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{"name": "Hyperbolic Functions 1", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Hyperbolic Functions 1", "tags": ["cosh", "hyperbolic equations", "hyperbolic functions", "logarithms", "quadratic equation", "sinh", "solving equations", "solving hyperbolic equations", "solving quadratic equation"], "advice": "\n    <p>Using \\[\\cosh(x)=\\frac{e^x+e^{-x}}{2},\\;\\;\\;\\sinh(x)=\\frac{e^x-e^{-x}}{2}\\] and substituting into the equation gives:</p>\n    <p>\\[\\simplify[std]{{a1+b1}e^x+{a1-b1}e^(-x)={2*c1}}\\]</p>\n    <p>On multiplying throughout by $e^x$, putting $y=e^x$ and tidying up the equation, we get:</p>\n    <p>\\[\\simplify[std]{{m1}* y ^ 2 + {m2} * y + {m3} = 0}\\]</p>\n    <p>This quadratic has solutions:</p>\n    <p>\\[y = \\simplify[std]{{d1}/{al1}},\\;\\;\\;y=\\simplify[std]{{d2}/{be1}}\\]</p>\n    <p>Since $y=e^x$ we see that the solutions in terms of $x$ are:<br/>\\[x = \\ln\\left(\\simplify[std]{{d1}/{al1}}\\right)=\\var{tans2},\\;\\;\\;x=\\ln\\left(\\simplify[std]{{d2}/{be1}}\\right)=\\var{tans3}\\]<br/>both to 3 decimal places.</p>\n    <p>Hence the minimum solution is $x=\\var{ans2}$ and the maximum solution is $x=\\var{ans3}$.</p>\n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n      \n      \n      <p>Input the solutions for $x$ here:(if the solutions are the same input the number twice)</p>\n      \n      \n      \n      <p>Least solution = [[0]]</p>\n      \n      \n      \n      <p>Greatest solution= [[1]]</p>\n      \n      \n      \n      <p>Input both to 3 decimal places.</p>\n      \n      \n        ", "gaps": [{"minvalue": "ans2-tol", "type": "numberentry", "maxvalue": "ans2+tol", "marks": 1.0, "showPrecisionHint": false}, {"minvalue": "ans3", "type": "numberentry", "maxvalue": "ans3", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    <p>Solve the following equation for $x$.</p>\n    <p>\\[\\simplify[std]{{a1}cosh(x)+{b1}sinh(x)={c1}}\\]</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"ans2": {"definition": "precround(ans12,3)", "name": "ans2"}, "ans3": {"definition": "precround(ans21,3)", "name": "ans3"}, "b1": {"definition": "s*(al1*be1-d1*d2)/2", "name": "b1"}, "d2": {"definition": "s6*random(1..8)", "name": "d2"}, "ans21": {"definition": "max(tans2,tans3)", "name": "ans21"}, "d1": {"definition": "if(al1*be1=d*d2,d+2,d)", "name": "d1"}, "al1": {"definition": "s5*random(2,4,6)", "name": "al1"}, "s6": {"definition": "random(1,-1)", "name": "s6"}, "s5": {"definition": 1.0, "name": "s5"}, "m1": {"definition": "round((a1+b1)/h)", "name": "m1"}, "m3": {"definition": "round((a1-b1)/h)", "name": "m3"}, "m2": {"definition": "round(-2*c1/h)", "name": "m2"}, "be1": {"definition": "s6*random(1..9)", "name": "be1"}, "tol": {"definition": 0.0, "name": "tol"}, "ans12": {"definition": "min(tans2,tans3)", "name": "ans12"}, "a2": {"definition": "(al1*be1+d1*d2)/2", "name": "a2"}, "c1": {"definition": "s*(d1*be1+d2*al1)/2", "name": "c1"}, "tans3": {"definition": "precround(ln(d2/be1),3)", "name": "tans3"}, "tans2": {"definition": "precround(ln(d1/al1),3)", "name": "tans2"}, "a1": {"definition": "abs(a2)", "name": "a1"}, "d": {"definition": "s5*random(2,4)", "name": "d"}, "h": {"definition": "gcf(gcf(abs(a1+b1),abs(a1-b1)),abs(2*c1))", "name": "h"}, "s": {"definition": "sign(a2)", "name": "s"}}, "metadata": {"notes": "\n                \t\t<p><strong>2/07/2012:</strong></p>\n                \t\t    <p>Added tags.</p>\n                \t\t    <p>Forced answers to be exactly to 3 decimal places, no tolerances via tolerance variable tol=0.</p>\n                \t\t    <p>Improved display in Advice. Checked calculations.</p>\n                \t\t    <p><strong>19/07/2012:</strong></p>\n                \t\t    <p>Added description. Rechecked calculations.</p>\n                \t\t    <p>&nbsp;</p>\n                \t\t    <p><strong>25/07/2012:</strong></p>\n                \t\t    <p>&nbsp;Added tags.</p>\n                \t\t    <p>&nbsp;Question appears to be working correctly.</p>\n                \t\t    <p>&nbsp;</p>\n                \t\t", "description": "<p>Solve for $x$: $a\\cosh(x)+b\\sinh(x)=c$. There are two solutions for this example.</p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}