// Numbas version: exam_results_page_options
{"name": "Find the area of a triangle given by 2 bearings and 1 side", "extensions": [], "custom_part_types": [], "resources": [["question-resources/park.png", "/srv/numbas/media/question-resources/park.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Find the area of a triangle given by 2 bearings and 1 side", "tags": [], "metadata": {"description": "<p>The student is given a triangle with one side running N-S. They are given bearings for the other two sides. They are given the length of the N-S side.</p>\n<p>The bearings and the length are randomised.</p>\n<p>They are then asked to find the area and the perimeter of the triangle.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A triangular block of land is to be declared a park. On the east, the land is bounded by a road that runs north-south. The length of the road-park boundary is&nbsp;{length} m.</p>\n<p>The bearing from the northernmost point of the park along the northwestern boundary is {dsp_bearing1}.</p>\n<p>The bearing from the southernmost point of the park along the southwestern boundary is {dsp_bearing2}.</p>\n<p><img src=\"resources/question-resources/park.png\" alt=\"an illustration of the park\"/></p>\n<p><em>not to scale</em></p>\n<p></p>", "advice": "<p>We can work out the angle at the northernmost tip of the park as $\\var{a1}$&deg;.</p>\n<p>We can work out the angle at the southernmost tip of the park as $\\var{a2}$&deg;.</p>\n<p>We can subtract these angles from 180&deg; to find the value of the angle at the westernmost tip of the park. This is $\\var{a3}$&deg;.</p>\n<p>Now we can use the sine rule to calculate the length of one of the other two boundaries.</p>\n<p>For example:</p>\n<p>$\\frac{NW \\, boundary}{\\sin (\\var{a2}&deg;)}=\\frac{\\var{length}}{\\sin (\\var{a3}&deg;)}$</p>\n<p>So NW boundary $= \\var{nw_len}$ m</p>\n<p>Now we can compute the area using $A = \\frac{1}{2}ab\\sin (C)$</p>\n<p>$Area = \\frac{1}{2} \\times \\var{nw_len} \\times \\var{length} \\times \\sin (\\var{a1}&deg;) = \\var{area3}$ m$^2$</p>\n<p>We can use the sine rule again to find the length of the other boundary:</p>\n<p>$\\frac{SW \\, boundary}{\\sin (\\var{a1}&deg;)}=\\frac{\\var{length}}{\\sin (\\var{a3}&deg;)}$</p>\n<p>So SW boundary $= \\var{sw_len}$ m</p>\n<p>Then the perimeter of the park = $\\var{length} + \\var{nw_len} + \\var{sw_len} = \\var{perimeter}$ m.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"length": {"name": "length", "group": "Ungrouped variables", "definition": "random(50..1000)", "description": "", "templateType": "anything", "can_override": false}, "is_compass": {"name": "is_compass", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "<p>0=true</p>\n<p>1=compass</p>", "templateType": "anything", "can_override": false}, "bearing1": {"name": "bearing1", "group": "Ungrouped variables", "definition": "random(200..250)", "description": "", "templateType": "anything", "can_override": false}, "bearing2": {"name": "bearing2", "group": "Ungrouped variables", "definition": "random(290..340)", "description": "", "templateType": "anything", "can_override": false}, "dsp_bearing1": {"name": "dsp_bearing1", "group": "Ungrouped variables", "definition": "if(is_compass=1,\"S\"+string(bearing1-180)+\"\u00b0W\",string(bearing1)+\"\u00b0\")", "description": "", "templateType": "anything", "can_override": false}, "dsp_bearing2": {"name": "dsp_bearing2", "group": "Ungrouped variables", "definition": "if(is_compass=1,\"N\"+string(360-bearing2)+\"\u00b0W\",string(bearing2)+\"\u00b0\")", "description": "", "templateType": "anything", "can_override": false}, "a1": {"name": "a1", "group": "Ungrouped variables", "definition": "bearing1-180", "description": "", "templateType": "anything", "can_override": false}, "a2": {"name": "a2", "group": "Ungrouped variables", "definition": "360-bearing2", "description": "", "templateType": "anything", "can_override": false}, "a3": {"name": "a3", "group": "Ungrouped variables", "definition": "180-a1-a2", "description": "", "templateType": "anything", "can_override": false}, "nw_len": {"name": "nw_len", "group": "Ungrouped variables", "definition": "precround(sin(radians(a2))/sin(radians(a3))*length,2)", "description": "", "templateType": "anything", "can_override": false}, "sw_len": {"name": "sw_len", "group": "Ungrouped variables", "definition": "precround(sin(radians(a1))/sin(radians(a3))*length,2)", "description": "", "templateType": "anything", "can_override": false}, "area1": {"name": "area1", "group": "Ungrouped variables", "definition": "round(0.5*nw_len*sw_len*sin(radians(a3)))", "description": "", "templateType": "anything", "can_override": false}, "area2": {"name": "area2", "group": "Ungrouped variables", "definition": "round(0.5*length*sw_len*sin(radians(a2)))", "description": "", "templateType": "anything", "can_override": false}, "area3": {"name": "area3", "group": "Ungrouped variables", "definition": "round(0.5*nw_len*length*sin(radians(a1)))", "description": "", "templateType": "anything", "can_override": false}, "area": {"name": "area", "group": "Ungrouped variables", "definition": "round((area1+area2+area3)/3)", "description": "", "templateType": "anything", "can_override": false}, "perimeter": {"name": "perimeter", "group": "Ungrouped variables", "definition": "round(length+nw_len+sw_len)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["length", "is_compass", "bearing1", "bearing2", "dsp_bearing1", "dsp_bearing2", "a1", "a2", "a3", "nw_len", "sw_len", "area1", "area2", "area3", "area", "perimeter"], "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>What is the area of the park, in square metres? Round your answer to the nearest square metre.</p>", "minValue": "{area}-1", "maxValue": "{area}+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>A&nbsp;fence needs to be built around the perimeter of the park.&nbsp;How many metres of fencing is required?</p>\n<p>Round your answer to the nearest metre.</p>", "minValue": "perimeter-1", "maxValue": "perimeter+1", "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/"}]}