Griechisch-römische Antike 3
", "licence": "Creative Commons Attribution 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "questions": [{"name": "Heron: Katroptika", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "Über den Weg des Lichts
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Wählen Sie die korrekte Aussage aus!
", "advice": "Heron nennt als Gesetz das Nehmen des kürzesten Wegs, dies entspricht innerhalkb desselben Mediums zwar dem schnellsten Weg, aber nicht mehr, wenn verschiedene Medien (z.B. Wasser und Luft) durchdrungen werden.
\nEinfallswinkel = Ausfallswinkel ist nicht das Gesetz, sondern eine Folgerung aus dem Gesetz, ebenso die geradlinige Ausbreitung.
", "rulesets": {}, "variables": {}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Heron nennt in seiner Katroptik das Gesetz:
", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": false, "choices": ["Licht nimmt immer den kürzesten Weg.", "Licht nimmt immer den schnellsten Weg.", "Licht breitet sich geradlinig aus.", "Einfallswinkel = Ausfallswinkel."], "matrix": ["2", 0, 0, 0], "distractors": ["", "Nein, das ist erst bei Fermat bekannt.", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Gleichungen bei Diophant", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Andreas Vohns", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3622/"}], "tags": [], "metadata": {"description": "Lösen einer quadratischen Gleichung im Stile Diophants
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Wir betrachten ein weiteres Problem der Form: Zwei Zahlen finden, deren Summe und deren Summe der Quadrate der Zahlen gegeben sind.
\nSumme sei $\\var{sum}$, Quadratsumme $\\var{qsum}$.
\nWir bezeichnen die Differenz der beiden Zahlen mit $2\\cdot x$.
\n", "advice": "
a) Es ist $a=\\frac{\\var{sum}}{2}-x=\\var{sum/2}-x$, $b=\\frac{\\var{sum}}{2}+x=\\var{sum/2}+x$.
\nb) Es ergibt sich: $\\var{qsum}=2\\cdot\\var{(sum/2)^2}+2x^2\\Leftrightarrow x^2=\\frac{\\var{qsum}-\\var{2*(sum/2)^2}}{2}=\\frac{\\var{(qsum-2*(sum/2)^2)}}{2}=\\var{(qsum-2*(sum/2)^2)/2}\\Leftrightarrow x=\\var{x}$
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\nJeweils einen Term in Abhängigkeit von $x$ eintragen!
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\n$x=$[[2]] , $a=$[[0]] , $b=$[[1]]
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Nach Nikomachos: Ermitteln der 5. Proportion
", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "Nach Nikomachos stehen drei Zahlen $a<b<c$ dann in der \"fünften Proportion\", wenn die mittlere Zahl sich zur kleinere Zahl verhält, wie sich deren Differenz zur Differenz der größeren Zahl und der mittleren Zahl verhält.
\n", "advice": "
a) Es gilt: $b:a=(b-a):(c-a)$
\nb) Durch probieren (oder auflösen der Gleichung oben) kann man mehrere Lösungen finden, eine mögliche Lösung ist $2, 4, 5$, sowie alle ganzzahligen Vielfachen dieser Lösung.
", "rulesets": {}, "variables": {"app": {"name": "app", "group": "Ungrouped variables", "definition": "geogebra_applet('https://www.geogebra.org/m/ezesj2ms')\n\n", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["app"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Beschreiben Sie die Proportion durch eine Gleichung in den Variablen $a,b,c$:
\n$\\Big($[[0]]$\\Big):\\Big($[[1]]$\\Big)=\\Big($[[2]]$\\Big):\\Big($[[3]]$\\Big)$
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Finden Sie drei natürliche Zahlen $a<b<c$, die in fünfter Proportion stehen:
\n{app}
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