Chèvre

Le\;problème\;de\;la\;chèvre

V

V'

Corde

A

P

B

O

Pré

On\;considère\;les\;points:V\left(S_{brout}\;,\;\alpha   \right) \;\et\;\;V'\left( S_{brout}\;,\;R  \right) \;
où:\
*\;R\;est\; la\;longueur\; de\; la\; corde\
*\;\alpha=\widehat{BPA}\;en\;rad\
*\;S_{brout}\;est\;la\;surface\;qui\;peut\;être\;\;broutée

Contrainte:\;S_{brout}=\frac{\pi}{2}

Conjectures:\
1^{ère} \;méthode:d'après\;les\;traces\;des\;points\;V\;et\;V':\
on\;a:\;\;\;\;\;\widehat{BPA}=\alpha=1.90\; rad\;\; \;\;et\;\;\;\;R=1.16\
2^{ème }\;méthode:\;d'après\;le\;tableur\
Ouvrir\;le\;menu\;''Affichage''\;et\;activer\
l'option\;''Tableur''   .\
A\;la\;ligne\;117\;de\;ce\;tableur\;on\;lit:\
pour\;S_{brout}=1.5736,\;on\;a:\
\alpha=1.90414\;rad\;\;et\;\;R=1.16
\ 

Le\;pas\;de\;la\;colonne\;R\;du\;tableur\;est\;10^{-2}:\
\;\;\;\;\;\;\;\;\;c'est\;l'incrément\; du \;curseur\;R

R = 1.22 (slider from 0 to 2 * r, step 0.01)
O = (0.7602563893928068, 0.31950457801775156)
a: y = y(O)
r = 1
c = Circle(O, 1)
P = Intersect(c, a, 1)
d = Circle(P, R)
B = Intersect(c, d, 1)
A = Intersect(c, d, 2)
α = Angle(B, P, A)
e = CircleArc(O, A, B)
f = CircleArc(P, B, A)
Sect_{circul} = CircleSector(P, B, A)
poly1, p, a_1, o = Polygon(A, O, P)
β = Angle(A, O, P)
Seg_{circul}: r^(2) * (β - sin(β)) / 2
S_{brout}: Sect_{circul} + 2 * Seg_{circul}
C = Point(Sect_{circul})
b = Segment(P, C)
V: (S_{brout}, α)
V': (S_{brout}, R)
g = OrthogonalLine(V, xAxis)
D = Intersect(g, xAxis)
h = OrthogonalLine(V, yAxis)
i = OrthogonalLine(V', yAxis)
E = Intersect(h, yAxis)
F = Intersect(i, yAxis)
j = Segment(V, E)
k = Segment(V', F)
l = Segment(V, D)
m = Segment(V', D)
texte5: "\alpha=" + (LaTeX[α]) + ""
texte6: "Long.\;cor:R=" + (LaTeX[R]) + ""
texte7: "Surface\; broutée:S_{brout}=" + (LaTeX[S_{brout}]) + ""
G = Midpoint(b)
A1 = Name(S_{brout})
B1 = Name(R)
C1 = Name(α)
B2 = 0.01
C2 = 3.131592611922653 rad
A2 = 1.5674629851332304E-4
B3 = 0.02
C3 = 3.1215923202414633 rad
A3 = 6.256518373838842E-4
B4 = 0.03
C4 = 3.111591528475866 rad
A4 = 0.001404716491600596
B5 = 0.04
C5 = 3.1015899864430145 rad
A5 = 0.002491939936095149
B6 = 0.05
C6 = 3.0915874437910698 rad
A6 = 0.003885321545630238
B7 = 0.06
C7 = 3.0815836499428375 rad
A7 = 0.005582860294586035
B8 = 0.07
C8 = 3.0715783540390946 rad
A8 = 0.007582554656610506
B9 = 0.08
C9 = 3.061571304881815 rad
A9 = 0.009882402504100692
B10 = 0.09
C10 = 3.0515622508771645 rad
A10 = 0.012480401007480256
B11 = 0.1
C11 = 3.041550939978255 rad
A11 = 0.01537454653424031
B12 = 0.11
C12 = 3.031537119627589 rad
A12 = 0.018562834547707147
B13 = 0.12
C13 = 3.021520536699236 rad
A13 = 0.022043259505501057
B14 = 0.13
C14 = 3.011500937440571 rad
A14 = 0.025813814757652428
B15 = 0.14
C15 = 3.0014780674136925 rad
A15 = 0.029872492444347053
B16 = 0.15
C16 = 2.991451671436359 rad
A16 = 0.03421728339324809
B17 = 0.16
C17 = 2.9814214935224763 rad
A17 = 0.038846177016378734
B18 = 0.17
C18 = 2.9713872768220457 rad
A18 = 0.04375716120650732
B19 = 0.18
C19 = 2.9613487635606033 rad
A19 = 0.0489482222330236
B20 = 0.19
C20 = 2.9513056949780134 rad
A20 = 0.05441734463724419
B21 = 0.2
C21 = 2.9412578112666727 rad
A21 = 0.06016251112712867
B22 = 0.21
C22 = 2.9312048515090163 rad
A22 = 0.06618170247135238
B23 = 0.22
C23 = 2.9211465536143 rad
A23 = 0.07247289739271222
B24 = 0.23
C24 = 2.911082654254638 rad
A24 = 0.0790340724608121
B25 = 0.24
C25 = 2.9010128888002176 rad
A25 = 0.08586320198399798
B26 = 0.25
C26 = 2.8909369912536627 rad
A26 = 0.09295825790050255
B27 = 0.26
C27 = 2.880854694183501 rad
A27 = 0.10031720966874258
B28 = 0.27
C28 = 2.8707657286566817 rad
A28 = 0.10793802415675192
B29 = 0.28
C29 = 2.860669824170082 rad
A29 = 0.11581866553067946
B30 = 0.29
C30 = 2.85056670858098 rad
A30 = 0.12395709514233845
B31 = 0.3
C31 = 2.8404561080364212 rad
A31 = 0.13235127141573277
B32 = 0.31
C32 = 2.830337746901422 rad
A32 = 0.14099914973253472
B33 = 0.32
C33 = 2.820211347685972 rad
A33 = 0.1498986823164637
B34 = 0.33
C34 = 2.8100766309707597 rad
A34 = 0.15904781811651209
B35 = 0.34
C35 = 2.799933315331584 rad
A35 = 0.16844450268897518
B36 = 0.35
C36 = 2.7897811172623603 rad
A36 = 0.1780866780782358
B37 = 0.36
C37 = 2.779619751096698 rad
A37 = 0.18797228269625282
B38 = 0.37
C38 = 2.7694489289279463 rad
A38 = 0.19809925120069388
B39 = 0.38
C39 = 2.759268360527674 rad
A39 = 0.2084655143716757
B40 = 0.39
C40 = 2.7490777532624864 rad
A40 = 0.2190689989870316
B41 = 0.4
C41 = 2.7388768120091314 rad
A41 = 0.22990762769608367
B42 = 0.41
C42 = 2.7286652390678086 rad
A42 = 0.24097931889182866
B43 = 0.42
C43 = 2.7184427340736015 rad
A43 = 0.2522819865815026
B44 = 0.43
C44 = 2.7082089939059713 rad
A44 = 0.26381354025545084
B45 = 0.44
C45 = 2.6979637125962044 rad
A45 = 0.2755718847542478
B46 = 0.45
C46 = 2.687706581232753 rad
A46 = 0.2875549201339917
B47 = 0.46
C47 = 2.6774372878643673 rad
A47 = 0.299760541529728
B48 = 0.47
C48 = 2.667155517400923 rad
A48 = 0.3121866390169105
B49 = 0.48
C49 = 2.656860951511867 rad
A49 = 0.3248310974708482
B50 = 0.49
C50 = 2.646553268522161 rad
A50 = 0.3376917964240548
B51 = 0.5
C51 = 2.636232143305636 rad
A51 = 0.3507666099214344
B52 = 0.51
C52 = 2.625897247175636 rad
A52 = 0.36405340637321787
B53 = 0.52
C53 = 2.615548247772855 rad
A53 = 0.37755004840557815
B54 = 0.53
C54 = 2.6051848089502423 rad
A54 = 0.3912543927088479
B55 = 0.54
C55 = 2.5948065906548465 rad
A55 = 0.4051642898832361
B56 = 0.55
C56 = 2.5844132488064924 rad
A56 = 0.4192775842819752
B57 = 0.56
C57 = 2.574004435173137 rad
A57 = 0.43359211385180296
B58 = 0.57
C58 = 2.5635797972427907 rad
A58 = 0.4481057099706912
B59 = 0.58
C59 = 2.5531389780918277 rad
A59 = 0.4628161972827003
B60 = 0.59
C60 = 2.542681616249577 rad
A60 = 0.477721393529905
B61 = 0.6
C61 = 2.532207345558999 rad
A61 = 0.4928191093812477
B62 = 0.61
C62 = 2.521715795033302 rad
A62 = 0.508107148258227
B63 = 0.62
C63 = 2.5112065887083443 rad
A63 = 0.5235833061573306
B64 = 0.63
C64 = 2.500679345490597 rad
A64 = 0.5392453714690508
B65 = 0.64
C65 = 2.490133679000533 rad
A65 = 0.5550911247934266
B66 = 0.65
C66 = 2.4795691974112035 rad
A66 = 0.5711183387519297
B67 = 0.66
C67 = 2.468985503281832 rad
A67 = 0.5873247777956075
B68 = 0.67
C68 = 2.4583821933861776 rad
A68 = 0.6037081980093386
B69 = 0.68
C69 = 2.447758858535469 rad
A69 = 0.6202663469120479
B70 = 0.69
C70 = 2.437115083395663 rad
A70 = 0.6369969632527711
B71 = 0.7
C71 = 2.4264504462987726 rad
A71 = 0.6538977768023889
B72 = 0.71
C72 = 2.4157645190480164 rad
A72 = 0.6709665081408916
B73 = 0.72
C73 = 2.405056866716513 rad
A73 = 0.6882008684400216
B74 = 0.73
C74 = 2.394327047439231 rad
A74 = 0.7055985592411038
B75 = 0.74
C75 = 2.3835746121978914 rad
A75 = 0.7231572722279146
B76 = 0.75
C76 = 2.3727991045985153 rad
A76 = 0.7408746889943921
B77 = 0.76
C77 = 2.3620000606412717 rad
A77 = 0.7587484808070125
B78 = 0.77
C78 = 2.35117700848229 rad
A78 = 0.7767763083616218
B79 = 0.78
C79 = 2.340329468187049 rad
A79 = 0.7949558215345398
B80 = 0.79
C80 = 2.329456951474973 rad
A80 = 0.8132846591276914
B81 = 0.8
C81 = 2.3185589614548165 rad
A81 = 0.8317604486075836
B82 = 0.81
C82 = 2.3076349923504047 rad
A82 = 0.8503808058378562
B83 = 0.82
C83 = 2.296684529216282 rad
A83 = 0.8691433348051999
B84 = 0.83
C84 = 2.2857070476427817 rad
A84 = 0.8880456273383545
B85 = 0.84
C85 = 2.2747020134500207 rad
A85 = 0.9070852628199573
B86 = 0.85
C86 = 2.2636688823702715 rad
A86 = 0.9262598078909358
B87 = 0.86
C87 = 2.252607099718155 rad
A87 = 0.9455668161471774
B88 = 0.87
C88 = 2.241516100048059 rad
A88 = 0.9650038278281587
B89 = 0.88
C89 = 2.2303953067981466 rad
A89 = 0.9845683694972335
B90 = 0.89
C90 = 2.219244131920286 rad
A90 = 1.0042579537132277
B91 = 0.9
C91 = 2.2080619754952 rad
A91 = 1.0240700786930206
B92 = 0.91
C92 = 2.1968482253320833 rad
A92 = 1.0440022279647236
B93 = 0.92
C93 = 2.1856022565518884 rad
A93 = 1.064051870011093
B94 = 0.93
C94 = 2.1743234311534527 rad
A94 = 1.0842164579027642
B95 = 0.94
C95 = 2.163011097561561 rad
A95 = 1.1044934289208923
B96 = 0.95
C96 = 2.151664590156004 rad
A96 = 1.1248802041687784
B97 = 0.96
C97 = 2.140283228780617 rad
A97 = 1.14537418817197
B98 = 0.97
C98 = 2.128866318231248 rad
A98 = 1.165972768466399
B99 = 0.98
C99 = 2.117413147721498 rad
A99 = 1.186673315174017
B100 = 0.99
C100 = 2.105922990325035 rad
A100 = 1.2074731805654009
B101 = 1
C101 = 2.0943951023931957 rad
A101 = 1.2283696986087564
B102 = 1.01
C102 = 2.082828722946495 rad
A102 = 1.24936018450472
B103 = 1.02
C103 = 2.0712230730385937 rad
A103 = 1.2704419342063378
B104 = 1.03
C104 = 2.059577355091149 rad
A104 = 1.291612223923555
B105 = 1.04
C105 = 2.047890752197905 rad
A105 = 1.3128683096115095
B106 = 1.05
C106 = 2.036162427396227 rad
A106 = 1.3342074264419037
B107 = 1.06
C107 = 2.0243915229041916 rad
A107 = 1.3556267882566533
B108 = 1.07
C108 = 2.0125771593212023 rad
A108 = 1.377123587003016
B109 = 1.08
C109 = 2.000718434789949 rad
A109 = 1.398694992149307
B110 = 1.09
C110 = 1.9888144241173986 rad
A110 = 1.4203381500802732
B111 = 1.1
C111 = 1.976864177852306 rad
A111 = 1.4420501834711792
B112 = 1.11
C112 = 1.9648667213165865 rad
A112 = 1.4638281906395347
B113 = 1.12
C113 = 1.9528210535876687 rad
A113 = 1.4856692448733932
B114 = 1.13
C114 = 1.940726146428747 rad
A114 = 1.507570393735034
B115 = 1.14
C115 = 1.9285809431636196 rad
A115 = 1.529528658338815
B116 = 1.15
C116 = 1.9163843574925485 rad
A116 = 1.551541032601862
B117 = 1.16
C117 = 1.904135272245291 rad
A117 = 1.5736044824662185
B118 = 1.17
C118 = 1.8918325380671759 rad
A118 = 1.595715945090947
B119 = 1.18
C119 = 1.8794749720337502 rad
A119 = 1.617872328012633
B120 = 1.19
C120 = 1.8670613561891856 rad
A120 = 1.6400705082725766
B121 = 1.2
C121 = 1.854590436003224 rad
A121 = 1.6623073315088908
B122 = 1.21
C122 = 1.8420609187410475 rad
A122 = 1.68457961101159
B123 = 1.22
C123 = 1.8294714717399478 rad
A123 = 1.7068841267386277
B124 = 1.23
C124 = 1.816820720586215 rad
A124 = 1.7292176242906745
B125 = 1.24
C125 = 1.8041072471850497 rad
A125 = 1.7515768138423318
B126 = 1.25
C126 = 1.79132958771573 rad
A126 = 1.7739583690272274
B127 = 1.26
C127 = 1.7784862304635594 rad
A127 = 1.7963589257743706
B128 = 1.27
C128 = 1.76557561351939 rad
A128 = 1.8187750810928196
B129 = 1.28
C129 = 1.7525961223366813 rad
A129 = 1.8412033918016362
B130 = 1.29
C130 = 1.7395460871351542 rad
A130 = 1.8636403732017408
B131 = 1.3
C131 = 1.7264237801390818 rad
A131 = 1.8860824976861432
B132 = 1.31
C132 = 1.713227412637156 rad
A132 = 1.9085261932846875
B133 = 1.32
C133 = 1.699955131849614 rad
A133 = 1.9309678421391339
B134 = 1.33
C134 = 1.6866050175869631 rad
A134 = 1.9534037789041525
B135 = 1.34
C135 = 1.673175078683075 rad
A135 = 1.975830289069326
B136 = 1.35
C136 = 1.6596632491837529 rad
A136 = 1.9982436071970107
B137 = 1.36
C137 = 1.6460673842699516 rad
A137 = 2.020639915070302
B138 = 1.37
C138 = 1.6323852558927183 rad
A138 = 2.0430153397450472
B139 = 1.38
C139 = 1.6186145480945269 rad
A139 = 2.0653659514991682
B140 = 1.39
C140 = 1.6047528519890388 rad
A140 = 2.087687761672101
B141 = 1.4
C141 = 1.5907976603682872 rad
A141 = 2.109976720386428
B142 = 1.41
C142 = 1.5767463619029352 rad
A142 = 2.1322287141431397
B143 = 1.42
C143 = 1.5625962348974491 rad
A143 = 2.154439563281124
B144 = 1.43
C144 = 1.5483444405577105 rad
A144 = 2.176605019290632
B145 = 1.44
C145 = 1.5339880157237333 rad
A145 = 2.1987207619695317
B146 = 1.45
C146 = 1.51952386501463 rad
A146 = 2.2207823964099895
B147 = 1.46
C147 = 1.504948752326674 rad
A147 = 2.2427854498021293
B148 = 1.47
C148 = 1.4902592916181474 rad
A148 = 2.264725368039775
B149 = 1.48
C149 = 1.4754519369064973 rad
A149 = 2.2865975121119178
B150 = 1.49
C150 = 1.4605229713939294 rad
A150 = 2.3083971542618453
B151 = 1.5
C151 = 1.445468495626831 rad
A151 = 2.330119473893925
B152 = 1.51
C152 = 1.4302844145820413 rad
A152 = 2.3517595532059437
B153 = 1.52
C153 = 1.4149664235586858 rad
A153 = 2.37331237252237
B154 = 1.53
C154 = 1.3995099927377843 rad
A154 = 2.394772805301259
B155 = 1.54
C155 = 1.3839103502526338 rad
A155 = 2.416135612784258
B156 = 1.55
C156 = 1.3681624635906582 rad
A156 = 2.4373954382556615
B157 = 1.56
C157 = 1.3522610191213227 rad
A157 = 2.4585468008723876
B158 = 1.57
C158 = 1.3362003995141536 rad
A158 = 2.479584089021981
B159 = 1.58
C159 = 1.3199746587749968 rad
A159 = 2.5005015531604875
B160 = 1.59
C160 = 1.3035774945862413 rad
A160 = 2.52129329807566
B161 = 1.6
C161 = 1.2870022175865685 rad
A161 = 2.541953274514033
B162 = 1.61
C162 = 1.2702417171660851 rad
A162 = 2.5624752701017783
B163 = 1.62
C163 = 1.2532884232814812 rad
A163 = 2.582852899479827
B164 = 1.63
C164 = 1.2361342637104293 rad
A164 = 2.603079593562236
B165 = 1.64
C165 = 1.2187706160615899 rad
A165 = 2.6231485878135548
B166 = 1.65
C166 = 1.2011882537321017 rad
A166 = 2.643052909425287
B167 = 1.66
C167 = 1.1833772848530877 rad
A167 = 2.6627853632529686
B168 = 1.67
C168 = 1.1653270830786302 rad
A168 = 2.6823385163534166
B169 = 1.68
C169 = 1.1470262088461933 rad
A169 = 2.7017046809353102
B170 = 1.69
C170 = 1.1284623194551637 rad
A170 = 2.7208758955047982
B171 = 1.7
C171 = 1.109622065960143 rad
A171 = 2.7398439039495726
B172 = 1.71
C172 = 1.0904909744372564 rad
A172 = 2.7586001322587443
B173 = 1.72
C173 = 1.0710533086287755 rad
A173 = 2.7771356625191945
B174 = 1.73
C174 = 1.051291910268727 rad
A174 = 2.795441203759829
B175 = 1.74
C175 = 1.0311880124921808 rad
A175 = 2.8135070591290496
B176 = 1.75
C176 = 1.0107210205683146 rad
A176 = 2.8313230887838396
B177 = 1.76
C177 = 0.9898682526817907 rad
A177 = 2.8488786677340916
B178 = 1.77
C178 = 0.9686046314918398 rad
A178 = 2.8661626377154654
B179 = 1.78
C179 = 0.9469023145441323 rad
A179 = 2.883163251945803
B180 = 1.79
C180 = 0.9247302480388545 rad
A180 = 2.899868111338648
B181 = 1.8
C181 = 0.9020536235925247 rad
A181 = 2.9162640903798396
B182 = 1.81
C182 = 0.8788332109125462 rad
A182 = 2.932337250387654
B183 = 1.82
C183 = 0.8550245298897389 rad
A183 = 2.9480727372281863
B184 = 1.83
C184 = 0.8305768122101796 rad
A184 = 2.9634546596772355
B185 = 1.84
C185 = 0.805431683161323 rad
A185 = 2.9784659434085916
B186 = 1.85
C186 = 0.779521465594949 rad
A186 = 2.9930881538915566
B187 = 1.86
C187 = 0.7527669646354556 rad
A187 = 3.007301279059755
B188 = 1.87
C188 = 0.725074524575948 rad
A188 = 3.0210834590848434
B189 = 1.88
C189 = 0.6963320425459221 rad
A189 = 3.034410645323902
B190 = 1.89
C190 = 0.6664034480411005 rad
A190 = 3.047256162436586
B191 = 1.9
C191 = 0.6351208585830429 rad
A191 = 3.0595901349012964
B192 = 1.91
C192 = 0.6022730911067443 rad
A192 = 3.071378718227684
B193 = 1.92
C193 = 0.5675882184166552 rad
A193 = 3.082583039358717
B194 = 1.93
C194 = 0.5307058991952994 rad
A194 = 3.093157686331765
B195 = 1.94
C195 = 0.49113103503058253 rad
A195 = 3.1030484639714047
B196 = 1.95
C196 = 0.4481505706036376 rad
A196 = 3.112188877502186
B197 = 1.96
C197 = 0.4006696846462404 rad
A197 = 3.1204942238662565
B198 = 1.97
C198 = 0.34684464219120814 rad
A198 = 3.1278506489817883
B199 = 1.98
C199 = 0.28307894664885364 rad
A199 = 3.1340916857646377
B200 = 1.99
C200 = 0.20008342722307618 rad
A200 = 3.1389333221031617
B201 = 2
C201 = 0.20008342722307618 rad
A201 = NaN
B202 = 1.99
C202 = 0.20008342722307618 rad
A202 = 3.1389333221031617
B203 = 1.98
C203 = 0.28307894664885364 rad
A203 = 3.1340916857646377
B204 = 1.97
C204 = 0.34684464219120814 rad
A204 = 3.1278506489817883
B205 = 1.96
C205 = 0.4006696846462404 rad
A205 = 3.1204942238662565
B206 = 1.95
C206 = 0.4481505706036376 rad
A206 = 3.112188877502186
B207 = 1.94
C207 = 0.49113103503058253 rad
A207 = 3.1030484639714047
B208 = 1.93
C208 = 0.5307058991952994 rad
A208 = 3.093157686331765
B209 = 1.92
C209 = 0.5675882184166552 rad
A209 = 3.082583039358717
B210 = 1.91
C210 = 0.6022730911067443 rad
A210 = 3.071378718227684
B211 = 1.9
C211 = 0.6351208585830429 rad
A211 = 3.0595901349012964
B212 = 1.89
C212 = 0.6664034480411005 rad
A212 = 3.047256162436586
B213 = 1.88
C213 = 0.6963320425459221 rad
A213 = 3.034410645323902
B214 = 1.87
C214 = 0.725074524575948 rad
A214 = 3.0210834590848434
B215 = 1.86
C215 = 0.7527669646354556 rad
A215 = 3.007301279059755
B216 = 1.85
C216 = 0.779521465594949 rad
A216 = 2.9930881538915566
B217 = 1.84
C217 = 0.805431683161323 rad
A217 = 2.9784659434085916
B218 = 1.83
C218 = 0.8305768122101796 rad
A218 = 2.9634546596772355
B219 = 1.82
C219 = 0.8550245298897389 rad
A219 = 2.9480727372281863
B220 = 1.81
C220 = 0.8788332109125462 rad
A220 = 2.932337250387654
B221 = 1.8
C221 = 0.9020536235925247 rad
A221 = 2.9162640903798396
B222 = 1.79
C222 = 0.9247302480388545 rad
A222 = 2.899868111338648
H = Point(c)
texte17: "S_{brout}=" + (LaTeX[S_{brout}]) + "\\
\widehat{BPA}=" + (LaTeX[α]) + "\\
R=" + (LaTeX[R]) + ""