// Numbas version: finer_feedback_settings
{"name": "Solving Hyperbolic Equations", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Solving Hyperbolic Equations", "tags": [], "metadata": {"description": "<p>Using double angle and pythagorean identities (2 parts)</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "<p>Find the <strong>largest</strong> solution to each of the following equations, giving your answers in terms of natural logarithms.</p>", "advice": "<p><strong>a)</strong></p>\n<p>We'd like the arguments of the cosh functions to be the same, so we use the identity</p>\n<p style=\"text-align: center;\">$\\cosh(2\\theta)=2\\cosh^2(\\theta)-1$</p>\n<p>To attain</p>\n<p style=\"text-align: center;\">$(2\\cosh^2(\\var{a}x)-1)-\\cosh(\\var{a}x)=\\var{c}$</p>\n<p style=\"text-align: center;\">$2\\cosh^2(\\var{a}x)\\cosh(\\var{a}x)-\\var{c+1}=0$.</p>\n<p>Proceed to solve this quadratic in $\\cosh(\\var{a}x)$ by factorisation or using the quadratic formula to attain</p>\n<p style=\"text-align: center;\">$\\cosh(\\var{a}x)=\\var{(k+1)/2}, \\var{-k/2} .$</p>\n<p>Taking the only valid solution $\\var{(k+1)/2}$ (since $\\cosh(x)\\ge1$) and using the definition of $\\text{arcosh}(\\var{a}x)$, we now have</p>\n<p style=\"text-align: center;\">$\\var{a}x=\\ln(\\var{(k+1)/2}+\\sqrt{(\\var{(k+1)/2})^2-1})$</p>\n<p style=\"text-align: center;\">$x=\\frac{1}{\\var{a}}\\ln(\\var{(k+1)/2}+\\sqrt{\\var{((k+1)/2)^2-1}})$</p>\n<p><strong>b)</strong></p>\n<p>We begin by making use of the identity</p>\n<p style=\"text-align: center;\">$\\cosh^2(x) - \\sinh^2(x) = 1$</p>\n<p style=\"text-align: left;\">That is</p>\n<p style=\"text-align: center;\">$\\sinh^2(x) = \\cosh^2(x) - 1$</p>\n<p style=\"text-align: left;\">We can substitute this into the main equation</p>\n<p style=\"text-align: center;\">$\\begin{aligned}<br/>\\var{D}\\sinh^2(x) - \\var{B}\\cosh(x) &amp;= \\var{B} \\\\<br/>\\var{D}(\\cosh^2(x) - 1) - \\var{B}\\cosh(x) - \\var{B} &amp;= 0 \\\\<br/>\\var{D}\\cosh^2(x) - \\var{B}\\cosh(x) - \\var{B + D} &amp;= 0<br/>\\end{aligned}$</p>\n<p style=\"text-align: left;\">This equation takes the form of a quadratic, so we subsitute $y = \\cosh(x)$ to get</p>\n<p style=\"text-align: center;\">$\\var{D}y^2 - \\var{B}y - \\var{B+D} = 0$</p>\n<p style=\"text-align: left;\">Then, by the quadratic formula (using $+$ to find the largest solution) we have</p>\n<p style=\"text-align: center;\">$\\begin{aligned}<br/>y &amp;= \\frac{\\var{B} + \\sqrt{\\var{B^2} + \\simplify{ -4*{D}*(-{B}-{D})}}}{\\var{2D}} \\\\<br/>&amp;= \\frac{\\var{B} + \\sqrt{\\simplify{{B^2} + -4*{D}*(-{B}-{D})}}}{\\var{2D}} \\\\<br/>&amp;= \\frac{\\var{B} + \\simplify{{({B^2} + -4*{D}*(-{B}-{D}))^(1/2)}}}{\\var{2D}} \\\\<br/>&amp;= \\simplify{({B} + {({B^2} + -4*{D}*(-{B}-{D}))^(1/2)})/({2*D})} \\\\<br/>\\end{aligned}$</p>\n<p style=\"text-align: left;\">Therefore, finally we have</p>\n<p style=\"text-align: center;\">$y = \\cosh(x) = \\frac{\\var{B + D}}{\\var{D}}$</p>\n<p style=\"text-align: left;\">Which gives us</p>\n<p style=\"text-align: center;\">$x = \\cosh^{-1}(y) = \\ln\\left( \\frac{\\var{B + D}}{\\var{D}} + \\sqrt{\\left(\\frac{\\var{B + D}}{\\var{D}}\\right)^2 - 1} \\right)$</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "k*(k+1)/2 - 1", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "d": {"name": "d", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "c", "k", "b", "d"], "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": "<p style=\"text-align: left;\">$\\cosh(\\var{2*a}x) - 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