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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": [{"name": "Differentiation 18 - Applications", "tags": [], "metadata": {"description": "<p>Another practical application of differentiation</p>", "licence": "Creative Commons Attribution 4.0 International"}, "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>", "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>", "rulesets": {}, "extensions": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(100..200)", "description": "", "templateType": "anything"}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "siground((2v)^(1/3),3)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v", "a"], "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": "<p>Describe the height of the tank in terms of&nbsp;$x$</p>\n<p>$h=$&nbsp;[[0]] m</p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{v}/x^2", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"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": "<p>Describe the surface area of the tank in terms of&nbsp;$x$</p>\n<p>$S=$ [[0]]&nbsp;$\\text{m}^2$</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "x^2+4{v}/x", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": true, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "x", "value": ""}]}], "sortAnswers": false}, {"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": "<p>Find the first derivative </p>\n<p>$S'(x)=$&nbsp;[[1]]</p>\n<p>and given that the surface area is a minimum, find the value of $x$. Therefore,</p>\n<p>$x=$&nbsp;[[0]] &nbsp; &nbsp;&nbsp;(give your answer to 2 decimal places)</p>\n<p>Find the second derivative </p>\n<p>$S''(x)=$&nbsp;[[2]]</p>\n<p>Check this is a minimum.</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, calculate the minimum area of metal used </p>\n<p>$A_{min}=$&nbsp;[[4]] &nbsp; &nbsp;&nbsp;(give your answer to 2 decimal places)</p>", "stepsPenalty": "1", "steps": [{"type": "information", "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": "<p><em>Hint:</em> find $x$ at 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