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{"name": "Differentiation: Polynomial Functions - Max Temperature", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Differentiation: Polynomial Functions - Max Temperature", "tags": [], "metadata": {"description": "<p>Finding the stationary point (maximum) of a quadratic equation in a contextualised problem.&nbsp;</p>", "licence": "None specified"}, "statement": "<p>The temperature, $T$&deg;C of the air at height $z$ metres inside a blast furnace is modelled by the equation:</p>\n<p>\\[ T=\\var{a}z^2+\\var{b}z+\\var{c} \\]</p>", "advice": "<p>a) We first need to differentiate the formula of the curve in respect of $z$.</p>\n<p style=\"font-weight: 400;\">\\[ \\frac{dT}{dz} = 2 \\times (\\var{a}) z+ \\var{b}+0 \\\\ \\implies \\frac{dT}{dz}= \\simplify{2*{a}}z+\\var{b} \\]</p>\n<p style=\"font-weight: 400;\">Now to find the gradient at the point where $z=\\var{z_1}$ we need to substitute the value in the equation we found in the previous step.</p>\n<p style=\"font-weight: 400;\">\\[ \\frac{dT}{dz}=\\simplify{2*{a}}\\times \\var{z_1} +\\var{b} \\\\ \\implies \\frac{dT}{dz}= \\simplify{2*{a}*{z_1}}+\\var{b} \\\\ \\implies \\frac{dT}{dz}= \\simplify{2*{a}*{z_1}+{b}} \\]</p>\n<p>b) The height at which the temperature reaches the maximum can be calculated by solving the equation:&nbsp;</p>\n<p style=\"font-weight: 400;\">\\[ \\frac{dT}{dz}=0\\]</p>\n<p style=\"font-weight: 400;\">Therefore:</p>\n<p style=\"font-weight: 400;\">\\[ \\simplify{2*{a}}z+\\var{b}=0 \\\\ \\implies \\simplify{2*{a}}z= - \\var{b} \\\\ \\implies z = \\frac{-\\var{b}}{\\simplify{2*{a}}} \\\\ \\implies z= \\simplify{{-{b}}/{2*{a}}} \\\\ z=\\var{ans_b_rounded} \\text{m (2.d.p.)} \\]</p>\n<p>c) Now that we know that $z=\\var{ans_b_rounded}$m is the height that results in the maximum temperature, we can substitute this value in the original equation of the curve and calculate the maximum temperature.&nbsp;</p>\n<p style=\"font-weight: 400;\">\\[ T_{max}= \\var{a} \\times \\var{ans_b_rounded}^2 + \\var{b} \\times \\var{ans_b_rounded} + \\var{c} \\\\ \\implies T_{max}=\\var{a} \\times \\var{ans_b_rounded^2}+ \\var{b} \\times \\var{ans_b_rounded}+\\var{c} \\\\ \\implies T_{max}=\\simplify{{a*ans_b_rounded^2}}+ \\simplify{{b*ans_b_rounded}}+\\var{c} \\\\ \\implies T_{max}=\\simplify{{a*ans_b_rounded^2+b*ans_b_rounded+c}}\\]</p>\n<p style=\"font-weight: 400;\">So, if rounded,&nbsp;&nbsp;$T_{max}=\\var{ans_c_rounded}$.</p>\n<p style=\"font-weight: 400;\">&nbsp;</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Problem posing", "definition": "-random ( 0.5 .. 1.55 #0.05 except 1)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", 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"variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Find the rate of decrease of the temperature with height at the point where $z=\\var{z_1}$.</p>\n<p>When $z=\\var{z_1}$ then $\\frac{dT}{dz}=$ [[0]].</p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans_a}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, 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temperarure is $T_{max}=$[[0]]$^\\circ C$.</p>\n<p><em>Give your answer rounded to 2 decimal places, if needed.</em></p>\n<p></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans_c_rounded}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": 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