// Numbas version: exam_results_page_options
{"name": "Centroid of a triangle by integration", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Centroid of a triangle by integration", "tags": ["centroid", "definite integral", "integrtion"], "metadata": {"description": "<p>Student calculates $\\bar{y}$ for a triangle. &nbsp;Must use similar triangles get element $dA$.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>{applet}</p>", "advice": "<p>Use similar triangles to express the width of the strip as a function of $y$:</p>\n<p style=\"padding-left: 40px;\">$\\dfrac{w}{h-y} = \\dfrac{b}{h}$</p>\n<p style=\"padding-left: 40px;\">$w = b\\dfrac{(h-y)}{h}$</p>\n<p>Therefore,</p>\n<p style=\"padding-left: 40px;\">$dA = w\\; dy = b\\dfrac{(h-y)}{h} dy$</p>\n<p>With this you can evaluate $Q_x$</p>\n<p style=\"padding-left: 40px;\">$\\begin{align} Q_x &amp;= \\int y\\; dA\\\\ &amp;=\\int_0^h y\\;b\\frac{(h-y)}{h}\\; dy\\\\&amp;= \\frac{b}{h}\\int_0^h (hy - y^2)\\; dy\\\\&amp; = \\frac{b}{h}\\left. \\left(\\frac{h y^2}{2} - \\frac{y^3}{3}\\right)\\right |_0^h \\\\ &amp;= bh^2\\left( \\frac{1}{2} - \\frac{1}{3}\\right)\\\\ &amp;= \\frac{bh^2}{6}\\end{align}$</p>\n<p>Finaly, find $\\bar{y}$ knowing the area of the triangle is $\\dfrac{bh}{2}$<br/><br/></p>\n<p style=\"padding-left: 40px;\">$\\begin{align} \\bar{y} &amp;= \\frac{Q_x}{A}\\\\ &amp;= \\frac{{bh^2}/{6}}{{bh}/{2}}\\\\ &amp;= \\frac{h}{3}\\end{align}$</p>", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [{"name": "dA", "value": "diff:A", "tex": "\\mathrm{d}A"}, {"name": "dx", "value": "diff:x", "tex": "\\mathrm{d}x"}, {"name": "dy", "value": "diff:y", "tex": "\\mathrm{d}y"}], "variables": {"applet": {"name": "applet", "group": "Ungrouped variables", "definition": "geogebra_applet(\"jyqxv2uf\",[B: vector(random(-1,1,2,4,5),3.5)])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["applet"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Questions ", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Use horizontal strips to prove by integration that for a triangle with base $b$ and height $h$, &nbsp;$\\bar{y} = h/3$.</p>\n<p>1. Derive expressions for $\\bar{y}_{el}$ and $dA$ in terms of $y$.</p>\n<p style=\"padding-left: 40px;\">$\\bar{y}_{el} =$&nbsp;[[1]]&nbsp;$\\qquad dA=$&nbsp;[[0]]</p>\n<p>2. Determine the limits of the integration.</p>\n<p style=\"padding-left: 40px;\">Lower limit [[2]]&nbsp; Upper limit [[3]]</p>\n<p>3. Evaluate $Q_x $ between your limits.</p>\n<p style=\"padding-left: 40px;\">$Q_x &nbsp;= \\int \\bar{y}_{el} dA = $ [[4]]&nbsp;</p>\n<p>4. &nbsp;Evaluate $\\bar{y}$ for the triangle.</p>\n<p style=\"padding-left: 40px;\">$\\bar{y} = \\dfrac{Q_x}{A} = $&nbsp;[[5]]</p>", "gaps": [{"type": "jme", "useCustomName": true, "customName": "dA", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "b*(h-y)/h dy", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "b", "value": ""}, {"name": "h", "value": ""}, {"name": "y", "value": ""}]}, {"type": "patternmatch", "useCustomName": true, "customName": "$\\bar{y}_{el}", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "y", "displayAnswer": "", "matchMode": "exact"}, {"type": "numberentry", "useCustomName": true, "customName": "lower", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "0", "maxValue": "0", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "patternmatch", "useCustomName": true, "customName": "upper", "marks": "5", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "h", "displayAnswer": "", "caseSensitive": true, "partialCredit": 0, "matchMode": "exact"}, {"type": "jme", "useCustomName": true, "customName": "Qx", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "b h^2 /6", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "b", "value": ""}, {"name": "h", "value": ""}]}, {"type": "jme", "useCustomName": true, "customName": "ybar", "marks": "10", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "h/3", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "h", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}]}]}], "contributors": [{"name": "William Haynes", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2530/"}]}