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{"name": "Differentiation 18 - Applications", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>This section draws on the skills learnt the previous parts of the&nbsp;<em>'Differentiation'</em> series of questions, and some geometry knowledge.</p>\n<p>The hint&nbsp;under the steps should be all the extra information you need.</p>", "parts": [{"unitTests": [], "customName": "", "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "{v}/x^2", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": "2", "useCustomName": false, "answerSimplification": "all"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>The height of the tank</p>\n<p>$h=$&nbsp;[[0]] m</p>\n<p></p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}, {"unitTests": [], "customName": "", "marks": 0, "gaps": [{"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "x^2+4{v}/x", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": "2", "useCustomName": false, "answerSimplification": "all"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>The surface area of the tank</p>\n<p>$S=$ [[0]]&nbsp;$\\text{m}^2$</p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}, {"unitTests": [], "customName": "", "steps": [{"unitTests": [], "customName": "", "marks": 0, "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p><em>Hint:</em> find $x$ at stationary point</p>", "showFeedbackIcon": true, "type": "information", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}], "marks": 0, "stepsPenalty": "1", "gaps": [{"precision": "2", "precisionPartialCredit": "50", "minValue": "{a}", "mustBeReduced": false, "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "allowFractions": false, "extendBaseMarkingAlgorithm": true, "precisionType": "dp", "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false, "correctAnswerFraction": false, "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "mustBeReducedPC": 0, "maxValue": "{a}", "customMarkingAlgorithm": "", "marks": "2", "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "correctAnswerStyle": "plain"}, {"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "2x-4{v}/x^2", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": 1, "useCustomName": false, "answerSimplification": "all"}, {"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "x"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": true, "answer": "2+8{v}/x^3", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": 1, "useCustomName": false, "answerSimplification": "all"}, {"checkingAccuracy": 0.001, "checkingType": "absdiff", "showFeedbackIcon": true, "type": "jme", "failureRate": 1, "extendBaseMarkingAlgorithm": true, "valuegenerators": [{"value": "", "name": "y"}], "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPreview": false, "answer": "Y", "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "vsetRange": [0, 1], "checkVariableNames": true, "customMarkingAlgorithm": "", "vsetRangePoints": 5, "marks": 1, "useCustomName": false, "answerSimplification": "all"}, {"precision": "2", "precisionPartialCredit": "50", "minValue": "{a}^2+4({v}/{a})", "mustBeReduced": false, "strictPrecision": false, "showFeedbackIcon": true, "type": "numberentry", "allowFractions": false, "extendBaseMarkingAlgorithm": true, "precisionType": "dp", "variableReplacementStrategy": "originalfirst", "scripts": {}, "showPrecisionHint": false, "correctAnswerFraction": false, "unitTests": [], "customName": "", "variableReplacements": [], "showCorrectAnswer": true, "mustBeReducedPC": 0, "maxValue": "{a}^2+4({v}/{a})", "customMarkingAlgorithm": "", "marks": "1", "precisionMessage": "You have not given your answer to the correct precision.", "notationStyles": ["plain", "en", "si-en"], "useCustomName": false, "correctAnswerStyle": "plain"}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": "<p>Given that the surface area is a minimum, find the value of $x$.</p>\n<p>$S'(x)=$&nbsp;[[1]]</p>\n<p>Therefore,</p>\n<p>$x=$&nbsp;[[0]] &nbsp; &nbsp;&nbsp;(give your answer to 2 decimal places)</p>\n<p>Check this is a minimum.</p>\n<p>$S''(x)=$&nbsp;[[2]]</p>\n<p>Substitute your value for $x$ into $S''(x)$ and determine whether is it a minimum.</p>\n<p>Type '$Y$' for yes, '$N$' for no, or '$U$' for undefined.</p>\n<p>[[3]]</p>\n<p></p>\n<p>Hence, the minimum area of metal used is</p>\n<p>$A_{min}=$&nbsp;[[4]] &nbsp; &nbsp;&nbsp;(give your answer to 2 decimal places)</p>", "sortAnswers": false, "showFeedbackIcon": true, "type": "gapfill", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "useCustomName": false}], "variable_groups": [], "ungrouped_variables": ["v", "a"], "rulesets": {}, "name": "Differentiation 18 - Applications", "extensions": [], "tags": [], "metadata": {"description": "<p>Another practical application of differentiation</p>", "licence": "Creative Commons Attribution 4.0 International"}, "functions": {}, "variables": {"v": {"definition": "random(100..200)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "v"}, "a": {"definition": "siground((2v)^(1/3),3)", "templateType": "anything", "group": "Ungrouped variables", "description": "", "name": "a"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "statement": "<p>An&nbsp;<em>open</em> metal tank of square base has a volume of $\\var{v}\\text{ m}^3$</p>\n<p><strong>Given that the square base has sides of length $x$ metres, find expressions, in terms of $x$, for the following.</strong></p>", "preamble": {"css": "", "js": ""}, "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}]}], "contributors": [{"name": "Jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/514/"}, {"name": "Xiaodan Leng", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2146/"}]}