Un \;rectangle\;variable\;dans\;un\;quart\;de\;cercle

A

O

M

B

On \;désigne\;par:\
*\;x=\alpha=\widehat{AOM}\;\;(en\;rad)\
*\;A:\;le\;point\;de\;coordonnées\;(x\;,\;a(x))\
*\;B:\;le\;point\;de\;coordonnées\;(x\;,\;p(x))

Conjecture:\
*\;1^{ère}\;méthode:d'après\;les\;traces\;des\;points\;A\;et\;P\
le\;maximum\;de\;a(x)\;est\;\;0.5\;et\;celui\;de\;p(x)\;est\;2.828\
ils \;sont\;atteints\;pour\;la\;même\;valeur\;de\;x=0.785\;rad\
*\;2^{ème}\;Méthode\:d'après \;un \;tableur \;de \;pas \;10^{-2} \;\;(ligne\;81)    \
le \;maximum \;de\; a(x) \;est\;\; 0.49998   \;et\;celui\; de\;p(x) \;est\;\;  2.8284  \
ils \;sont \;atteints\; pour\;la \;même\;valeur\;de\; x=0.79\;rad

C

S

α = 0.12 (slider from 0 to 1.5707963267948966, step 0.01)
O = (0, 0)
A' = (1, 0)
B = (0, 1)
c = CircleArc(O, A', B)
M = Rotate(A', α, O)
α_1 = Angle(A', O, M)
b = Segment(O, M)
d = OrthogonalLine(M, xAxis)
e = OrthogonalLine(M, yAxis)
S = Intersect(e, yAxis)
C = Intersect(d, xAxis)
poly1, o, c_1, m, s = Polygon(O, C, M, S)
périmètrepoly1 = Perimeter(poly1)
Textepoly1: "Périmètre de  " + (Name[O]) + (Name[C]) + (Name[M]) + (Name[S]) + " = " + périmètrepoly1
P: (α, périmètrepoly1)
A: (α, poly1)
f = OrthogonalLine(P, yAxis)
g = OrthogonalLine(P, xAxis)
h = OrthogonalLine(A, yAxis)
D = Intersect(f, yAxis)
E = Intersect(h, yAxis)
F = Intersect(g, xAxis)
i = Segment(P, D)
j = Segment(P, F)
k = Segment(A, E)
texte7: "P(" + (LaTeX[α]) + "\;,\;" + (LaTeX[périmètrepoly1]) + ")"
texte8: "A(" + (LaTeX[α]) + "\;,\;" + (LaTeX[poly1]) + ")"
texte9: "p(x)=" + (LaTeX[périmètrepoly1]) + ""
texte10: "a(x)=" + (LaTeX[poly1]) + ""
texte11: "x=" + (LaTeX[α]) + ""
G = PointIn(poly1)
texte12: "\widehat{AOM}=" + (LaTeX[α]) + "\;rad"
texte13: "Pour\;x=" + (LaTeX[α]) + ":\;\;\;a(x)=" + (LaTeX[poly1]) + "\;\;\;et\;\;\;p(x)=" + (LaTeX[périmètrepoly1]) + ""
A1 = Name(α)
B1 = Name(poly1)
C1 = Name(périmètrepoly1)
A2 = 0
B2 = 0
C2 = 2
A3 = 0.01
B3 = 0.00999933334666654
C3 = 2.019899667501664
A4 = 0.02
B4 = 0.01999466709331708
C4 = 2.039597346719822
A5 = 0.03
B5 = 0.0299820032397223
C5 = 2.0590910679029664
A6 = 0.04
B6 = 0.03995734698458634
C6 = 2.0783788816952242
A7 = 0.05
B7 = 0.04991670832341408
C7 = 2.0974588593312893
A8 = 0.06
B8 = 0.05985610364445968
C8 = 2.1163290928292975
A9 = 0.07
B9 = 0.06977155732211825
C9 = 2.134987695181625
A10 = 0.08
B10 = 0.07965910330712299
C10 = 2.1534328005435843
A11 = 0.09
B11 = 0.08951478671291208
C11 = 2.1716625644200107
A12 = 0.1
B12 = 0.09933466539753062
C12 = 2.189675163849708
A13 = 0.11
B13 = 0.10911481154043466
C13 = 2.2074687975877434
A14 = 0.12
B14 = 0.11885131321356729
C14 = 2.225041686285571
A15 = 0.13
B15 = 0.12854027594607756
C15 = 2.2423920726689657
A16 = 0.14
B16 = 0.13817782428205688
C16 = 2.2595182217137473
A17 = 0.15
B17 = 0.14776010333066977
C17 = 2.276418420819283
A18 = 0.16
B18 = 0.15728328030805888
C18 = 2.2930909799797456
A19 = 0.17
B19 = 0.16674354607040723
C19 = 2.309534231953114
A20 = 0.18
B20 = 0.176137116637545
C20 = 2.325746532427891
A21 = 0.19
B21 = 0.18546023470649134
C21 = 2.3417262601875417
A22 = 0.2
B22 = 0.19470917115432523
C22 = 2.3574718172726055
A23 = 0.21
B23 = 0.20388022652978507
C23 = 2.3729816291404955
A24 = 0.22
B24 = 0.2129697325329998
C24 = 2.3882541448229495
A25 = 0.23
B25 = 0.22197405348275992
C25 = 2.403287837081127
A26 = 0.24
B26 = 0.23088958777074145
C26 = 2.4180812025583283
A27 = 0.25
B27 = 0.2397127693021015
C27 = 2.432632761930335
A28 = 0.26
B28 = 0.24844006892186837
C28 = 2.446941060053337
A29 = 0.27
B29 = 0.2570679958265566
C29 = 2.461004666109443
A30 = 0.28
B30 = 0.2655930989604417
C30 = 2.4748221737497693
A31 = 0.29
B31 = 0.2740119683959368
C31 = 2.4883922012350657
A32 = 0.3
B32 = 0.28232123669751763
C32 = 2.501713391573891
A33 = 0.31
B33 = 0.29051758026865254
C33 = 2.514784412658314
A34 = 0.32
B34 = 0.298597720681196
C34 = 2.527603957397117
A35 = 0.33
B35 = 0.30655842598671695
C35 = 2.540170743846511
A36 = 0.34
B36 = 0.31439651200923424
C36 = 2.552483515338321
A37 = 0.35
B37 = 0.3221088436188455
C37 = 2.5645410406056603
A38 = 0.36
B38 = 0.32969233598573655
C38 = 2.5763421139060494
A39 = 0.37
B39 = 0.3371439558140726
C39 = 2.587885555141993
A40 = 0.38
B40 = 0.34446072255527566
C40 = 2.5991702099789857
A41 = 0.39
B41 = 0.35163970960020513
C41 = 2.6101949499609494
A42 = 0.4
B42 = 0.3586780454497614
C42 = 2.6209586726230714
A43 = 0.41
B43 = 0.36557291486344795
C43 = 2.631460301602056
A44 = 0.42
B44 = 0.3723215599854296
C44 = 2.6416987867437567
A45 = 0.43
B45 = 0.3789212814476385
C45 = 2.6516731042081916
A46 = 0.44
B46 = 0.38536943944948465
C46 = 2.661382256571926
A47 = 0.45
B47 = 0.3916634548137417
C47 = 2.6708252729278144
A48 = 0.46
B48 = 0.397800810018183
C48 = 2.68000120898209
A49 = 0.47
B49 = 0.40377905020255717
C49 = 2.6889091471487943
A50 = 0.48
B50 = 0.40959578415049913
C50 = 2.697548196641534
A51 = 0.49
B51 = 0.4152486852459852
C51 = 2.705917493562559
A52 = 0.5
B52 = 0.42073549240394825
C52 = 2.7140162009891515
A53 = 0.51
B53 = 0.4260540109746815
C53 = 2.7218435090573174
A54 = 0.52
B54 = 0.4312021136216692
C54 = 2.7293986350427732
A55 = 0.53
B55 = 0.43617774117249314
C55 = 2.7366808234392157
A56 = 0.54
B56 = 0.4409789034424738
C56 = 2.7436893460338743
A57 = 0.55
B57 = 0.4456036800307177
C57 = 2.75042350198033
A58 = 0.56
B58 = 0.4500502210882525
C58 = 2.756882617868599
A59 = 0.57
B59 = 0.45431674805794164
C59 = 2.763066047792476
A60 = 0.58
B60 = 0.4584015543858835
C60 = 2.768973173414121
A61 = 0.59
B61 = 0.4623030062040102
C61 = 2.7746034040258944
A62 = 0.6
B62 = 0.4660195429836132
C62 = 2.7799561766094274
A63 = 0.61
B63 = 0.4695496781595338
C63 = 2.7850309558919215
A64 = 0.62
B64 = 0.4728919997247695
C64 = 2.789827234399678
A65 = 0.63
B65 = 0.4760451707952579
C65 = 2.794344532508843
A66 = 0.64
B66 = 0.47900793014461257
C66 = 2.7985823984933695
A67 = 0.65
B67 = 0.4817790927085965
C67 = 2.802540408570191
A68 = 0.66
B68 = 0.4843575500591326
C68 = 2.8062181669415978
A69 = 0.67
B69 = 0.4867422708476597
C69 = 2.809615305834818
A70 = 0.68
B70 = 0.4889323012176581
C70 = 2.812731485538793
A71 = 0.69
B71 = 0.49092676518617984
C71 = 2.8155663944381493
A72 = 0.7
B72 = 0.4927248649942301
C72 = 2.8181197490443592
A73 = 0.71
B73 = 0.49432588142585987
C73 = 2.82039129402409
A74 = 0.72
B74 = 0.4957291740958432
C74 = 2.8223808022247363
A75 = 0.73
B75 = 0.49693418170582243
C75 = 2.8240880746971366
A76 = 0.74
B76 = 0.49794042226882
C76 = 2.825512940715466
A77 = 0.75
B77 = 0.49874749330202717
C77 = 2.82665525779431
A78 = 0.76
B78 = 0.4993550719877915
C78 = 2.827514911702913
A79 = 0.77
B79 = 0.4997629153027395
C79 = 2.8280918164766
A80 = 0.78
B80 = 0.4999708601149831
C80 = 2.828385914425375
A81 = 0.79
B81 = 0.49997882324937
C81 = 2.8283971761396876
A82 = 0.8
B82 = 0.49978680152075255
C82 = 2.8281256004933764
A83 = 0.81
B83 = 0.499394871735262
C83 = 2.827571214643779
A84 = 0.82
B84 = 0.4988031906595868
C84 = 2.8267340740290186
A85 = 0.83
B85 = 0.49801199495826837
C85 = 2.8256142623624596
A86 = 0.84
B86 = 0.497021601099038
C86 = 2.824211891624335
A87 = 0.85
B87 = 0.49583240522623434
C87 = 2.82252710205055
A88 = 0.86
B88 = 0.4944448830023508
C88 = 2.820560062118658
A89 = 0.87
B89 = 0.49285958941777674
C89 = 2.8183109685310126
A90 = 0.88
B90 = 0.49107715856880924
C90 = 2.815780046195099
A91 = 0.89
B91 = 0.4890983034040223
C91 = 2.8129675482010414
A92 = 0.9
B92 = 0.4869238154390976
C92 = 2.8098737557962954
A93 = 0.91
B93 = 0.4845545644402282
C93 = 2.8064989783575243
A94 = 0.92
B94 = 0.48199149807622405
C94 = 2.802843553359658
A95 = 0.93
B95 = 0.4792356415394571
C95 = 2.798907846342151
A96 = 0.94
B96 = 0.4762880971357977
C96 = 2.794692250872425
A97 = 0.95
B97 = 0.47315004384370724
C97 = 2.7901971885065144
A98 = 0.96
B98 = 0.46982273684266246
C98 = 2.78542310874691
A99 = 0.97
B99 = 0.4663075070111003
C99 = 2.780370488997609
A100 = 0.98
B100 = 0.46260576039408413
C100 = 2.7750398345163756
A101 = 0.99
B101 = 0.4587189776409049
C101 = 2.7694316783642163
A102 = 1
B102 = 0.4546487134128409
C102 = 2.7635465813520725
A103 = 1.01
B103 = 0.4503965957613137
C103 = 2.7573851319847416
A104 = 1.02
B104 = 0.4459643254766898
C104 = 2.7509479464020252
A105 = 1.03
B105 = 0.44135367540798703
C105 = 2.7442356683171174
A106 = 1.04
B106 = 0.4365664897537583
C106 = 2.7372489689522337
A107 = 1.05
B107 = 0.4316046833244369
C107 = 2.7299885469714877
A108 = 1.06
B108 = 0.4264702407764381
C108 = 2.7224551284110277
A109 = 1.07
B109 = 0.42116521581832284
C109 = 2.7146494666064314
A110 = 1.08
B110 = 0.4156917303893416
C110 = 2.706572342117375
A111 = 1.09
B111 = 0.41005197381068703
C111 = 2.698224562649576
A112 = 1.1
B112 = 0.40424820190979505
C112 = 2.6896069629740254
A113 = 1.11
B113 = 0.39828273611804327
C113 = 2.680720404843509
A114 = 1.12
B114 = 0.39215796254220997
C114 = 2.6715657769064345
A115 = 1.13
B115 = 0.38587633101006297
C115 = 2.6621439946179666
A116 = 1.14
B116 = 0.379440354090461
C116 = 2.652456000148483
A117 = 1.15
B117 = 0.37285260608836013
C117 = 2.642502762289357
A118 = 1.16
B118 = 0.36611572201512566
C118 = 2.6322852763560802
A119 = 1.17
B119 = 0.3592323965345631
C119 = 2.6218045640887317
A120 = 1.18
B120 = 0.3522053828850881
C120 = 2.6110616735498042
A121 = 1.19
B121 = 0.3450374917784682
C121 = 2.6000576790193994
A122 = 1.2
B122 = 0.3377315902755755
C122 = 2.5887936808878
A123 = 1.21
B123 = 0.33029060063960036
C123 = 2.5772708055454325
A124 = 1.22
B124 = 0.3227174991671854
C124 = 2.5654902052702293
A125 = 1.23
B125 = 0.3150153149979461
C125 = 2.5534530581124004
A126 = 1.24
B126 = 0.3071871289028558
C126 = 2.5411605677766302
A127 = 1.25
B127 = 0.2992360720519783
C127 = 2.5286139635017095
A128 = 1.26
B128 = 0.29116532476204093
C128 = 2.51581449993761
A129 = 1.27
B129 = 0.2829781152243514
C129 = 2.502763457020022
A130 = 1.28
B130 = 0.27467771821356335
C130 = 2.489462139842361
A131 = 1.29
B131 = 0.2662674537778105
C131 = 2.4759118785252605
A132 = 1.3
B132 = 0.2577506859107321
C132 = 2.4621140280835605
A133 = 1.31
B133 = 0.24913082120591926
C133 = 2.448069968290807
A134 = 1.32
B134 = 0.2404113074943241
C134 = 2.433781103541276
A135 = 1.33
B135 = 0.23159563246517256
C135 = 2.4192488627095354
A136 = 1.34
B136 = 0.22268732227093557
C136 = 2.4044746990075576
A137 = 1.35
B137 = 0.2136899401169149
C137 = 2.3894600898394014
A138 = 1.36
B138 = 0.20460708483600867
C138 = 2.374206536653471
A139 = 1.37
B139 = 0.1954423894492261
C139 = 2.358715564792374
A140 = 1.38
B140 = 0.18619951971252785
C140 = 2.342988723340388
A141 = 1.39
B141 = 0.17688217265057155
C141 = 2.327027584968552
A142 = 1.4
B142 = 0.16749407507795255
C142 = 2.310833745777402
A143 = 1.41
B143 = 0.1580389821085269
C143 = 2.294408825137363
A144 = 1.42
B144 = 0.14852067565341617
C144 = 2.277754465526811
A145 = 1.43
B145 = 0.1389429629082934
C145 = 2.260872332367829
A146 = 1.44
B146 = 0.1293096748305554
C146 = 2.243764113859664
A147 = 1.45
B147 = 0.11962466460699121
C147 = 2.22643152080991
A148 = 1.46
B148 = 0.10989180611255848
C148 = 2.208876286463429
A149 = 1.47
B149 = 0.10011499236088527
C149 = 2.1911001663290253
A150 = 1.48
B150 = 0.09029813394711647
C150 = 2.1731049380038994
A151 = 1.49
B151 = 0.0804451574837279
C151 = 2.1548924009958883
A152 = 1.5
B152 = 0.0705600040299336
C152 = 2.1364643765435147
A153 = 1.51
B153 = 0.060646627515314876
C153 = 2.1178227074338682
A154 = 1.52
B154 = 0.050708993158300936
C154 = 2.0989692578183243
A155 = 1.53
B155 = 0.04075107588013457
C155 = 2.079905913026134
A156 = 1.54
B156 = 0.03077685871495657
C156 = 2.0606345793758902
A157 = 1.55
B157 = 0.020790331216645246
C157 = 2.041157183984899
A158 = 1.56
B158 = 0.01079548786304798
C158 = 2.0214756745764673
A159 = 1.57
B159 = 7.963264582434141E-4
C159 = 2.0015920192851357