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Zuletzt geändert von Dirk Tebbe am 2025/10/14 12:32
Von Version 23.1
bearbeitet von Dirk Tebbe
am 2025/10/14 08:37
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Auf Version 36.1
bearbeitet von Dirk Tebbe
am 2025/10/14 12:22
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Inhalt
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11 11  
12 12  == Elementargeometrie ==
13 13  
14 -{{aufgabe id="Zelt" afb="III" kompetenzen="K2,K3" quelle="KMap" cc="BY-SA" niveau="g" links="[[Interaktives Erkunden>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Optimieren#erkunden]]"}}
14 +{{aufgabe id="Optimierungsaufgabe beschreiben" afb="I" kompetenzen="K6" quelle="Martin Stern, Dirk Tebbe" cc="BY-SA" zeit="10"}}
15 15  
16 +Gegeben ist eine Funktion {{formula}}f{{/formula}}. Ihr Schaubild ist {{formula}}K_f{{/formula}}.
17 +
18 +Die Kurve und die x-Achse schließen eine Fläche ein. Zeichnen Sie in diese Fläche ein achsenparalleles Rechteck ein, von dem zwei Eckpunkte auf der Kurve und zwei Eckpunkte auf der x-Achse liegen.
19 +Erläutern Sie, wie man das Rechteck mit maximalem Flächeninhalt ermitteln kann.
20 +
21 +[[image:Optimierung_Beschreibung.svg||width="450"]]
22 +
23 +
24 +{{/aufgabe}}
25 +
26 +{{aufgabe id="Abstand zweier Kurvenpunkte berechnen" afb="I" kompetenzen="K4,K5" quelle="Martin" cc="BY-SA" zeit="15"}}
27 +
28 +Gegeben sind zwei Funktionen {{formula}}f{{/formula}} und {{formula}}g{{/formula}} durch {{formula}}f(x)=-e^{-0,25x}-0,5x+2{{/formula}} und {{formula}}g(x)=-0,5x+1{{/formula}}. Ihre Graphen sind {{formula}}K_f{{/formula}} und {{formula}}K_g{{/formula}}.
29 +Eine Gerade mit der Gleichung {{formula}}x=u{{/formula}} und {{formula}}-6\leq u \leq 3{{/formula}} schneidet {{formula}}K_f{{/formula}} im Punkt {{formula}}P{{/formula}} und {{formula}}K_g{{/formula}} im Punkt {{formula}}Q{{/formula}}. Berechnen Sie den maximalen Abstand der Punkte {{formula}}P{{/formula}} und {{formula}}Q{{/formula}}.
30 +[[image:Optimieren_Streckenlänge.svg||width="450"]]
31 +
32 +
33 +{{/aufgabe}}
34 +
35 +{{aufgabe id="Zelt" afb="III" kompetenzen="K2,K5" quelle="KMap" cc="BY-SA" zeit="15" links="[[Interaktives Erkunden>>https://kmap.eu/app/browser/Mathematik/Differentialrechnung/Optimieren#erkunden]]"}}
36 +
16 16  Für ein Zelt ist vorgegeben, dass es die Form einer senkrechten Pyramide mit quadratischer Grundfläche haben soll. Für diese Form soll nun bei einer gegebenen Zeltstangenlänge von 2,5 m das Volumen V maximiert werden, indem die Kantenlänge a der Grundfläche variiert wird. Folgende Formel gilt für das Volumen einer Pyramide:
17 17  
18 18  {{formula}}V= \frac{1}{3}\cdot a^2 \cdot h{{/formula}}
... ... @@ -22,15 +22,15 @@
22 22  1. Maximiere das Volumen! Gib dafür die Kantenlänge //a//, das Volumen //V// und die Höhe //h// an!
23 23  {{/aufgabe}}
24 24  
25 -{{aufgabe id="Zaun" afb="II" kompetenzen="K2,K3,K5" quelle="Martin Stern" cc="BY-SA" niveau="g"}}
46 +{{aufgabe id="Zaun" afb="II" kompetenzen="K4,K5" quelle="Martin Stern" cc="BY-SA" zeit="10"}}
26 26  Du hast 110 Meter Zaun zur Verfügung und möchtest eine Wiese für eine Schafweide einzäunen. Du wählst dafür eine Rechteckfläche. Auf einer Seite steht dafür eine Mauer zur Verfügung. Berechne die Längen der Zaunseiten so, dass der Flächeninhalt der eingezäunten Fläche maximal ist.
27 27  {{/aufgabe}}
28 28  
29 -{{aufgabe id="Rechteck unter Parabel" afb="II" kompetenzen="K2,K3,K5" quelle="Martin Stern" cc="BY-SA" niveau="g"}}
50 +{{aufgabe id="Rechteck unter Parabel" afb="II" kompetenzen="K4,K5" quelle="Martin Stern" cc="BY-SA" zeit="10"}}
30 30  Zwei Eckpunkte eines symmetrisch zur y-Achse liegenden Rechtecks sind auf der x-Achse, zwei Eckpunkte auf der Parabel mit der Gleichung {{formula}}y=-1,25x^2+5 {{/formula}} für {{formula}}-2<x<2 {{/formula}}. Der Flächeninhalt soll maximal sein. Wie lang müssen die Seiten des Rechtecks sein?
31 31  {{/aufgabe}}
32 32  
33 -{{aufgabe id="Lampen" afb="II" kompetenzen="K2,K3,K5" quelle="Martin Stern" cc="BY-SA" niveau="g"}}
54 +{{aufgabe id="Lampen" afb="III" kompetenzen="K3,K4,K5" quelle="Martin Stern" cc="BY-SA" zeit="20"}}
34 34  Eine Baumarktkette verkauft monatlich 1100 Stück einer Lampe zum Stückpreis von 30 €. Die Marketingabteilung hat durch eine Untersuchung festgestellt, dass sich der monatliche Absatz bei jeder Senkung des Preises um 1 € um 50 Stück erhöhen würde.
35 35  
36 36  1. Berechne den Stückpreis, bei dem die monatlichen Einnahmen am größten sind.
... ... @@ -37,13 +37,13 @@
37 37  1. Wie hoch sind die Einnahmen in diesem Fall?
38 38  {{/aufgabe}}
39 39  
40 -{{aufgabe id="Fluß" afb="III" kompetenzen="K2,K3,K5,K6" quelle="Martin Stern" cc="BY-SA" niveau="e"}}
61 +{{aufgabe id="Fluß" afb="III" kompetenzen="K2,K3,K5,K6" quelle="Martin Stern" cc="BY-SA" niveau="e" zeit="25"}}
41 41  Ein gut trainierter Sportler sonnt sich an einem Fluss, als er per Handy einen Hilferuf von seiner Freundin erhält. Diese befindet sich am anderen Ufer 1000 Meter flussabwärts. Er möchte möglichst schnell zu ihr gelangen. Der Fluss ist 500 Meter breit und verläuft in diesem Abschnitt gerade. Auf der anderen Seite des Flusses ist ein Weg. Der Sportler kann auf solch einem Weg 300 Meter in einer Minute zurücklegen. Schwimmend erreicht er eine Geschwindigkeit von 50 m/min.
42 42  
43 43  Bestimme die minimale Zeit sowie die Länge der zugehörigen Gesamtstrecke, die unser Held zu seiner Freundin zurücklegt. Vernachlässige hierbei die Strömungsgeschwindigkeit des Flusses.
44 44  {{/aufgabe}}
45 45  
46 -{{aufgabe id="Fenster" afb="III" kompetenzen="K2,K3,K5,K6" quelle="Martin Stern" cc="BY-SA" niveau="e"}}
67 +{{aufgabe id="Fenster" afb="III" kompetenzen="K2,K3,K5,K6" quelle="Martin Stern" cc="BY-SA" niveau="e" zeit="30"}}
47 47  Ein Fenster besteht aus einem Rechteck mit einem aufgesetzten Halbkreis. Aus bautechnischen Gründen darf der Umfang des Fensters die Länge 3,50 m nicht übersteigen. Für das Rechteck und den Halbkreis werden verschiedene Glassorten verwendet, die 10 % (Rechteck) und 30 % (Halbkreis) des einfallenden Lichtes absorbieren.
48 48  
49 49  Ziel: Es soll das optimale Fenster gefunden werden, bei dem möglichst viel Licht einfällt.
... ... @@ -50,4 +50,3 @@
50 50  {{/aufgabe}}
51 51  
52 52  {{seitenreflexion/}}
53 -
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