// Numbas version: exam_results_page_options
{"name": "linear depreciation from graph", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "linear depreciation from graph", "tags": [], "metadata": {"description": "<p>Students are shown a graph of the value of a machine over time. The line equation is randomised.</p>\n<p>They are asked to evaluate&nbsp;value at a given time, and the time at which a given value is reached. They are asked when the machine has no value, and the range of times over which the model is valid. They are also asked to explain the physical meaning of the gradient.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A machine was purchased for a cost of {b} {unit} dollars.</p>\n<p>Its value depreciates by a fixed amount each year, as shown in this graph. $x$ represents the time in years, and $y$ represents the cost in {unit} dollars.</p>\n<p>{geogebra_applet('https://www.geogebra.org/m/wxv4ykqj',defs)}</p>", "advice": "<p>To work out what the machine is worth after $\\var{worthless/2}$ years we look at the graph.</p>\n<p>Find $\\var{worthless/2}$ on the horizontal axis. Follow this point up vertically until it meets the line, then go horizontally left until you meet the $y$-axis.</p>\n<p>If this point is on a line you can read off the value.</p>\n<p>If, however, it is not on a line then we need to use another method. If we look at the horizontal axis, we notice that the line cuts it at $\\var{worthless}$, which is $2 \\times \\var{worthless/2}$.</p>\n<p>This means that the value depreciates from $\\var{b}$ to $0$ over $\\var{worthless}$ years.&nbsp;</p>\n<p>And so, over&nbsp;$\\var{worthless} \\div 2 = \\var{worthless/2}$ years the value depreciates to $\\var{b} \\div 2 = \\var{b/2}$.</p>\n<p>That is, the value of the machine at $\\var{worthless/2}$ years is $\\var{b/2}$ {unit} dollars.</p>\n<p>The machine has no monetary value at the point where the line cuts the $x$-axis. This is at $\\var{worthless}$ years.</p>\n<p>The gradient of the line is calculated as (vertical rise) $\\div$ (horizontal run). For this graph, this is equal to $\\var{m}$.</p>\n<p>The gradient is the amount of money (in {unit} dollar lots) that the machine depreciates by each year.</p>\n<p>Reading from the graph, after $\\var{x}$ years, the machine will be worth $\\var{y}$ {unit} dollars.</p>\n<p>If you cannot read this from the graph, you can use the formula for the line: $y = \\var{m}x + \\var{b}$. Put in $x=\\var{x}$ and calculate $y$.</p>\n<p>The model is only valid from the time that the machine was bought (when $x = 0$ years) until the time the machine has no monetary value (when $x = \\var{worthless}$ years). The value of the machine cannot go negative so the model is no longer relevant after this time.</p>", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(-0.5, -1, -1.5, -2)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(30,60,90,120,150,180,210)", "description": "", "templateType": "anything", "can_override": false}, "unit": {"name": "unit", "group": "Ungrouped variables", "definition": "random('','thousand','million')", "description": "", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Ungrouped variables", "definition": "[\n  ['a',m],['b',b],['xmin',0],['xmax',-b/m],['ymin',0],['ymax',b]\n  ]", "description": "", "templateType": "anything", "can_override": false}, "y": {"name": "y", "group": "Ungrouped variables", "definition": "b*0.4", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "(y-b)/m", "description": "", "templateType": "anything", "can_override": false}, "bon2": {"name": "bon2", "group": "Ungrouped variables", "definition": "b/2", "description": "", "templateType": "anything", "can_override": false}, "worthless": {"name": "worthless", "group": "Ungrouped variables", "definition": "-b/m", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "b", "unit", "defs", "y", "x", "bon2", "worthless"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>How many {unit} dollars is the machine worth&nbsp;after $\\var{-b/m/2}$ years?</p>", "minValue": "b/2", "maxValue": "b/2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>After how many years will the machine have no monetary value?</p>", "minValue": "-b/m", "maxValue": "-b/m", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>What is the gradient of the line that relates $x$ and $y$?</p>", "minValue": "m", "maxValue": "m", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, 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