// Numbas version: exam_results_page_options
{"name": "Bounds with area", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Bounds with area", "tags": [], "metadata": {"description": "Working out upper and lower bounds for area given length and width, and for length given area and width.", "licence": "Creative Commons Attribution 4.0 International"}, "statement": "<p>Give all answers to 2 decimal places:</p>", "advice": "<p><strong>(a)</strong></p>\n<p>The given width = $\\var{width}$ cm and length = $\\var{length}$ cm are each rounded to the nearest 0.1cm. To find the upper and lower bounds we must add/subtract half of this value.</p>\n<p>Since half of 0.1cm is 0.05cm we obtain:</p>\n<p>Lower bound for width $=&nbsp;\\var{width}-0.05=\\var{width-0.05}$</p>\n<p>Upper bound for width $=&nbsp;\\var{width}+0.05=\\var{width+0.05}$</p>\n<p>Lower bound for length $=&nbsp;\\var{length}-0.05=\\var{length-0.05}$</p>\n<p>Upper bound for&nbsp;length&nbsp;$=&nbsp;\\var{length}+0.05=\\var{length+0.05}$</p>\n<p>&nbsp;</p>\n<p>Since area = length $\\times$ width, we have:</p>\n<p>lower bound&nbsp;area =&nbsp;lower bound&nbsp;length $\\times$ lower bound width $=\\var{length-0.05} \\times \\var{width-0.05} = \\var{minarea}$ cm$^2=\\var{precround(minarea,2)}$ cm$^2$ to 2d.p.</p>\n<p>and</p>\n<p>upper bound area = upper&nbsp;bound length $\\times$ upper&nbsp;bound width&nbsp;$=\\var{length+0.05} \\times\\var{width+0.05} = \\var{maxarea}$&nbsp;cm$^2=\\var{precround(maxarea,2)}$ cm$^2$ to 2d.p.</p>\n<p></p>\n<p><strong>(b)</strong></p>\n<p>The given width = $\\var{width2}$ cm is rounded to the nearest 1cm. To find the upper and lower bounds we must add/subtract half of this value.</p>\n<p>Since half of 1cm is 0.5cm we obtain:</p>\n<p>Lower bound for width $=&nbsp;\\var{width2}-0.5=\\var{width2-0.5}$</p>\n<p>Upper bound for width $=&nbsp;\\var{width2}+0.5=\\var{width2+0.5}$</p>\n<p></p>\n<p></p>\n<p>The given area = $\\var{area}$ cm is rounded to the nearest 10cm$^2$. To find the upper and lower bounds we must add/subtract half of this value.</p>\n<p>Since half of 10cm$^2$ is 5cm$^2$ we obtain:</p>\n<p>Lower bound for area $=&nbsp;\\var{area}-5=\\var{area-5}$</p>\n<p>Upper bound for area $=&nbsp;\\var{area}+5=\\var{area+5}$</p>\n<p></p>\n<p>Since&nbsp;length = area $\\div$ width, we have:</p>\n<p>lower bound length=&nbsp;lower bound area $\\div$ upper&nbsp;bound width&nbsp; &nbsp; &nbsp; (to get a small answer we want to be dividing a small number by a big number)</p>\n<p>$=\\var{area-5} \\div \\var{width2+0.5} = \\var{minlength}$ cm$=\\var{precround(minlength,2)}$ cm to&nbsp;2d.p.</p>\n<p></p>\n<p>and</p>\n<p>upper bound length = upper&nbsp;bound&nbsp;area $\\div$ lower&nbsp;bound width&nbsp; &nbsp; &nbsp;(to get a large answer we want to be dividing a large number by a small number)</p>\n<p>$=\\var{area+5} \\div \\var{width2-0.5} = \\var{maxlength}$ cm$=\\var{precround(maxlength,2)}$ cm to&nbsp;2d.p.</p>", "rulesets": {}, "extensions": [], "variables": {"width": {"name": "width", "group": "Ungrouped variables", "definition": "random (2..4#0.1)", "description": "", "templateType": "anything"}, "length": {"name": "length", "group": "Ungrouped variables", "definition": "random (5..7#0.1)", "description": "", "templateType": "anything"}, "minarea": {"name": "minarea", "group": "Ungrouped variables", "definition": "(width-0.05)*(length-0.05)", "description": "", "templateType": "anything"}, "maxarea": {"name": "maxarea", "group": "Ungrouped variables", "definition": "(width+0.05)*(length+0.05)", "description": "", "templateType": "anything"}, "width2": {"name": "width2", "group": "Ungrouped variables", "definition": "random(25..65)", "description": "", "templateType": "anything"}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "random(200..500#10)", "description": "", "templateType": "anything"}, "minlength": {"name": "minlength", "group": "Ungrouped variables", "definition": "(area-5)/(width2+0.5)", "description": "", "templateType": "anything"}, "maxlength": {"name": "maxlength", "group": "Ungrouped variables", "definition": "(area+5)/(width2-0.5)", "description": "", "templateType": "anything"}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["width", "length", "minarea", "maxarea", "width2", "area", "minlength", "maxlength"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>A rectangle has a width of $\\var{width}$ cm and length of $\\var{length}$ cm, each rounded to the nearest mm.</p>\n<p>What is:</p>\n<p>(i) the minimum possible area of the rectangle in&nbsp;cm$^2$ [[0]]&nbsp;</p>\n<p>(ii) the maximum possible area of the rectangle in cm$^2$ [[1]]&nbsp;</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "minarea", "maxValue": "minarea", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "maxarea", "maxValue": "maxarea", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "prompt": "<p>Another rectangle has a width of $\\var{width2}$ cm (rounded to the nearest 1cm) and area of $\\var{area}$ cm$^2$ (rounded to the nearest 10 cm$^2)$.</p>\n<p>What is:</p>\n<p>(i) the minimum possible length of the rectangle in&nbsp;cm&nbsp;[[0]]&nbsp;</p>\n<p>(ii) the maximum possible length of the rectangle in cm&nbsp;[[1]]&nbsp;</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "showCorrectAnswer": true, "showFeedbackIcon": true, "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "adaptiveMarkingPenalty": 0, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "minValue": "minlength", "maxValue": "minlength", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "precisionType": "dp", "precision": "2", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", 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