\n\u2666 Existe-t-il une position pour le point B o\u00f9 l’aire de l’enclos est la plus grande ?\n\t\t\t\t
![\"EnclosImageMiseEnAvant\"](\"https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-1.png\" "\\"EnclosImageMiseEnAvant\\"")
\n\t\t\t\n\t\t\t\t\t\tRetour Panier \u00e0 probl\u00e8mes\n\t\t\t\t\t<\/a>\n\n\t\t\tLes math\u00e9matiques travaill\u00e9es ou \u00e0 travailler\t<\/h2>\n\t\u2666 Production de formules et\/ou de m\u00e9thodes en utilisant le calcul litt\u00e9ral (expression alg\u00e9brique d’une fonction)
\n\u2666 Formule d’aire et de p\u00e9rim\u00e8tre du rectangle
\n\u2666 Utilisation du tableur (tableau de valeurs d’une fonction)
\n\u2666 Utilisation du grapheur (repr\u00e9sentation graphique d’une fonction) Ce registre sera normalement moins utilis\u00e9 par les \u00e9l\u00e8ves car la solution d\u00e9cimale reste relativement simple \u00e0 trouver : 5+1\/4\n\n\t\t\tSolutions possibles\t<\/h2>\n\n\t\t\tSolution experte\t<\/h3>\n\tIl s’agit d’\u00e9tudier ici la fonction f d\u00e9finie, sur l’intervalle [0; 21\/2 ] par : f(x) = x(21 – 2x)
\nUne \u00e9tude des variations de f montre qu’elle admet un maximum en 5,25 et qu’en ce point la valeur de cette fonction est 55,125.
\nConclusion : L’aire est maximale quand le piquet B est plac\u00e9 \u00e0 5,25 m du mur et dans ce cas l’aire est 55,125 m2<\/sup>.\n\n\t\t\tAnalyse a priori des strat\u00e9gies des \u00e9l\u00e8ves\t<\/h3>\n\t\u2666 Par essais successifs : On arrive facilement \u00e0 la conclusion que la position du piquet B influe sur l’aire de l’enclos et celle-ci est assez grande s’il est \u00e9loign\u00e9 \u00e0 5 m du poteau.
\n\u2666 Mod\u00e9lisation de la situation la situation \u00e0 l’aide d’un tableur en saisissant comme formule < Bi =Ai*(21-2*Ai) >. Le pas de la colonne A est \u00e0 la charge de l’\u00e9l\u00e8ve.
\n\u2666 En plus du travail sur le tableur, l’\u00e9l\u00e8ve peut en faire l’interpr\u00e9tation graphique et visualiser le maximum.
\n\u2666 Utilisation de la sym\u00e9trie : les valeurs extr\u00eames (0 et 10,5) donnant lieu \u00e0 des situations identiques, le cas m\u00e9dian (5,25) est optimal (car on se trouve implicitement \u00e0 \u00e9tudier une fonction du second degr\u00e9 ayant pour axe de sym\u00e9trie la droite d’\u00e9quation x=5,25).\n\t\t\t\n\t\t\t\t\t\t\tT\u00e9l\u00e9charcher le document complet\n\t\t\t<\/a>\n\n","protected":false},"excerpt":{"rendered":"Avec 21 m de grillage et un mur comme quatri\u00e8me c\u00f4t\u00e9, o\u00f9 placer le point B pour obtenir l\u2019enclos rectangulaire le plus grand ? L\u2019aire varie-t-elle selon sa position ?<\/p>\n","protected":false},"author":3,"featured_media":2542,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[107],"tags":[],"niveau":[113],"type_situation":[116],"theme_maths":[121,119],"probleme_associe":[],"type-expe":[],"class_list":{"0":"post-3173","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-situation-probleme","8":"niveau-cycle-4","9":"type_situation-situation-probleme","10":"theme_maths-grandeurs-et-mesures","11":"theme_maths-ogd-fonctions","12":"czr-hentry"},"rttpg_featured_image_url":{"full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"landscape":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"portraits":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"thumbnail":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-150x150.png",150,150,true],"medium":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-300x218.png",300,218,true],"large":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"1536x1536":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"2048x2048":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"tc-grid-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-722x444.png",722,444,true],"tc-grid":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-570x350.png",570,350,true],"tc-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-270x250.png",270,250,true],"slider-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-722x200.png",722,200,true],"slider":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-722x200.png",722,200,true],"tc-sq-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-510x510.png",510,510,true],"tc-ws-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"tc-ws-small-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-528x297.png",528,297,true],"tc-slider-small":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-517x94.png",517,94,true]},"rttpg_author":{"display_name":"St\u00e9phanie CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"Situation Probl\u00e8me<\/a>","rttpg_excerpt":"Avec 21 m de grillage et un mur comme quatri\u00e8me c\u00f4t\u00e9, o\u00f9 placer le point B pour obtenir l\u2019enclos rectangulaire le plus grand ? L\u2019aire varie-t-elle selon sa position ?","_links":{"self":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3173"}],"version-history":[{"count":11,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3173\/revisions"}],"predecessor-version":[{"id":3829,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3173\/revisions\/3829"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/media\/2542"}],"wp:attachment":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3173"},{"taxonomy":"niveau","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fniveau&post=3173"},{"taxonomy":"type_situation","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype_situation&post=3173"},{"taxonomy":"theme_maths","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftheme_maths&post=3173"},{"taxonomy":"probleme_associe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fprobleme_associe&post=3173"},{"taxonomy":"type-expe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype-expe&post=3173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\t\t\tSolution experte\t<\/h3>\n\tIl s’agit d’\u00e9tudier ici la fonction f d\u00e9finie, sur l’intervalle [0; 21\/2 ] par : f(x) = x(21 – 2x)
\nUne \u00e9tude des variations de f montre qu’elle admet un maximum en 5,25 et qu’en ce point la valeur de cette fonction est 55,125.
\nConclusion : L’aire est maximale quand le piquet B est plac\u00e9 \u00e0 5,25 m du mur et dans ce cas l’aire est 55,125 m2<\/sup>.\n\n\t\t\tAnalyse a priori des strat\u00e9gies des \u00e9l\u00e8ves\t<\/h3>\n\t\u2666 Par essais successifs : On arrive facilement \u00e0 la conclusion que la position du piquet B influe sur l’aire de l’enclos et celle-ci est assez grande s’il est \u00e9loign\u00e9 \u00e0 5 m du poteau.
\n\u2666 Mod\u00e9lisation de la situation la situation \u00e0 l’aide d’un tableur en saisissant comme formule < Bi =Ai*(21-2*Ai) >. Le pas de la colonne A est \u00e0 la charge de l’\u00e9l\u00e8ve.
\n\u2666 En plus du travail sur le tableur, l’\u00e9l\u00e8ve peut en faire l’interpr\u00e9tation graphique et visualiser le maximum.
\n\u2666 Utilisation de la sym\u00e9trie : les valeurs extr\u00eames (0 et 10,5) donnant lieu \u00e0 des situations identiques, le cas m\u00e9dian (5,25) est optimal (car on se trouve implicitement \u00e0 \u00e9tudier une fonction du second degr\u00e9 ayant pour axe de sym\u00e9trie la droite d’\u00e9quation x=5,25).\n\t\t\t\n\t\t\t\t\t\t\tT\u00e9l\u00e9charcher le document complet\n\t\t\t<\/a>\n\n","protected":false},"excerpt":{"rendered":"Avec 21 m de grillage et un mur comme quatri\u00e8me c\u00f4t\u00e9, o\u00f9 placer le point B pour obtenir l\u2019enclos rectangulaire le plus grand ? L\u2019aire varie-t-elle selon sa position ?<\/p>\n","protected":false},"author":3,"featured_media":2542,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[107],"tags":[],"niveau":[113],"type_situation":[116],"theme_maths":[121,119],"probleme_associe":[],"type-expe":[],"class_list":{"0":"post-3173","1":"post","2":"type-post","3":"status-publish","4":"format-standard","5":"has-post-thumbnail","7":"category-situation-probleme","8":"niveau-cycle-4","9":"type_situation-situation-probleme","10":"theme_maths-grandeurs-et-mesures","11":"theme_maths-ogd-fonctions","12":"czr-hentry"},"rttpg_featured_image_url":{"full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"landscape":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"portraits":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"thumbnail":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-150x150.png",150,150,true],"medium":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-300x218.png",300,218,true],"large":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"1536x1536":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"2048x2048":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"tc-grid-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-722x444.png",722,444,true],"tc-grid":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-570x350.png",570,350,true],"tc-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-270x250.png",270,250,true],"slider-full":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-722x200.png",722,200,true],"slider":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-722x200.png",722,200,true],"tc-sq-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-510x510.png",510,510,true],"tc-ws-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant.png",722,525,false],"tc-ws-small-thumb":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-528x297.png",528,297,true],"tc-slider-small":["https:\/\/math.univ-lyon1.fr\/dream\/wp-content\/uploads\/2023\/02\/EnclosImageMiseEnAvant-517x94.png",517,94,true]},"rttpg_author":{"display_name":"St\u00e9phanie CROQUELOIS","author_link":"https:\/\/math.univ-lyon1.fr\/dream\/?author=3"},"rttpg_comment":0,"rttpg_category":"Situation Probl\u00e8me<\/a>","rttpg_excerpt":"Avec 21 m de grillage et un mur comme quatri\u00e8me c\u00f4t\u00e9, o\u00f9 placer le point B pour obtenir l\u2019enclos rectangulaire le plus grand ? L\u2019aire varie-t-elle selon sa position ?","_links":{"self":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3173","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/users\/3"}],"replies":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=3173"}],"version-history":[{"count":11,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3173\/revisions"}],"predecessor-version":[{"id":3829,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/posts\/3173\/revisions\/3829"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=\/wp\/v2\/media\/2542"}],"wp:attachment":[{"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=3173"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=3173"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=3173"},{"taxonomy":"niveau","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fniveau&post=3173"},{"taxonomy":"type_situation","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype_situation&post=3173"},{"taxonomy":"theme_maths","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftheme_maths&post=3173"},{"taxonomy":"probleme_associe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Fprobleme_associe&post=3173"},{"taxonomy":"type-expe","embeddable":true,"href":"https:\/\/math.univ-lyon1.fr\/dream\/index.php?rest_route=%2Fwp%2Fv2%2Ftype-expe&post=3173"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n\u2666 Mod\u00e9lisation de la situation la situation \u00e0 l’aide d’un tableur en saisissant comme formule < Bi =Ai*(21-2*Ai) >. Le pas de la colonne A est \u00e0 la charge de l’\u00e9l\u00e8ve.
\n\u2666 En plus du travail sur le tableur, l’\u00e9l\u00e8ve peut en faire l’interpr\u00e9tation graphique et visualiser le maximum.
\n\u2666 Utilisation de la sym\u00e9trie : les valeurs extr\u00eames (0 et 10,5) donnant lieu \u00e0 des situations identiques, le cas m\u00e9dian (5,25) est optimal (car on se trouve implicitement \u00e0 \u00e9tudier une fonction du second degr\u00e9 ayant pour axe de sym\u00e9trie la droite d’\u00e9quation x=5,25).\n\t\t\t\n\t\t\t\t\t\t\tT\u00e9l\u00e9charcher le document complet\n\t\t\t<\/a>\n\n","protected":false},"excerpt":{"rendered":"
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