// Numbas version: exam_results_page_options
{"name": "Maximum/minimum Cylinder", "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": [{"name": "Maximum/minimum Cylinder", "tags": [], "metadata": {"description": "<p>Problem on a&nbsp;closed cylindrical tank having minimum surface area</p>", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>A closed cylindrical tank is to be built having a volume of \\(\\var{v}\\) cm<sup>3</sup>.</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>", "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>", "rulesets": {}, "extensions": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(50 .. 300#5)", "description": "", "templateType": "randrange"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v"], "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>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", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, 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