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{"name": "Simon's copy of Maximum/minimum2", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Untitled2.jpg", "/srv/numbas/media/question-resources/Untitled2.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"tags": [], "statement": "<p>A closed cylindrical tank is to be built having a volume of \\(\\var{v}\\) cc.</p>\n<p>Determine the required height, \\(h\\), and radius, \\(r\\), if the total surface area is to be a minimum.</p>\n<p style=\"padding-left: 120px;\"><img src=\"resources/question-resources/Untitled2.jpg\"/></p>", "preamble": {"css": "", "js": ""}, "ungrouped_variables": ["v"], "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "advice": "<p>\\(\\pi r^2h=\\var{v}\\)</p>\n<p>\\(h=\\frac{\\var{v}}{\\pi r^2}\\)</p>\n<p>The total surface area is to be a minimum.</p>\n<p>Lid + curved surface area + base</p>\n<p style=\"padding-left: 30px;\">\\(A=\\pi r^2+2\\pi rh+\\pi r^2\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=2\\pi r^2+2\\pi r\\left(\\frac{\\var{v}}{\\pi r^2}\\right)\\)</p>\n<p style=\"padding-left: 30px;\">\\(A=2\\pi r^2+\\simplify{2*{v}}r^{-1}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dA}{dr}=4\\pi r-\\simplify{2{v}}r^{-2}=0\\)</p>\n<p style=\"padding-left: 30px;\">\\(4\\pi r=\\simplify{2*{v}}/{r^2}\\)</p>\n<p style=\"padding-left: 30px;\">\\(r^3=\\frac{\\var{v}}{2\\pi}\\)</p>\n<p style=\"padding-left: 30px;\">\\(r=\\simplify{({v}/(2*pi))^(1/3)}\\)</p>\n<p>From the second line we have the relation&nbsp;\\(h=\\frac{\\var{v}}{\\pi r^2}\\) to get</p>\n<p style=\"padding-left: 30px;\">\\(h=2*\\simplify{({v}/(2*pi))^(1/3)}\\)</p>\n<p></p>", "variables": {"v": {"group": "Ungrouped variables", "templateType": "randrange", "description": "", "name": "v", "definition": "random(50..300#5)"}}, "parts": [{"marks": 0, "showCorrectAnswer": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>Input the cyinder height, correct to two decimal places.</p>\n<p style=\"padding-left: 30px;\">\\(h = \\)&nbsp;[[0]]</p>\n<p>Input the required cylinder radius, correct to two decimal places.</p>\n<p style=\"padding-left: 30px;\">\\(r = \\)&nbsp;[[1]]</p>", "gaps": [{"type": "numberentry", "minValue": "2*({v}/(2*pi))^(1/3)", "strictPrecision": false, "allowFractions": false, "maxValue": "2*({v}/(2*pi))^(1/3)", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "marks": 1, "showCorrectAnswer": true, "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "precision": "2", "variableReplacements": [], "scripts": {}}, {"type": "numberentry", "minValue": "({v}/(2*pi))^(1/3)", "strictPrecision": false, "allowFractions": false, "maxValue": "({v}/(2*pi))^(1/3)", "precisionMessage": "You have not given your answer to the correct precision.", "precisionType": "dp", "marks": 1, "showCorrectAnswer": true, "precisionPartialCredit": 0, "correctAnswerFraction": false, "variableReplacementStrategy": "originalfirst", "showPrecisionHint": false, "precision": "2", "variableReplacements": [], "scripts": {}}], "variableReplacements": [], "type": "gapfill", "scripts": {}}], "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Problem on a&nbsp;closed cylindrical tank having minimum surface area</p>"}, "extensions": [], "name": "Simon's copy of Maximum/minimum2", "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}