// Numbas version: exam_results_page_options
{"name": "Maximum/minimum", "extensions": [], "custom_part_types": [], "resources": [["question-resources/Untitled.jpg", "/srv/numbas/media/question-resources/Untitled.jpg"], ["question-resources/Untitled_qCawkyB.jpg", "/srv/numbas/media/question-resources/Untitled_qCawkyB.jpg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["l", "w"], "name": "Maximum/minimum", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>The length of the box is \\(\\var{l}-2x\\), the width is \\(\\var{w}-2x\\) and the height is \\(x\\).</p>\n<p>The volume is then given by</p>\n<p style=\"padding-left: 30px;\">\\(V=(\\var{l}-2x).(\\var{w}-2x).x\\)</p>\n<p style=\"padding-left: 30px;\">\\(V=\\simplify{4x^3-2*({w}+{l})x^2+{l}*{w}x}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{dV}{dx}=\\simplify{12x^2-4*({w}+{l})x+{l}*{w}}\\)</p>\n<p>This is a quadratic equation.</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{\\simplify{4*({w}+{l})}\\pm\\sqrt(\\simplify{16*({w}+{l})^2-48*{w}*{l}})}{24}\\)</p>\n<p style=\"padding-left: 30px;\">\\(x=\\frac{\\simplify{{w}+{l}}\\pm\\sqrt(\\simplify{{w}^2-{w}*{l}+{l}^2})}{6}\\)</p>\n<p style=\"padding-left: 30px;\">\\(\\frac{d^2V}{dx^2}=\\simplify{24x-4*({w}+{l})}\\)</p>\n<p>when &nbsp;\\(x=\\simplify{({w}+{l}-sqrt({w}^2-{w}*{l}+{l}^2))/6}\\) &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; \\(\\frac{d^2V}{dx^2}&lt;0\\) &nbsp; &nbsp; &nbsp;and therefore is the value that gives a maximum.</p>", "rulesets": {}, "parts": [{"prompt": "<p>Input the value for \\(x\\) correct to one decimal place.</p>\n<p style=\"padding-left: 30px;\">\\(x = \\)&nbsp;[[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"precisionType": "dp", "precisionMessage": "You have not given your answer to the correct precision.", "allowFractions": false, "variableReplacements": [], "maxValue": "({w}+{l}-sqrt({w}^2+{l}^2-{w}*{l}))/6", "strictPrecision": false, "minValue": "({w}+{l}-sqrt({w}^2+{l}^2-{w}*{l}))/6", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "1", "scripts": {}, "marks": 1, "showPrecisionHint": false, "type": "numberentry"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>A rectangular sheet of metal of length = \\(\\var{l}cm\\) and width &nbsp;= \\(\\var{w}cm\\) has a square of side \\(x\\,cm\\) cut from each corner. The ends and sides will be folded upwards to form an open box.</p>\n<p>Determine the value of \\(x\\) that will maximise the volume of this box.</p>\n<p style=\"padding-left: 120px;\"><img src=\"resources/question-resources/Untitled_qCawkyB.jpg\" width=\"386\" height=\"200\"/></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"l": {"definition": "random(30..100#5)", "templateType": "randrange", "group": "Ungrouped variables", "name": "l", "description": ""}, "w": {"definition": "random(10..28#2)", "templateType": "randrange", "group": "Ungrouped variables", "name": "w", "description": ""}}, "metadata": {"description": "<p>Maximising the volume of a rectangular box</p>", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}]}], "contributors": [{"name": "Frank Doheny", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/789/"}]}