This question asks students to find the distance from an aircraft to a given marker. Angle of depression of 2 markers from the aircraft are given and the distance between the markers on the ground (all randomised). Students need to use the sine rule to find the answer. The workding of the question makes googling the answer difficult.
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "From the note we know that the angle at \\(A\\) in the triangle is \\(\\var{angA}^\\circ\\) and the angle at \\(B\\) is \\(\\var{angB}^\\circ\\).
\nUsing the fact that the angle sum of a triangle is \\(180^\\circ\\) we have that third angle of the triangle (at the plane) must be \\(180^\\circ-\\var{angA}^\\circ-\\var{angB}^\\circ=\\var{180-angA-angB}^\\circ\\).
\nLetting \\(a\\) be the distance of the plane to point \\(A\\) we have \\begin{align}\\frac{a}{\\sin \\var{angB}^\\circ}&=\\frac{\\var{dist}}{\\sin\\var{180-angA-angB}^\\circ}\\\\a&=\\frac{\\var{dist}\\sin\\var{angB}^\\circ}{\\sin \\var{180-angA-angB}^\\circ}\\\\&\\approx\\var{precround(answer,1)}\\text{ km}\\end{align}
\nHence the plane is approximately \\(\\var{precround(answer,1)}\\) km from point \\(A\\).
\nLetting \\(b\\) be the distance of the plane to point \\(B\\) we have \\begin{align}\\frac{b}{\\sin \\var{angA}^\\circ}&=\\frac{\\var{dist}}{\\sin\\var{180-angA-angB}^\\circ}\\\\b&=\\frac{\\var{dist}\\sin\\var{angA}^\\circ}{\\sin \\var{180-angA-angB}^\\circ}\\\\&\\approx\\var{precround(answer,1)}\\text{ km}\\end{align}
\nHence the plane is approximately \\(\\var{precround(answer,1)}\\) km from point \\(B\\).
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", "templateType": "anything", "can_override": false}, "dist": {"name": "dist", "group": "Ungrouped variables", "definition": "random(3..6)", "description": "Distance in km between markers A and B, random integer 3, 4, 5 or 6.
", "templateType": "anything", "can_override": false}, "marker": {"name": "marker", "group": "Ungrouped variables", "definition": "[latex(\"A\"),latex(\"B\")][selector]", "description": "Chosen marker which students are asked to find the distance to, value is determined by selector variable, 0 = A, 1 = B.
", "templateType": "anything", "can_override": false}, "anga": {"name": "anga", "group": "Ungrouped variables", "definition": "random(30..50)", "description": "Angle of depression in degrees of marker A from aircraft, random value between 30 and 50.
", "templateType": "anything", "can_override": false}, "diagram": {"name": "diagram", "group": "Ungrouped variables", "definition": "jessiecode(400,300,[0,6,8,0],\"\"\"\n point(0.5,0.5) << id:'A', fixed: true, snapToGrid: true, withLabel: true, showInfobox: false>>;\n point(7.5,0.5) << id:'B', fixed: true, snapToGrid: true, withLabel: true, showInfobox: false>>;\n segment(A,B) <JSXGraph diagram of problem, not drawn to scale.
", "templateType": "anything", "can_override": false}, "selector": {"name": "selector", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "Variable to select which marker students are asked to find distance to.
", "templateType": "anything", "can_override": false}, "answer": {"name": "answer", "group": "Ungrouped variables", "definition": "[dist*sin(angB*pi/180)/sin((180-angA-angB)*pi/180),dist*sin(angA*pi/180)/sin((180-angA-angB)*pi/180)][selector]", "description": "List of answers for distance to each marker, value is determined by selector variable, corresponds to chosen marker. Provides correct answer for question.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["angb", "dist", "marker", "anga", "diagram", "selector", "answer"], "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 plane is flying over a straight highway. The angle of depression to two distance markers, \\(A\\) and \\(B\\) are \\(\\var{angA}^\\circ\\) and \\(\\var{angB}^\\circ\\) respectively. Note that the angle of elevation of the plane from the distance markers equals the angle of depression of the marker from the plane. The distance markers are \\(\\var{dist}\\) km apart (see diagram below).
\n{diagram}
\nHow far is the plane from marker \\(\\var{marker}\\)?
\nDistance to marker \\(\\var{marker}=\\;\\)[[0]] km.
", "gaps": [{"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, "minValue": "{answer}", "maxValue": "{answer}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "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}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}]}]}], "contributors": [{"name": "Don Shearman", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/680/"}]}