// Numbas version: exam_results_page_options
{"name": "Newton's law of cooling", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Newton's law of cooling", "tags": [], "metadata": {"description": "", "licence": "None specified"}, "statement": "<p>For water in a particular boiler, the temperature $T$, in $^\\circ$C, as it cools down, is given by Newton's Law of Cooling:</p>\n<p>\\[T=\\simplify{{Ta}+{T0-Ta}e^({-k}*t)}\\]</p>\n<p>where $t$ is the time in seconds.</p>", "advice": "<h4>Part a)</h4>\n<p>{t1/60} minutes is {t1} seconds, so we substitute $t=\\var{t1}$ into the temperature formula:</p>\n<p>\\[T=\\var{Ta}+\\var{T0-Ta}e^{\\var{-k}\\times\\var{t1}}=\\var{precround(Temp1,4)}\\]</p>\n<p>which rounds to {precround(Temp1,1)}$^\\circ$C to 1 decimal place.</p>\n<h4>Part b)</h4>\n<p>Start by swapping sides:</p>\n<p>\\[\\simplify{{Ta}+{T0-Ta}e^({-k}*t)}=T\\]</p>\n<p>Subtract $\\var{Ta}$:</p>\n<p>\\[\\simplify{{T0-Ta}e^({-k}*t)}=T-\\var{Ta}\\]</p>\n<p>Divide by $\\var{T0-Ta}$:</p>\n<p>\\[\\simplify{e^({-k}*t)}=\\frac{T-\\var{Ta}}{\\var{T0-Ta}}\\]</p>\n<p>Take natural log:</p>\n<p>\\[\\simplify{{-k}*t}=\\ln\\left(\\frac{T-\\var{Ta}}{\\var{T0-Ta}}\\right)\\]</p>\n<p>And divide by $\\var{-k}$. A convenient way to do this, is to multiply by $-\\frac{1}{\\var{k}}$:</p>\n<p>\\[t=\\simplify{-1/{k}*ln((T-{Ta})/({T0}-{Ta}))}\\]</p>\n<h4>Part c)</h4>\n<p>Substitute $T=\\var{Temp2}$ into this formula:</p>\n<p>\\[t=-\\frac{1}{\\var{k}}\\times\\ln\\left(\\frac{\\var{Temp2}-\\var{Ta}}{\\var{T0-Ta}}\\right)=\\var{precround(t2,4)}\\]</p>\n<p>This rounds to {precround(t2,0)} seconds.</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"T0": {"name": "T0", "group": "Ungrouped variables", "definition": "random(50..90)", "description": "", "templateType": "anything", "can_override": false}, "Ta": {"name": "Ta", "group": "Ungrouped variables", "definition": "random(8..28)", "description": "", "templateType": "anything", "can_override": false}, "k": {"name": "k", "group": "Ungrouped variables", "definition": "random(0.002..0.007 #0.0001)", "description": "", "templateType": "anything", "can_override": false}, "t1": {"name": "t1", "group": "Ungrouped variables", "definition": "random(120..800 #60)", "description": "", "templateType": "anything", "can_override": false}, "Temp1": {"name": "Temp1", "group": "Ungrouped variables", "definition": "Ta+(T0-Ta)*exp(-k*t1)", "description": "", "templateType": "anything", "can_override": false}, "Temp2": {"name": "Temp2", "group": "Ungrouped variables", "definition": "floor(Ta+random(0.1..0.9 #0.001)*(T0-Ta))", "description": "", "templateType": "anything", "can_override": false}, "t2": {"name": "t2", "group": "Ungrouped variables", "definition": "-1/k*ln((Temp2-Ta)/(T0-Ta))", "description": "", "templateType": "anything", "can_override": false}, "Temp1m": {"name": "Temp1m", "group": "Ungrouped variables", "definition": "Ta+(T0-Ta)*exp(-k*t1/60)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["T0", "Ta", "k", "t1", "Temp1", "Temp2", "t2", "Temp1m"], "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>Calculate the temperature after {t1/60} minutes, correct to 1 decimal place.</p>\n<p>$T=$ [[0]]$^\\circ$C</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "Poor rounding", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>Incorrect rounding/accuracy.</p>", "useAlternativeFeedback": false, "minValue": "Temp1-1", "maxValue": "Temp1+1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "Used mins", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>Did not convert minutes into seconds.</p>", "useAlternativeFeedback": false, "minValue": "Temp1m", "maxValue": "Temp1m", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": "100", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "minValue": "Temp1", "maxValue": "Temp1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": "50", "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": false, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "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>Select the correctly rearranged formula to make $t$ the subject.</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": true, "displayType": "radiogroup", "displayColumns": "1", "showCellAnswerState": true, "choices": ["$t=\\simplify{-1/{k}*ln((T-{Ta})/({T0}-{Ta}))}$", "$t=\\simplify{1/{k}*ln(({T0}-{Ta})/(T-{Ta}))}$", "$t=\\simplify{-{k}*ln(T/({T0}-{Ta})-{Ta})}$", "$t=\\simplify{(ln(T/{k})-{Ta})/({T0}-{Ta})}$", "$t=\\simplify{{k}*(ln(T)/({T0}-{Ta})-{Ta})}$"], "matrix": ["1", 0, 0, 0, 0], "distractors": ["", "", "", "", ""]}, {"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>Hence calculate the time (to the nearest second) when the temperature reaches $\\var{Temp2}^\\circ$C.</p>\n<p>$t=$ [[0]] seconds</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": "2", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "alternatives": [{"type": "numberentry", "useCustomName": true, "customName": "Poor rounding", "marks": "1", "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "alternativeFeedbackMessage": "<p>Incorrect 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