// Numbas version: exam_results_page_options
{"name": "Clare's copy of Apply the cosine rule", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["c4", "s3", "cc0", "temp2", "temp1", "b0", "cc3", "b3", "u2", "q0", "q3", "c0", "cc5", "s2", "s0", "cc1", "u0", "u3", "cc2", "aa5", "aa4", "aa1", "aa0", "aa3", "aa2", "x2", "c31", "c32", "a0", "a3", "bb0", "s5", "c3", "c2", "c1", "x1", "c02", "x4", "x5", "p3", "p0", "r0", "r3", "bb3", "t5", "t2", "t3", "t0", "u5", "c5", "cc4", "c01", "bb5", "bb4", "check2", "bb2", "bb1", "check1"], "name": "Clare's copy of Apply the cosine rule", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>a) Use the Cosine Rule to find $a$: $a^2=b^2+c^2-2bc \\cos A$. &nbsp;&nbsp;</p>\n<p>\\[a^2=\\var{b0}^2+\\var{c0}^2-2 \\times \\var{b0}\\times\\var{c0} \\times \\cos (\\var{aa0})=\\var{b0^2}+\\var{c0^2}-\\var{2*b0*c0} \\times \\var{cos (aa0)}\\]</p>\n<p>\\[=\\var{b0^2+c0^2-2*b0*c0* cos (aa0)}.\\]</p>\n<p>Hence $a=\\sqrt{\\var{b0^2+c0^2-2*b0*c0* cos (aa0)}}=\\var{sqrt(b0^2+c0^2-2*b0*c0* cos (aa0))}$. To the nearest integer, this is $\\var{a0}$.</p>\n<p>b) Use the Cosine Rule to find $b$: $b^2=a^2+c^2-2ac \\cos B$. &nbsp;&nbsp;</p>\n<p>\\[b^2=\\var{a3}^2+\\var{c3}^2-2 \\times \\var{a3}\\times\\var{c3} \\times \\cos (\\var{bb3})=\\var{a3^2}+\\var{c3^2}-\\var{2*a3*c3} \\times \\var{cos (bb3)}\\]</p>\n<p>\\[=\\var{a3^2+c3^2-2*a3*c3* cos (bb3)}.\\]</p>\n<p>Hence $b=\\sqrt{\\var{a3^2+c3^2-2*a3*c3* cos (bb3)}}=\\var{sqrt(a3^2+c3^2-2*a3*c3* cos (bb3))}$. To the nearest integer, this is $\\var{b3}$.</p>", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers", "!noLeadingMinus"]}, "parts": [{"stepsPenalty": 1, "prompt": "<p>$A=\\var{AA0}$, $b=\\var{b0}$, $c=\\var{c0}$</p>\n<p>Side length $a=$ [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "<p>Use the Cosine Rule to find $a$: $a^2=b^2+c^2-2bc \\cos A$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{a0}", "minValue": "{a0}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"prompt": "<p>$B=\\var{BB3}$, $a=\\var{a3}$, $c=\\var{c3}$</p>\n        <p>Side length $b=$ [[0]]</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "{b3}", "minValue": "{b3}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 2, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "extensions": [], "statement": "<p>Suppose that $\\Delta ABC$ is a triangle with all interior angles $&lt; \\dfrac{\\pi}{2}$ (in other words, an acute triangle). Here all angles are expressed in radians. Suppose also that standard naming conventions are used as indicated in the picture below (not necessarily an accurate picture of $\\Delta ABC$).</p>\n<p style=\"margin: 0px;\">Given the following two sides and an angle, determine the third side length. Write down the side length as a whole number.</p>\n<p style=\"margin: 0px;\"></p>\n<p style=\"margin: 0px;\"></p>\n<p style=\"margin: 0px;\"><img src=\"resources/images/Triangle_1.PNG\"/></p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"cc0": {"definition": "precround(arccos(r0),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc0", "description": ""}, "temp2": {"definition": "b0-temp1", "templateType": "anything", "group": "Ungrouped variables", "name": "temp2", "description": ""}, "temp1": {"definition": "a0*t0/s0", "templateType": "anything", "group": "Ungrouped variables", "name": "temp1", "description": ""}, "b0": {"definition": "random(10..25)", "templateType": "anything", "group": "Ungrouped variables", "name": "b0", "description": ""}, "u3": {"definition": "sin(CC3)", "templateType": "anything", "group": "Ungrouped variables", "name": "u3", "description": ""}, "b3": {"definition": "random(7..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "b3", "description": ""}, "u2": {"definition": "sin(cc2)", "templateType": "anything", "group": "Ungrouped variables", "name": "u2", "description": ""}, "q0": {"definition": "(a0^2+c0^2-b0^2)/(2*a0*c0)", "templateType": "anything", "group": "Ungrouped variables", "name": "q0", "description": ""}, "q3": {"definition": "(a3^2+c3^2-b3^2)/(2*a3*c3)", "templateType": "anything", "group": "Ungrouped variables", "name": "q3", "description": ""}, "x1": {"definition": "abs(a0^2-b0^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "x1", "description": ""}, "cc5": {"definition": "precround(CC4,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc5", "description": ""}, "s2": {"definition": "sin(aa2)", "templateType": "anything", "group": "Ungrouped variables", "name": "s2", "description": ""}, "s0": {"definition": "sin(aa0)", "templateType": "anything", "group": "Ungrouped variables", "name": "s0", "description": ""}, "cc1": {"definition": "pi-aa0-bb0", "templateType": "anything", "group": "Ungrouped variables", "name": "cc1", "description": ""}, "u0": {"definition": "sin(cc0)", "templateType": "anything", "group": "Ungrouped variables", "name": "u0", "description": ""}, "cc3": {"definition": "precround(arccos(r3),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc3", "description": ""}, "cc2": {"definition": "precround(cc1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "cc2", "description": ""}, "aa5": {"definition": "precround(AA4,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "aa5", "description": ""}, "aa4": {"definition": "pi-BB3-CC3", "templateType": "anything", "group": "Ungrouped variables", "name": "aa4", "description": ""}, "aa1": {"definition": "pi-bb0-cc0", "templateType": "anything", "group": "Ungrouped variables", "name": "aa1", "description": ""}, "aa0": {"definition": "precround(arccos(p0),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "aa0", "description": ""}, "aa3": {"definition": "precround(arccos(p3),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "aa3", "description": ""}, "aa2": {"definition": "precround(aa1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "aa2", "description": ""}, "x5": {"definition": "a3^2+b3^2", "templateType": "anything", "group": "Ungrouped variables", "name": "x5", "description": ""}, "c31": {"definition": "ceil(sqrt(x4))", "templateType": "anything", "group": "Ungrouped variables", "name": "c31", "description": ""}, "c32": {"definition": "ceil(min(a3,b3)*0.05)", "templateType": "anything", "group": "Ungrouped variables", "name": "c32", "description": ""}, "c3": {"definition": "random(c4..c5 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c3", "description": ""}, "check1": {"definition": "pi-AA0-BB0-CC0", "templateType": "anything", "group": "Ungrouped variables", "name": "check1", "description": ""}, "a0": {"definition": "random(10..25)", "templateType": "anything", "group": "Ungrouped variables", "name": "a0", "description": ""}, "a3": {"definition": "random(7..20)", "templateType": "anything", "group": "Ungrouped variables", "name": "a3", "description": ""}, "s5": {"definition": "sin(AA5)", "templateType": "anything", "group": "Ungrouped variables", "name": "s5", "description": ""}, "x2": {"definition": "a0^2+b0^2", "templateType": "anything", "group": "Ungrouped variables", "name": "x2", "description": ""}, "c2": {"definition": "floor(sqrt(x2))", "templateType": "anything", "group": "Ungrouped variables", "name": "c2", "description": ""}, "c1": {"definition": "max(c01,c02)", "templateType": "anything", "group": "Ungrouped variables", "name": "c1", "description": ""}, "c0": {"definition": "random(c1..c2 except 0)", "templateType": "anything", "group": "Ungrouped variables", "name": "c0", "description": ""}, "bb5": {"definition": "precround(BB4,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "bb5", "description": ""}, "x4": {"definition": "abs(a3^2-b3^2)", "templateType": "anything", "group": "Ungrouped variables", "name": "x4", "description": ""}, "c4": {"definition": "max(c31,c32)", "templateType": "anything", "group": "Ungrouped variables", "name": "c4", "description": ""}, "p3": {"definition": "(c3^2+b3^2-a3^2)/(2*c3*b3)", "templateType": "anything", "group": "Ungrouped variables", "name": "p3", "description": ""}, "p0": {"definition": "(c0^2+b0^2-a0^2)/(2*c0*b0)", "templateType": "anything", "group": "Ungrouped variables", "name": "p0", "description": ""}, "r0": {"definition": "(a0^2+b0^2-c0^2)/(2*a0*b0)", "templateType": "anything", "group": "Ungrouped variables", "name": "r0", "description": ""}, "r3": {"definition": "(a3^2+b3^2-c3^2)/(2*a3*b3)", "templateType": "anything", "group": "Ungrouped variables", "name": "r3", "description": ""}, "check2": {"definition": "pi-AA3-BB3-CC3", "templateType": "anything", "group": "Ungrouped variables", "name": "check2", "description": ""}, "t5": {"definition": "sin(BB5)", "templateType": "anything", "group": "Ungrouped variables", "name": "t5", "description": ""}, "t2": {"definition": "sin(bb2)", "templateType": "anything", "group": "Ungrouped variables", "name": "t2", "description": ""}, "t3": {"definition": "sin(BB3)", "templateType": "anything", "group": "Ungrouped variables", "name": "t3", "description": ""}, "t0": {"definition": "sin(bb0)", "templateType": "anything", "group": "Ungrouped variables", "name": "t0", "description": ""}, "u5": {"definition": "sin(CC5)", "templateType": "anything", "group": "Ungrouped variables", "name": "u5", "description": ""}, "s3": {"definition": "sin(AA3)", "templateType": "anything", "group": "Ungrouped variables", "name": "s3", "description": ""}, "c5": {"definition": "floor(sqrt(x5))", "templateType": "anything", "group": "Ungrouped variables", "name": "c5", "description": ""}, "cc4": {"definition": "pi-AA3-BB3", "templateType": "anything", "group": "Ungrouped variables", "name": "cc4", "description": ""}, "c01": {"definition": "ceil(sqrt(x1))", "templateType": "anything", "group": "Ungrouped variables", "name": "c01", "description": ""}, "c02": {"definition": "ceil(min(a0,b0)*0.05)", "templateType": "anything", "group": "Ungrouped variables", "name": "c02", "description": ""}, "bb4": {"definition": "pi-AA3-CC3", "templateType": "anything", "group": "Ungrouped variables", "name": "bb4", "description": ""}, "bb3": {"definition": "precround(arccos(q3),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "bb3", "description": ""}, "bb2": {"definition": "precround(bb1,3)", "templateType": "anything", "group": "Ungrouped variables", "name": "bb2", "description": ""}, "bb1": {"definition": "pi-aa0-cc0", "templateType": "anything", "group": "Ungrouped variables", "name": "bb1", "description": ""}, "bb0": {"definition": "precround(arccos(q0),4)", "templateType": "anything", "group": "Ungrouped variables", "name": "bb0", "description": ""}}, "metadata": {"description": "<p>Two questions testing the application of the Cosine Rule when given two sides and an angle. In these questions, the triangle is always acute and both of the given side lengths are adjacent to the given angle.</p>", "licence": "All rights reserved"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}]}]}], "contributors": [{"name": "Clare Lundon", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/492/"}]}