This question is an application of a quadratic equation. Student is given dimensions of a rectangular area, and an area of pavers that are available. They are asked to calculate the width of a border that can be paved around the given rectangle (assuming border is the same width on all 4 sides). The equation for the area of the border is given in terms of the unknown border width. Students need to recognise that only one solution of the quadratic gives a physically possible solution.
\nThe dimensions of the rectangle, available area of tiles and type of space are randomised. Numeric variables are constructed so that resulting quadratic equation has one positive and one negative root.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "We are told that the area of the border is \\[A=4x^2+\\var{2*(width+breadth)}x\\] where \\(x\\) is the required width of the border. Hence we need to solve \\[\\var{area}=4x^2+\\var{2*(width+breadth)}x\\] for \\(x\\).
\nRearranging this equation we have \\[4x^2+\\var{2*(width+breadth)}x-\\var{area}=0\\] which can be solved using the quadratic formula as
\n\\begin{align}x&=\\frac{\\var{-2*(width+breadth)}\\pm\\sqrt{\\var{2*(width+breadth)}^2-4\\times 4\\times \\var{area}}}{2\\times 4}\\\\&=\\frac{\\var{-2*(width+breadth)}\\pm\\sqrt{\\var{(2*(width+breadth))^2-16*area}}}{8}\\\\&\\approx \\var{border}\\text{ or }\\var{negativeRoot}\\end{align}
\nSince the border cannot have negative width, we can ignore the negative solution and conclude that the width of the border will be approximately \\(\\var{border}\\) metres wide.
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", "templateType": "anything", "can_override": false}, "breadth": {"name": "breadth", "group": "Ungrouped variables", "definition": "random(width+1,width+2)", "description": "breadth of rectangular area in question - depends on generated width.
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", "templateType": "anything", "can_override": false}, "space": {"name": "space", "group": "Ungrouped variables", "definition": "random(\"garden bed\", \"pond\", \"decorative floor\",\"storage area\")", "description": "Type of space specified in question, irrelevant to the question but gives different applications.
", "templateType": "anything", "can_override": false}, "border": {"name": "border", "group": "Ungrouped variables", "definition": "precround((-(width+breadth)+sqrt((width+breadth)^2+4*area))/4 ,2) ", "description": "Answer to question - width of required border.
", "templateType": "anything", "can_override": false}, "negativeRoot": {"name": "negativeRoot", "group": "Ungrouped variables", "definition": "precround((-(width+breadth)-sqrt((width+breadth)^2+4*area))/4 ,2) ", "description": "other solution to quadratic equation - always negative
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["width", "breadth", "area", "space", "border", "negativeRoot"], "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": "A rectangular {space} {width} metres by {breadth} metres is to have a border built around it. The border is to be rectangular with the {space} in the middle. You have {area} square metres of material for the border. How wide will the border need to be? Give your answer correct to 2 decimal palces and show full working on your hand written notes.
\nNote that the area, \\(A\\), of the border is given by the formula
\n\\[A=4x^2+\\var{2*(width+breadth)}x\\]
\nwhere \\(x\\) is the width of the border.
\nThe border will be [[0]] metres wide.
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