// Numbas version: exam_results_page_options
{"name": "Geometric series - practical application 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["dur", "n", "m1", "m2", "m", "dayn"], "name": "Geometric series - practical application 2", "tags": [], "preamble": {"css": "", "js": ""}, "advice": "<p>The $n$<sup>th</sup> term in a geometric progression is given by $a_1 \\times r^{n-1}$ where $a_1$ is the first term and $r$ is the common ratio.&nbsp;</p>\n<p>We are given that the first term is $\\var{n}$ and the $\\var{dayn}$<sup>th</sup> term is $\\simplify{{n}*{m}^({dayn}-1)}$</p>\n<p>Hence we know that $r^{\\var{dayn}-1} = \\frac{\\simplify{{n}*{m}^({dayn}-1)}}{\\var{n}} =&nbsp;\\simplify{{m}^({dayn}-1)}$</p>\n<p>So we find that $r=\\var{m}$</p>\n<p></p>\n<p>If the pump were to extract air indefinitely, we look to the formula $S_n = \\frac{a_1}{1-r}$ to find $S_n = \\frac{\\var{n}}{1-\\var{m}} = \\simplify{{n}/(1-{m})}$</p>", "rulesets": {}, "parts": [{"vsetrangepoints": 5, "prompt": "<p>Determine the common ratio of the geometric series</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all,fractionnumbers", "scripts": {}, "answer": "{m}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"vsetrangepoints": 5, "prompt": "<p>Calculate the total amount of air that could be extracted from the bottle if the pump were to extract air indefinitely</p>", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{n}/(1-{m})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "<p>A pump is used to extract air from a bottle. The first operation of the pump extracts $\\var{n}$ cm$^3$ of air and subsequent extractions follow a geometric series. Extraction number $\\var{dayn}$ of the pump extracts $\\simplify{{n}*{m}^({dayn}-1)}$ cm$^3$ of air.</p>", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": "{dayn}<{dur}"}, "variables": {"m": {"definition": "1-1/m1", "templateType": "anything", "group": "Ungrouped variables", "name": "m", "description": ""}, "n": {"definition": "random(10..50)", "templateType": "anything", "group": "Ungrouped variables", "name": "n", "description": ""}, "dayn": {"definition": "random(3..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "dayn", "description": ""}, "m1": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "m1", "description": ""}, "m2": {"definition": "random(4..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "m2", "description": ""}, "dur": {"definition": "random(10..50)", "templateType": "anything", "group": "Ungrouped variables", "name": "dur", "description": ""}}, "metadata": {"notes": "", "description": "", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}]}], "contributors": [{"name": "joshua boddy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/557/"}]}