// Numbas version: exam_results_page_options
{"name": "2 triangle problem with angle of depression", "extensions": [], "custom_part_types": [], "resources": [["question-resources/angleofdepression_problem.svg", "/srv/numbas/media/question-resources/angleofdepression_problem.svg"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "2 triangle problem with angle of depression", "tags": [], "metadata": {"description": "<p>Students are given a diagram with 2 triangles. They are given 2 randomised lengths, and a randomised angle of depression.</p>\n<p>They need to compute an angle by subtracting the angle of depression from 90&deg;. Then they need to use the sine rule to calculate a second angle. Then they need to use the alternate angles on parallel lines theorem to work out a third angle. They use these to calculate a third angle, which they use in the right-angle triangle with the sine ratio to compute the third side. They then use the cos ratio to compute the length of the third side.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A person standing on the top of a building at $J$ looks down to a garden on the ground at point $M$. The angle of depression from $J$ to $M$ is $\\var{aod}$&deg;. There is a window in the building at $K$,&nbsp;$\\var{JK}$ metres below $J$. The distance from $M$ to $K$ is&nbsp;$\\var{KM}$ metres.</p>\n<p><img src=\"resources/question-resources/angleofdepression_problem.svg\"/></p>\n<p><em>not to scale</em></p>\n<p></p>", "advice": "<p>$\\angle MJK = 90&deg; - \\var{aod}&deg; = \\var{aMJK}&deg;$</p>\n<p>In $\\triangle JKM$, by the sine rule, $\\frac{JK}{\\sin (\\angle M)}=\\frac{KM}{\\sin (\\angle J)}$</p>\n<p>$\\frac{\\var{JK}}{\\sin (\\angle M)}=\\frac{\\var{KM}}{\\sin \\var{aMJK}&deg;}$</p>\n<p>$\\angle M = \\angle JMK = \\var{aJMK}&deg;$</p>\n<p></p>\n<p>Now $\\angle JML = \\var{aod}$&deg; (alternate angles on parallel lines)</p>\n<p>So $\\angle KML = \\angle JML - \\angle JMK = \\var{aod}&deg; - \\var{aJMK}&deg; = \\var{aKML}&deg; $</p>\n<p>The angle of elevation from $M$ to $K$ is $\\var{aKML}&deg; $.</p>\n<p></p>\n<p>$\\triangle KML$ is a right-angle triangle, and $\\angle KML = 90&deg;$</p>\n<p>So to find $KL$ we can use the sine ratio: $\\sin(angle)=\\frac{opposite}{hypotenuse}$</p>\n<p>$\\sin(\\var{aKML})&deg; = \\frac{KL}{\\var{KM}}$</p>\n<p>$KL = \\var{KM} \\times \\sin(\\var{aKML}&deg;) = \\var{KL}$&nbsp;m</p>\n<p>The distance from the ground to $K$ is $\\var{KL}$&nbsp;m.</p>\n<p></p>\n<p>To find $LM$ we can use the cosine ratio: $\\cos(angle)=\\frac{adjacent}{hypotenuse}$</p>\n<p>$\\cos(\\var{aKML})&deg; = \\frac{LM}{\\var{KM}}$</p>\n<p>$LM = \\var{KM} \\times \\cos(\\var{aKML}&deg;) = \\var{LM}$ m</p>\n<p>The distance from the pond to the building is $\\var{LM}$ m.</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"aod": {"name": "aod", "group": "Ungrouped variables", "definition": "random(50..75)", "description": "", "templateType": "anything", "can_override": false}, "JK": {"name": "JK", "group": "Ungrouped variables", "definition": "random(2..25)", "description": "", "templateType": "anything", "can_override": false}, "KM": {"name": "KM", "group": "Ungrouped variables", "definition": "random(JK..JK*2)", "description": "", "templateType": "anything", "can_override": false}, "aMJK": {"name": "aMJK", "group": "Ungrouped variables", "definition": "90-aod", "description": "", "templateType": "anything", "can_override": false}, "aJMK": {"name": "aJMK", "group": "Ungrouped variables", "definition": "round(degrees(arcsin(JK*sin(radians(aMJK))/KM )))", "description": "", "templateType": "anything", "can_override": false}, "KL": {"name": "KL", "group": "Ungrouped variables", "definition": "round(KM * sin(radians(aKML)))", "description": "", "templateType": "anything", "can_override": false}, "aKML": {"name": "aKML", "group": "Ungrouped variables", "definition": "aod-aJMK", "description": "", "templateType": "anything", "can_override": false}, "KLmin": {"name": "KLmin", "group": "Ungrouped variables", "definition": "round(KM * sin(radians(aKML-1))-0.1)", "description": "", "templateType": "anything", "can_override": false}, "KLmax": {"name": "KLmax", "group": "Ungrouped variables", "definition": "round(KM * sin(radians(aKML+1))+0.1)", "description": "", "templateType": "anything", "can_override": false}, "LM": {"name": "LM", "group": "Ungrouped variables", "definition": "round(KM * cos(radians(aKML)))", "description": "", "templateType": "anything", "can_override": false}, "LMmin": {"name": "LMmin", "group": "Ungrouped variables", "definition": "round(KM * cos(radians(aKML-1))-0.1)", "description": "", "templateType": "anything", "can_override": false}, "LMmax": {"name": "LMmax", "group": "Ungrouped variables", "definition": "round(KM * cos(radians(aKML+1))+0.1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["aod", "JK", "KM", "aMJK", "aJMK", "aKML", "KL", "KLmin", "KLmax", "LM", "LMmin", "LMmax"], "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>Calculate the angle of elevation from $M$ to $K$. Give your answer to the nearest degree.</p>", "minValue": "aKML-1", "maxValue": "aKML+1", "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 distance, in metres, from the ground (at $L$) to the window (at $K$)? Give your answer to the nearest metre.</p>", "minValue": "{KLmin}", "maxValue": "{KLmax}", "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>How far is it, in metres, from the building to the garden (i.e. what is the length of $LM$)?</p>\n<p>Give your answer to the nearest metre.</p>", "minValue": "{LMmin}", "maxValue": "{LMmax}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}]}], "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}]}