Géométrie dans l'espace

Pour aller plus loin (Ancien programme) - Mathématiques 5e

Exercice 1 : Lire les coordonnées géographique d'une ville sur terre à partir des parallèles et méridiens

Sur le schéma ci-dessous, le point D symbolise la ville de Chifeng (Chine).

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{"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": 15.320888862379553, "y": 25.71150438746157, "z": 26.53655792675754}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}

On pourra faire pivoter la figure en cliquant dessus.

Légende

Méridien d'origine Greenwich 0°
Méridiens (espacés de 20°)
Équateur 0°
Parallèles (espacés de 20°)
N : Pôle nord
S : Pôle sud
D : Ville de Chifeng

Donner les coordonnées géographique de la ville de Chifeng (Chine).
Par exemple:

Pour la ville de São José dos Campos (Brésil) on donnera une réponse de la forme \(\left(23° S; 46° O\right)\)

Pour la ville de Nouakchott (Mauritanie) on donnera une réponse de la forme \(\left(18° N; 16° O\right)\)

Exercice 2 : Calcul du volume d'une boule

Soit une boule de \(15 cm\) de rayon.

Calculer son volume.
On donnera le résultat exact.

Exercice 3 : Calculer le rayon d'un parallèle

On assimile la Terre à une sphère de rayon \( 6\:400\:\text{km} \).

On pourra faire pivoter la figure en cliquant dessus.

Légende

Équateur 0°
Parallèle passant par D
N : Pôle nord
S : Pôle sud
D : Ville d'Ujjain

Déterminer le rayon du parallèle passant à Ujjain de coordonnées (23° N, 76° E).
On arrondira donnera un résultat arrondi au \( km \).

Exercice 4 : Lire les coordonnées d'un point dans un plan 3D

Dans le repère ci-dessous, donner les coordonnées du point M sous la forme \((x ; y ; z)\).

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"label_render": {"visible": false}}, "Segment3D_89": {"type": "Segment", "pt1": {"x": 60.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 60.0, "y": 0.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_90": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 60.0}, "point_render": {"visible": false}}, "Label_for_Point3D_90": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 60.0}, "text": "10", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}}, "Segment3D_91": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 60.0}, "pt2": {"x": 0.0, "y": 1.2, "z": 60.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_92": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 60.0}, "pt2": {"x": 1.2, "y": 0.0, "z": 60.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_93": {"type": "Point", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}}, "Label_for_Point3D_93": {"type": "Label", "position": {"x": 0.0, "y": 0.0, "z": 0.0}, "text": "0", "position_offset": {"x": 0.0, "y": -3.52, "z": 0.0}}, "Segment3D_94": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 1.2, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_95": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 1.2, "y": 0.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Segment3D_96": {"type": "Segment", "pt1": {"x": 0.0, "y": 0.0, "z": 0.0}, "pt2": {"x": 0.0, "y": 0.0, "z": 1.2}, "point_render": {"visible": false}, "label_render": {"visible": false}}, "Point3D_97": {"type": "Point", "position": {"x": 36.0, "y": 42.0, "z": 12.0}, "point_render": {"color": 204}, "label_render": {"color": 204}}, "Label_for_Point3D_97": {"type": "Label", "position": {"x": 36.0, "y": 42.0, "z": 12.0}, "text": "M", "position_offset": {"x": 0.0, "y": 3.52, "z": 0.0}, "label_render": {"color": 204}}, "Point3D_98": {"type": "Point", "position": {"x": 36.0, "y": 0.0, "z": 12.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_99": {"type": "Point", "position": {"x": 36.0, "y": 42.0, "z": 0.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Point3D_100": {"type": "Point", "position": {"x": 0.0, "y": 42.0, "z": 12.0}, "point_render": {"color": 204}, "label_render": {"visible": false}}, "Segment3D_101": {"type": "Segment", "pt1": {"x": 36.0, "y": 42.0, "z": 12.0}, "pt2": {"x": 36.0, "y": 0.0, "z": 12.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_102": {"type": "Segment", "pt1": {"x": 36.0, "y": 42.0, "z": 12.0}, "pt2": {"x": 36.0, "y": 42.0, "z": 0.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}, "Segment3D_103": {"type": "Segment", "pt1": {"x": 36.0, "y": 42.0, "z": 12.0}, "pt2": {"x": 0.0, "y": 42.0, "z": 12.0}, "point_render": {"visible": false}, "label_render": {"visible": false}, "line_render": {"color": 204, "dashed": true}}}}

Exercice 5 : Lire les coordonnées géographiques d'une ville sur terre à partir des angles

Sur le schéma ci-dessous, le point D symbolise la ville de Puno (Pérou).

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3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_E": {"type": "Point", "position": {"x": -13.680805733026753, "y": 0.0, "z": -37.587704831436334}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_E": {"type": "Label", "position": {"x": -13.680805733026753, "y": 0.0, "z": -37.587704831436334}, "text": "E", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_A": {"type": "Point", "position": {"x": -38.450467837532756, "y": -11.025494232679966, "z": 0.0}, "point_render": {"color": 0, "size": 64}, "label_render": {"color": 0}}, "Label_for_Point3D_A": {"type": "Label", "position": {"x": -38.450467837532756, "y": -11.025494232679966, "z": 0.0}, "text": "A", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 0}}, "Point3D_D": {"type": "Point", "position": {"x": -13.150834520731973, "y": -11.025494232679966, "z": -36.13162089269543}, "point_render": {"color": 16711680, "size": 128}, "label_render": {"color": 16711680}}, "Label_for_Point3D_D": {"type": "Label", "position": {"x": -13.150834520731973, "y": -11.025494232679966, "z": -36.13162089269543}, "text": "D", "position_offset": {"x": 0.0, "y": 3.3941125496954276, "z": 3.3941125496954276}, "label_render": {"color": 16711680}}}}

On pourra faire pivoter la figure en cliquant dessus.

Légende

Méridien d'origine Greenwich 0°
Méridien du point D
Équateur 0°
Parallèle du point D
N : Pôle nord
S : Pôle sud
C : Centre de la terre
D : Ville de Puno

Sachant que \( \widehat{SCE} = 90 ° \), \( \widehat{DCS} = 74 ° \), \( \widehat{ECD} = 16 ° \) et \( \widehat{ABD} = 70 ° \), donner les coordonnées géographique de la ville de Puno (Pérou).
Par exemple:

Pour la ville de Le Cap (Afrique du Sud) on donnera une réponse de la forme \(\left(34° S; 18° E\right)\)

Pour la ville de Setúbal (Portugal) on donnera une réponse de la forme \(\left(39° N; 9° O\right)\)

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