 529599 H = 15 F(H= 176715 H=2 F(HJ = 418879 H = 2 5 F(H=818123 H=3 F(H=141372 H=35 F(HJ = 224493 H = 4 FIHI=335103 H = 4 5 F(H=477129 H = 5 F(HJ = 654498 H = 55 F(HJ=871137 H = 6 F(HJ = 11 3097 H = 65 F(HJ = 143793 H = 7 F(H)=179594 H = 7 5 F(HJ = 220893 H=8 F(HJ = 268083 H = 85 F(HJ= 321 555 H=9 F(HJ =381 708 H = 95 F(HJ=44892 H = 10 F(HJ = 523599
H = O F(HJ=O H=5 F(HJ = 10472 H = 1 F(HJ = 209439 H = 15 F(HJ=31 4159 H = 2 F(HJ = 41 8879 H = 2 5 F(H = 523599 H =3 F(HJ=628318 H =35 F(H) = 733038 H = 4 F(H = 887758 H = 4 5 F(H = 942477 H =5 F(H) = 10472 H = 55 F(H)= 145608 H = 6 FH) = 188496 H =65 F(HJ = 230383 H = 7 F(HJ = 272271 H = 7 5 F(H)=314159 H = 8 F(HJ = 356047 H = 85 F(HJ= 397935 H=9 FIHI=439823 H=95 F(HJ = 481711 H = 10 F(HJ = 523599
H = O FIHJ = O H=5 F(H = 709476 H = 1 F(HJ = 127758 H = 15 F(HJ = 172002 H =2 F(H = 205251 H = 25 F(HJ = 229074 H = 3 F(HJ = 245044 H =35 F(Hl = 254731 H = 4 F(H) = 259705 H = 45 F(Hl = 261538 H = 5 FIH) = 261 799 H = 5 5 F(H) = 262061 H = 6 F(H) = 263894 H = 65 F(Hl = 286868 H = 7 F(Hl = 278554 H = 7 5 F(Hl = 284524 H = 8 F(Hl = 318348 H =85 FHl = 351 597 H = 9 F(H) = 395841 H = 9 5 F(H) = 452651 H = 10 FHl = 523599
H = O FHl = O H =5 F(H) = 746783 H =1 F(H) = 141 895 H = 15 F(H) = 202044 H = 2 F(H) = 255516 H = 2 5 FIHl = 302705 H = 3 F(Hl = 344004 H = 3 5 F(H) = 379805 H = 4 F(H) = 410501 H = 4 5 F(H) = 436485 H=5 F(H) = 458149 H = 5 5 F(H) = 475886 H bull 6 FIH) = 490088 H = 6 5 F(H) = 501149 H = 7 F(H) = 509462 H = 7 5 F(H) = 515417 H=8 F(H) = 51941 H = 85 F(H) = 521882 H = 9 F(H) = 523075 H = 9 5 FIH) = 528588 H =10 F(H) = 523599
H = O H=5 H = 1 H = 15 H = 2 H =25 H =3 H = 35 H = 4 H = 4 5 H =5 H = 5 5 H = 6 H =65 H bull 7 H = 7 5 H = 8 H = 85 H =9 H = 95 H = 10
F(Hl=O F(H) = 41 8879 F(H) = 837758 F(H) = 425664 FIHl = 167552 F(H) = 209439 F(Hl = 251327 F(H) = 293215 F(H) = 335103 FIH)=376991 F(H) = 418879 F(H) = 429351 F(H) = 489823 F(H) = 450295 F(H) = 460767 F(H) = 471 239 F(H) = 481 711 F(H) = 492183 F(H) = 502655 F(H) = 513127 F(H) = 523599
52
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
Activiteacute II On preacutesente les dessins de lannexe 1 et les conventions On fournit
- les courbes donnant le volume du liquide (en litres) en fonction de la hauteur (en dm) (annexe Il)
- les formules dans le deacutesordre V = f(h) (annexe III) - six tableaux donnant les valeurs de V en fonction de h (annexe IV)
(eacutegalement en deacutesordre)
Question Qui va avec qui
Ces activiteacutes peuvent ecirctre proposeacutees agrave une classe entiegravere des groupes deacutelegraveves un club de math
Pour lactiviteacute II les mecircmes problegravemes vont se poser sous des formes sans doute diffeacuterentes
Utilisation possible bull En seconde bull
- calculs de volumes r epreacutesentations graphiques taux de variation reacutesolutions deacutequations avec ou sans graphiques
bull En Premiegravere ou Terminale - lien entre fonction et courbe - rapiditeacute de croissance - deacuteriveacutee et tangente agrave la courbe -eacutequation
Et pour la formation scientifique divers proceacutedeacutes de repreacutesentation dun pheacutenomegravene
48
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE 1
~ ---- -- ----
I __
49
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE II
50
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE III Volume de liquide en fonction de la hauteur h
I v(h) = ___ 7rh3
6
II v(h) = 57rh2 - ___ 7rh3
3
III v(h) = 50 7rh 3
IV ~hltS hgt5
v(h)= 80 7rh 3
v(h) =10071 + 2 3 degn-h
v v(h) = JI (h 3 - 30h2 + 300h)
6
VI v(h) = 250 7r- 2 7r (5- h)3
3 3
VII ~hltS hgt5
v(h)= 20 7rh 3
v(h) = 80 1rh -1007r 3
51
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
c
ANNEXE IV Six tableaux de valeurs (sur 7)
BA
D
FE
H = O F(H)=O H = 5 F(H)=379609 H = 1 F(H)= 146608 H = 15 F(H) = 31 8086 H=2 F(H) = 544543 H= 25 F(H)=818123 H=3 F(H)=113097 H= 3 5 F(H)= 147524 H=4 F(H) = 184307 H = 45 F(H) = 22266 H = 5 F(H)=261 799 H = 55 F(H) = 300938 H = 6 F(H) = 339292 H = 65 F(H) = 376075 H=7 F(H)=410501 H = 75 F(H) = 441 786 H = 8 F(H) = 469144 H = 8 5 F(H) = 491 79 H=9 F(H) = 508938 H = 9 5 F(H) = 519803 H = 10 F(H) = 523599
H = O F(H)=O H=5 F(HJ = 0654498 H = 1 F(HJ = 529599 H = 15 F(H= 176715 H=2 F(HJ = 418879 H = 2 5 F(H=818123 H=3 F(H=141372 H=35 F(HJ = 224493 H = 4 FIHI=335103 H = 4 5 F(H=477129 H = 5 F(HJ = 654498 H = 55 F(HJ=871137 H = 6 F(HJ = 11 3097 H = 65 F(HJ = 143793 H = 7 F(H)=179594 H = 7 5 F(HJ = 220893 H=8 F(HJ = 268083 H = 85 F(HJ= 321 555 H=9 F(HJ =381 708 H = 95 F(HJ=44892 H = 10 F(HJ = 523599
H = O F(HJ=O H=5 F(HJ = 10472 H = 1 F(HJ = 209439 H = 15 F(HJ=31 4159 H = 2 F(HJ = 41 8879 H = 2 5 F(H = 523599 H =3 F(HJ=628318 H =35 F(H) = 733038 H = 4 F(H = 887758 H = 4 5 F(H = 942477 H =5 F(H) = 10472 H = 55 F(H)= 145608 H = 6 FH) = 188496 H =65 F(HJ = 230383 H = 7 F(HJ = 272271 H = 7 5 F(H)=314159 H = 8 F(HJ = 356047 H = 85 F(HJ= 397935 H=9 FIHI=439823 H=95 F(HJ = 481711 H = 10 F(HJ = 523599
H = O FIHJ = O H=5 F(H = 709476 H = 1 F(HJ = 127758 H = 15 F(HJ = 172002 H =2 F(H = 205251 H = 25 F(HJ = 229074 H = 3 F(HJ = 245044 H =35 F(Hl = 254731 H = 4 F(H) = 259705 H = 45 F(Hl = 261538 H = 5 FIH) = 261 799 H = 5 5 F(H) = 262061 H = 6 F(H) = 263894 H = 65 F(Hl = 286868 H = 7 F(Hl = 278554 H = 7 5 F(Hl = 284524 H = 8 F(Hl = 318348 H =85 FHl = 351 597 H = 9 F(H) = 395841 H = 9 5 F(H) = 452651 H = 10 FHl = 523599
H = O FHl = O H =5 F(H) = 746783 H =1 F(H) = 141 895 H = 15 F(H) = 202044 H = 2 F(H) = 255516 H = 2 5 FIHl = 302705 H = 3 F(Hl = 344004 H = 3 5 F(H) = 379805 H = 4 F(H) = 410501 H = 4 5 F(H) = 436485 H=5 F(H) = 458149 H = 5 5 F(H) = 475886 H bull 6 FIH) = 490088 H = 6 5 F(H) = 501149 H = 7 F(H) = 509462 H = 7 5 F(H) = 515417 H=8 F(H) = 51941 H = 85 F(H) = 521882 H = 9 F(H) = 523075 H = 9 5 FIH) = 528588 H =10 F(H) = 523599
H = O H=5 H = 1 H = 15 H = 2 H =25 H =3 H = 35 H = 4 H = 4 5 H =5 H = 5 5 H = 6 H =65 H bull 7 H = 7 5 H = 8 H = 85 H =9 H = 95 H = 10
F(Hl=O F(H) = 41 8879 F(H) = 837758 F(H) = 425664 FIHl = 167552 F(H) = 209439 F(Hl = 251327 F(H) = 293215 F(H) = 335103 FIH)=376991 F(H) = 418879 F(H) = 429351 F(H) = 489823 F(H) = 450295 F(H) = 460767 F(H) = 471 239 F(H) = 481 711 F(H) = 492183 F(H) = 502655 F(H) = 513127 F(H) = 523599
52
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE 1
~ ---- -- ----
I __
49
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE II
50
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE III Volume de liquide en fonction de la hauteur h
I v(h) = ___ 7rh3
6
II v(h) = 57rh2 - ___ 7rh3
3
III v(h) = 50 7rh 3
IV ~hltS hgt5
v(h)= 80 7rh 3
v(h) =10071 + 2 3 degn-h
v v(h) = JI (h 3 - 30h2 + 300h)
6
VI v(h) = 250 7r- 2 7r (5- h)3
3 3
VII ~hltS hgt5
v(h)= 20 7rh 3
v(h) = 80 1rh -1007r 3
51
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
c
ANNEXE IV Six tableaux de valeurs (sur 7)
BA
D
FE
H = O F(H)=O H = 5 F(H)=379609 H = 1 F(H)= 146608 H = 15 F(H) = 31 8086 H=2 F(H) = 544543 H= 25 F(H)=818123 H=3 F(H)=113097 H= 3 5 F(H)= 147524 H=4 F(H) = 184307 H = 45 F(H) = 22266 H = 5 F(H)=261 799 H = 55 F(H) = 300938 H = 6 F(H) = 339292 H = 65 F(H) = 376075 H=7 F(H)=410501 H = 75 F(H) = 441 786 H = 8 F(H) = 469144 H = 8 5 F(H) = 491 79 H=9 F(H) = 508938 H = 9 5 F(H) = 519803 H = 10 F(H) = 523599
H = O F(H)=O H=5 F(HJ = 0654498 H = 1 F(HJ = 529599 H = 15 F(H= 176715 H=2 F(HJ = 418879 H = 2 5 F(H=818123 H=3 F(H=141372 H=35 F(HJ = 224493 H = 4 FIHI=335103 H = 4 5 F(H=477129 H = 5 F(HJ = 654498 H = 55 F(HJ=871137 H = 6 F(HJ = 11 3097 H = 65 F(HJ = 143793 H = 7 F(H)=179594 H = 7 5 F(HJ = 220893 H=8 F(HJ = 268083 H = 85 F(HJ= 321 555 H=9 F(HJ =381 708 H = 95 F(HJ=44892 H = 10 F(HJ = 523599
H = O F(HJ=O H=5 F(HJ = 10472 H = 1 F(HJ = 209439 H = 15 F(HJ=31 4159 H = 2 F(HJ = 41 8879 H = 2 5 F(H = 523599 H =3 F(HJ=628318 H =35 F(H) = 733038 H = 4 F(H = 887758 H = 4 5 F(H = 942477 H =5 F(H) = 10472 H = 55 F(H)= 145608 H = 6 FH) = 188496 H =65 F(HJ = 230383 H = 7 F(HJ = 272271 H = 7 5 F(H)=314159 H = 8 F(HJ = 356047 H = 85 F(HJ= 397935 H=9 FIHI=439823 H=95 F(HJ = 481711 H = 10 F(HJ = 523599
H = O FIHJ = O H=5 F(H = 709476 H = 1 F(HJ = 127758 H = 15 F(HJ = 172002 H =2 F(H = 205251 H = 25 F(HJ = 229074 H = 3 F(HJ = 245044 H =35 F(Hl = 254731 H = 4 F(H) = 259705 H = 45 F(Hl = 261538 H = 5 FIH) = 261 799 H = 5 5 F(H) = 262061 H = 6 F(H) = 263894 H = 65 F(Hl = 286868 H = 7 F(Hl = 278554 H = 7 5 F(Hl = 284524 H = 8 F(Hl = 318348 H =85 FHl = 351 597 H = 9 F(H) = 395841 H = 9 5 F(H) = 452651 H = 10 FHl = 523599
H = O FHl = O H =5 F(H) = 746783 H =1 F(H) = 141 895 H = 15 F(H) = 202044 H = 2 F(H) = 255516 H = 2 5 FIHl = 302705 H = 3 F(Hl = 344004 H = 3 5 F(H) = 379805 H = 4 F(H) = 410501 H = 4 5 F(H) = 436485 H=5 F(H) = 458149 H = 5 5 F(H) = 475886 H bull 6 FIH) = 490088 H = 6 5 F(H) = 501149 H = 7 F(H) = 509462 H = 7 5 F(H) = 515417 H=8 F(H) = 51941 H = 85 F(H) = 521882 H = 9 F(H) = 523075 H = 9 5 FIH) = 528588 H =10 F(H) = 523599
H = O H=5 H = 1 H = 15 H = 2 H =25 H =3 H = 35 H = 4 H = 4 5 H =5 H = 5 5 H = 6 H =65 H bull 7 H = 7 5 H = 8 H = 85 H =9 H = 95 H = 10
F(Hl=O F(H) = 41 8879 F(H) = 837758 F(H) = 425664 FIHl = 167552 F(H) = 209439 F(Hl = 251327 F(H) = 293215 F(H) = 335103 FIH)=376991 F(H) = 418879 F(H) = 429351 F(H) = 489823 F(H) = 450295 F(H) = 460767 F(H) = 471 239 F(H) = 481 711 F(H) = 492183 F(H) = 502655 F(H) = 513127 F(H) = 523599
52
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE II
50
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE III Volume de liquide en fonction de la hauteur h
I v(h) = ___ 7rh3
6
II v(h) = 57rh2 - ___ 7rh3
3
III v(h) = 50 7rh 3
IV ~hltS hgt5
v(h)= 80 7rh 3
v(h) =10071 + 2 3 degn-h
v v(h) = JI (h 3 - 30h2 + 300h)
6
VI v(h) = 250 7r- 2 7r (5- h)3
3 3
VII ~hltS hgt5
v(h)= 20 7rh 3
v(h) = 80 1rh -1007r 3
51
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
c
ANNEXE IV Six tableaux de valeurs (sur 7)
BA
D
FE
H = O F(H)=O H = 5 F(H)=379609 H = 1 F(H)= 146608 H = 15 F(H) = 31 8086 H=2 F(H) = 544543 H= 25 F(H)=818123 H=3 F(H)=113097 H= 3 5 F(H)= 147524 H=4 F(H) = 184307 H = 45 F(H) = 22266 H = 5 F(H)=261 799 H = 55 F(H) = 300938 H = 6 F(H) = 339292 H = 65 F(H) = 376075 H=7 F(H)=410501 H = 75 F(H) = 441 786 H = 8 F(H) = 469144 H = 8 5 F(H) = 491 79 H=9 F(H) = 508938 H = 9 5 F(H) = 519803 H = 10 F(H) = 523599
H = O F(H)=O H=5 F(HJ = 0654498 H = 1 F(HJ = 529599 H = 15 F(H= 176715 H=2 F(HJ = 418879 H = 2 5 F(H=818123 H=3 F(H=141372 H=35 F(HJ = 224493 H = 4 FIHI=335103 H = 4 5 F(H=477129 H = 5 F(HJ = 654498 H = 55 F(HJ=871137 H = 6 F(HJ = 11 3097 H = 65 F(HJ = 143793 H = 7 F(H)=179594 H = 7 5 F(HJ = 220893 H=8 F(HJ = 268083 H = 85 F(HJ= 321 555 H=9 F(HJ =381 708 H = 95 F(HJ=44892 H = 10 F(HJ = 523599
H = O F(HJ=O H=5 F(HJ = 10472 H = 1 F(HJ = 209439 H = 15 F(HJ=31 4159 H = 2 F(HJ = 41 8879 H = 2 5 F(H = 523599 H =3 F(HJ=628318 H =35 F(H) = 733038 H = 4 F(H = 887758 H = 4 5 F(H = 942477 H =5 F(H) = 10472 H = 55 F(H)= 145608 H = 6 FH) = 188496 H =65 F(HJ = 230383 H = 7 F(HJ = 272271 H = 7 5 F(H)=314159 H = 8 F(HJ = 356047 H = 85 F(HJ= 397935 H=9 FIHI=439823 H=95 F(HJ = 481711 H = 10 F(HJ = 523599
H = O FIHJ = O H=5 F(H = 709476 H = 1 F(HJ = 127758 H = 15 F(HJ = 172002 H =2 F(H = 205251 H = 25 F(HJ = 229074 H = 3 F(HJ = 245044 H =35 F(Hl = 254731 H = 4 F(H) = 259705 H = 45 F(Hl = 261538 H = 5 FIH) = 261 799 H = 5 5 F(H) = 262061 H = 6 F(H) = 263894 H = 65 F(Hl = 286868 H = 7 F(Hl = 278554 H = 7 5 F(Hl = 284524 H = 8 F(Hl = 318348 H =85 FHl = 351 597 H = 9 F(H) = 395841 H = 9 5 F(H) = 452651 H = 10 FHl = 523599
H = O FHl = O H =5 F(H) = 746783 H =1 F(H) = 141 895 H = 15 F(H) = 202044 H = 2 F(H) = 255516 H = 2 5 FIHl = 302705 H = 3 F(Hl = 344004 H = 3 5 F(H) = 379805 H = 4 F(H) = 410501 H = 4 5 F(H) = 436485 H=5 F(H) = 458149 H = 5 5 F(H) = 475886 H bull 6 FIH) = 490088 H = 6 5 F(H) = 501149 H = 7 F(H) = 509462 H = 7 5 F(H) = 515417 H=8 F(H) = 51941 H = 85 F(H) = 521882 H = 9 F(H) = 523075 H = 9 5 FIH) = 528588 H =10 F(H) = 523599
H = O H=5 H = 1 H = 15 H = 2 H =25 H =3 H = 35 H = 4 H = 4 5 H =5 H = 5 5 H = 6 H =65 H bull 7 H = 7 5 H = 8 H = 85 H =9 H = 95 H = 10
F(Hl=O F(H) = 41 8879 F(H) = 837758 F(H) = 425664 FIHl = 167552 F(H) = 209439 F(Hl = 251327 F(H) = 293215 F(H) = 335103 FIH)=376991 F(H) = 418879 F(H) = 429351 F(H) = 489823 F(H) = 450295 F(H) = 460767 F(H) = 471 239 F(H) = 481 711 F(H) = 492183 F(H) = 502655 F(H) = 513127 F(H) = 523599
52
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE III Volume de liquide en fonction de la hauteur h
I v(h) = ___ 7rh3
6
II v(h) = 57rh2 - ___ 7rh3
3
III v(h) = 50 7rh 3
IV ~hltS hgt5
v(h)= 80 7rh 3
v(h) =10071 + 2 3 degn-h
v v(h) = JI (h 3 - 30h2 + 300h)
6
VI v(h) = 250 7r- 2 7r (5- h)3
3 3
VII ~hltS hgt5
v(h)= 20 7rh 3
v(h) = 80 1rh -1007r 3
51
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
c
ANNEXE IV Six tableaux de valeurs (sur 7)
BA
D
FE
H = O F(H)=O H = 5 F(H)=379609 H = 1 F(H)= 146608 H = 15 F(H) = 31 8086 H=2 F(H) = 544543 H= 25 F(H)=818123 H=3 F(H)=113097 H= 3 5 F(H)= 147524 H=4 F(H) = 184307 H = 45 F(H) = 22266 H = 5 F(H)=261 799 H = 55 F(H) = 300938 H = 6 F(H) = 339292 H = 65 F(H) = 376075 H=7 F(H)=410501 H = 75 F(H) = 441 786 H = 8 F(H) = 469144 H = 8 5 F(H) = 491 79 H=9 F(H) = 508938 H = 9 5 F(H) = 519803 H = 10 F(H) = 523599
H = O F(H)=O H=5 F(HJ = 0654498 H = 1 F(HJ = 529599 H = 15 F(H= 176715 H=2 F(HJ = 418879 H = 2 5 F(H=818123 H=3 F(H=141372 H=35 F(HJ = 224493 H = 4 FIHI=335103 H = 4 5 F(H=477129 H = 5 F(HJ = 654498 H = 55 F(HJ=871137 H = 6 F(HJ = 11 3097 H = 65 F(HJ = 143793 H = 7 F(H)=179594 H = 7 5 F(HJ = 220893 H=8 F(HJ = 268083 H = 85 F(HJ= 321 555 H=9 F(HJ =381 708 H = 95 F(HJ=44892 H = 10 F(HJ = 523599
H = O F(HJ=O H=5 F(HJ = 10472 H = 1 F(HJ = 209439 H = 15 F(HJ=31 4159 H = 2 F(HJ = 41 8879 H = 2 5 F(H = 523599 H =3 F(HJ=628318 H =35 F(H) = 733038 H = 4 F(H = 887758 H = 4 5 F(H = 942477 H =5 F(H) = 10472 H = 55 F(H)= 145608 H = 6 FH) = 188496 H =65 F(HJ = 230383 H = 7 F(HJ = 272271 H = 7 5 F(H)=314159 H = 8 F(HJ = 356047 H = 85 F(HJ= 397935 H=9 FIHI=439823 H=95 F(HJ = 481711 H = 10 F(HJ = 523599
H = O FIHJ = O H=5 F(H = 709476 H = 1 F(HJ = 127758 H = 15 F(HJ = 172002 H =2 F(H = 205251 H = 25 F(HJ = 229074 H = 3 F(HJ = 245044 H =35 F(Hl = 254731 H = 4 F(H) = 259705 H = 45 F(Hl = 261538 H = 5 FIH) = 261 799 H = 5 5 F(H) = 262061 H = 6 F(H) = 263894 H = 65 F(Hl = 286868 H = 7 F(Hl = 278554 H = 7 5 F(Hl = 284524 H = 8 F(Hl = 318348 H =85 FHl = 351 597 H = 9 F(H) = 395841 H = 9 5 F(H) = 452651 H = 10 FHl = 523599
H = O FHl = O H =5 F(H) = 746783 H =1 F(H) = 141 895 H = 15 F(H) = 202044 H = 2 F(H) = 255516 H = 2 5 FIHl = 302705 H = 3 F(Hl = 344004 H = 3 5 F(H) = 379805 H = 4 F(H) = 410501 H = 4 5 F(H) = 436485 H=5 F(H) = 458149 H = 5 5 F(H) = 475886 H bull 6 FIH) = 490088 H = 6 5 F(H) = 501149 H = 7 F(H) = 509462 H = 7 5 F(H) = 515417 H=8 F(H) = 51941 H = 85 F(H) = 521882 H = 9 F(H) = 523075 H = 9 5 FIH) = 528588 H =10 F(H) = 523599
H = O H=5 H = 1 H = 15 H = 2 H =25 H =3 H = 35 H = 4 H = 4 5 H =5 H = 5 5 H = 6 H =65 H bull 7 H = 7 5 H = 8 H = 85 H =9 H = 95 H = 10
F(Hl=O F(H) = 41 8879 F(H) = 837758 F(H) = 425664 FIHl = 167552 F(H) = 209439 F(Hl = 251327 F(H) = 293215 F(H) = 335103 FIH)=376991 F(H) = 418879 F(H) = 429351 F(H) = 489823 F(H) = 450295 F(H) = 460767 F(H) = 471 239 F(H) = 481 711 F(H) = 492183 F(H) = 502655 F(H) = 513127 F(H) = 523599
52
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
c
ANNEXE IV Six tableaux de valeurs (sur 7)
BA
D
FE
H = O F(H)=O H = 5 F(H)=379609 H = 1 F(H)= 146608 H = 15 F(H) = 31 8086 H=2 F(H) = 544543 H= 25 F(H)=818123 H=3 F(H)=113097 H= 3 5 F(H)= 147524 H=4 F(H) = 184307 H = 45 F(H) = 22266 H = 5 F(H)=261 799 H = 55 F(H) = 300938 H = 6 F(H) = 339292 H = 65 F(H) = 376075 H=7 F(H)=410501 H = 75 F(H) = 441 786 H = 8 F(H) = 469144 H = 8 5 F(H) = 491 79 H=9 F(H) = 508938 H = 9 5 F(H) = 519803 H = 10 F(H) = 523599
H = O F(H)=O H=5 F(HJ = 0654498 H = 1 F(HJ = 529599 H = 15 F(H= 176715 H=2 F(HJ = 418879 H = 2 5 F(H=818123 H=3 F(H=141372 H=35 F(HJ = 224493 H = 4 FIHI=335103 H = 4 5 F(H=477129 H = 5 F(HJ = 654498 H = 55 F(HJ=871137 H = 6 F(HJ = 11 3097 H = 65 F(HJ = 143793 H = 7 F(H)=179594 H = 7 5 F(HJ = 220893 H=8 F(HJ = 268083 H = 85 F(HJ= 321 555 H=9 F(HJ =381 708 H = 95 F(HJ=44892 H = 10 F(HJ = 523599
H = O F(HJ=O H=5 F(HJ = 10472 H = 1 F(HJ = 209439 H = 15 F(HJ=31 4159 H = 2 F(HJ = 41 8879 H = 2 5 F(H = 523599 H =3 F(HJ=628318 H =35 F(H) = 733038 H = 4 F(H = 887758 H = 4 5 F(H = 942477 H =5 F(H) = 10472 H = 55 F(H)= 145608 H = 6 FH) = 188496 H =65 F(HJ = 230383 H = 7 F(HJ = 272271 H = 7 5 F(H)=314159 H = 8 F(HJ = 356047 H = 85 F(HJ= 397935 H=9 FIHI=439823 H=95 F(HJ = 481711 H = 10 F(HJ = 523599
H = O FIHJ = O H=5 F(H = 709476 H = 1 F(HJ = 127758 H = 15 F(HJ = 172002 H =2 F(H = 205251 H = 25 F(HJ = 229074 H = 3 F(HJ = 245044 H =35 F(Hl = 254731 H = 4 F(H) = 259705 H = 45 F(Hl = 261538 H = 5 FIH) = 261 799 H = 5 5 F(H) = 262061 H = 6 F(H) = 263894 H = 65 F(Hl = 286868 H = 7 F(Hl = 278554 H = 7 5 F(Hl = 284524 H = 8 F(Hl = 318348 H =85 FHl = 351 597 H = 9 F(H) = 395841 H = 9 5 F(H) = 452651 H = 10 FHl = 523599
H = O FHl = O H =5 F(H) = 746783 H =1 F(H) = 141 895 H = 15 F(H) = 202044 H = 2 F(H) = 255516 H = 2 5 FIHl = 302705 H = 3 F(Hl = 344004 H = 3 5 F(H) = 379805 H = 4 F(H) = 410501 H = 4 5 F(H) = 436485 H=5 F(H) = 458149 H = 5 5 F(H) = 475886 H bull 6 FIH) = 490088 H = 6 5 F(H) = 501149 H = 7 F(H) = 509462 H = 7 5 F(H) = 515417 H=8 F(H) = 51941 H = 85 F(H) = 521882 H = 9 F(H) = 523075 H = 9 5 FIH) = 528588 H =10 F(H) = 523599
H = O H=5 H = 1 H = 15 H = 2 H =25 H =3 H = 35 H = 4 H = 4 5 H =5 H = 5 5 H = 6 H =65 H bull 7 H = 7 5 H = 8 H = 85 H =9 H = 95 H = 10
F(Hl=O F(H) = 41 8879 F(H) = 837758 F(H) = 425664 FIHl = 167552 F(H) = 209439 F(Hl = 251327 F(H) = 293215 F(H) = 335103 FIH)=376991 F(H) = 418879 F(H) = 429351 F(H) = 489823 F(H) = 450295 F(H) = 460767 F(H) = 471 239 F(H) = 481 711 F(H) = 492183 F(H) = 502655 F(H) = 513127 F(H) = 523599
52
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
ANNEXE V Des reacuteponses (H = 10 dm)
Q) Cest la sphegravere qui donne le volume V= 50f7r f 5236 f
IcircLe cylindre (ndeg 1) 5007r =7rR2 H dougrave R 2 = 50 R= 5f6 ~ 3 3 3
Volume du liquide h -+V(h) = 507r h 3
Le cocircne (ndeg 4) 5007r =__7rR2H d ougrave R2 =50 R=5fi 3 3
Volume du liquide h -+V(h) = __7rr2h 3
avec r= Yi h on a V(h)=__7rh3 bull
2 6
Le cocircne (no 5) on a encore R=SYi
Volume du liquide h-+ V(h) = 5007r - __7r(10 - h)3 (voir 3) 3 6
dougrave V(h)= ~ (h3 - 30h2 +300h)
250(iLe sablier (ndeg 6) Les deux parties ont le mecircme volume 7r 3
la mecircme hauteur 5 et le mecircme rayon 5fi
Volume du liquide
si h~5 V(h) = 2507r- 27r (5-h)3
3 3
si hgt5 V(h)= 2507r + Z7r (h-5)3 3 3
V(h) = 2507r _ 27r (5 _ h)3 En deacutefinitive pour tout h 3 3
Le double cyclindre (2) r = 2r 5007r = 57rr 2 + 57rr2 = 257rr2 dougrave r 2 = 60 r= 2ff5 middotr = 4ff5
3 33 9
Volume du liquide
~si h~5 V(h) = 8~7r h
~si hgt5 V(h)= 4007r + Z07r (h-5)= 1007r + Z07r h 3 3 3
53
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969
J)L autre double cylindre (3) mecircmes rayons
si h~5 middot V(h) = 20
371 h
si h gt 5 V(h) = 10071 + S01r (h - 5) = S07r h -10071 3 3 3
regLa calotte spheacuterique R = 5
V(h) = ___7rh 1(15- h) =57rh2 - ___7rh3
bull
3 3
Qui va avec qui
Reacutecipient Courbe Formule Tableaux de valeurs
Cylindre (1) n o 4 (III) agrave faire
Double cylindre
(2) ndeg 2 (IV) F
Double cylindre
(3) n deg 5 (VII) E
Cocircne renverseacute (4) n deg 7 (1) c Cocircne (5) ndeg 1 (V) D
Sablier (6) n deg 6 (VI) B
Sphegravere (7) ndeg 3 (Il) A
54
Activiteacutes second cycle - Seconde - Premiegravere -Terminale (ndeg 69) - APMEP 1969