// Numbas version: exam_results_page_options
{"name": "Manipulation of exponential formula", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"metadata": {"description": "<p>Manipulation of an exponential function</p>", "licence": "Creative Commons Attribution 4.0 International"}, "ungrouped_variables": ["k", "c", "m", "d"], "statement": "<p>Rearrange the following expression to make \\(x\\) the subject:</p>\n<p>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;\\(y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\\)</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "name": "Manipulation of exponential formula", "functions": {}, "variable_groups": [], "tags": [], "extensions": [], "variables": {"m": {"definition": "random(1.6..5#0.2)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "m"}, "c": {"definition": "random(2..10#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "c"}, "k": {"definition": "random(2..12#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "k"}, "d": {"definition": "random(1..14#1)", "description": "", "group": "Ungrouped variables", "templateType": "randrange", "name": "d"}}, "rulesets": {}, "advice": "<p style=\"text-align: left; padding-left: 60px;\">\\(y=\\var{k}(1-\\var{c}e^{\\var{m}x+\\var{d}})\\)</p>\n<p style=\"text-align: left;\">Working from the outside in,&nbsp;we divide across by \\(\\var{k}\\) &nbsp;&nbsp;</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\frac{y}{\\var{k}}=1-\\var{c}e^{\\var{m}x+\\var{d}}\\)</p>\n<p style=\"text-align: left;\">We can bring the \\(x\\) variable to the left hand side and move the \\(y\\) variable to the right hand side</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\var{c}e^{\\var{m}x+\\var{d}}=1-\\frac{y}{\\var{k}}\\)</p>\n<p style=\"text-align: left;\">Again working from the outside in we divide across by \\(\\var{c}\\)</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(e^{\\var{m}x+\\var{d}}=\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\)</p>\n<p style=\"text-align: left;\">Taking the natural log of both&nbsp;sides eliminates the \\(e\\) from the left hand side.&nbsp;</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\var{m}x+\\var{d}=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)\\)</p>\n<p style=\"text-align: left;\">Subtract \\(\\var{d}\\) from both sides</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(\\var{m}x=ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}\\)</p>\n<p style=\"text-align: left;\">and finally divide by \\(\\var{m}\\) to get</p>\n<p style=\"text-align: left; padding-left: 60px;\">\\(x=\\frac{ln\\left(\\frac{1-\\frac{y}{\\var{k}}}{\\var{c}}\\right)-\\var{d}}{\\var{m}}\\)</p>", "preamble": {"css": "", "js": ""}, "parts": [{"type": "gapfill", "scripts": {}, "variableReplacements": [], "marks": 0, "gaps": [{"type": "jme", "checkingtype": "absdiff", "marks": "2", "showpreview": true, "checkingaccuracy": 0.001, "expectedvariablenames": [], "showFeedbackIcon": true, "answer": "(ln((1-y/{k})/{c})-{d})/{m}", "scripts": {}, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "vsetrangepoints": 5, "vsetrange": [0, 1], "showCorrectAnswer": true, "checkvariablenames": false}], "variableReplacementStrategy": "originalfirst", "prompt": "<p>\\(x =\\)&nbsp;[[0]]</p>", "showCorrectAnswer": true, "showFeedbackIcon": true}], "type": "question", "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}]}], "contributors": [{"name": "Nigel Atkins", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/275/"}]}