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{"name": "Exponential and logarithms: Introduction -  Population Doubling Time", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Exponential and logarithms: Introduction -  Population Doubling Time", "tags": [], "metadata": {"description": "<p>Knowing the doubling time of a population and the population on day $t$, calculate the initial population and the number of days required for the population to reach a threshold.&nbsp;</p>", "licence": "None specified"}, "statement": "<p>The doubling time of a population of flies is $\\var{d}$ days.</p>\n<p>$\\var{days[j]}$ days after the start of an experiment, there were $\\var{d1}$ flies in the population.</p>", "advice": "<p>a) We know that $\\var{ndays[j]}$ days after the start of the experience there are $\\var{d1}$ flies. We also know that the doubling time is ${\\var{d}}$ days.</p>\n<p>In those $\\var{ndays[j]}$ days, the populations would have doubled $\\frac{\\var{ndays[j]}}{\\var{d}}=\\simplify{{ndays[j]}/{d}}$ times. Which means that the initial population, let's call it $n_0$, would have been multiplied by $2^{\\var{int}}$ to result in $\\var{d1}$ flies. So,&nbsp;</p>\n<p>\\[ \\begin{split}&nbsp; \\simplify[!all]{n_0*2^{ndays[j]/d}}&amp;=\\var{d1} \\\\&nbsp; \\simplify{n_0*2^{ndays[j]/d}}&amp;=\\var{d1} \\end{split} \\]</p>\n<p>If we solve the equation we created, we can find the initial population of flies. Therefore,&nbsp;</p>\n<p>\\[ \\begin{split} \\simplify{n_0*2^{ndays[j]/d}}&amp;=&nbsp;&nbsp;\\var{d1} \\\\&nbsp; n_0 &amp;=\\frac{\\var{d1}}{\\var{2^int}}&nbsp; \\end{split} \\]</p>\n<p>Which means that there are $n_0=\\frac{\\var{d1}}{\\var{2^int}}=\\var{n0}~ $ flies at the start of the experiment.&nbsp;</p>\n<p></p>\n<p>b) To calculate the number of days for the population to reach $\\var{n1}$ flies, we need an expression that connects the number of days and the size of the population.</p>\n<p>Let's say it takes $t$ days for the fly population to reach $\\var{n1}$. Since the populatioin has doubling time of $\\var{d}$ days, this means that in $t$ days the population would have doubled $ \\frac{t}{\\var{d}}$ times. In other words, the initial population will have been multiplied by $2^{\\frac{t}{\\var{d}}}$.</p>\n<p>So, we can now substitute this expression into our growth equation:</p>\n<p>\\[ \\begin{split} \\var{n0}\\times 2^{\\frac{t}{\\var{d}}} &amp;= \\var{n1}&nbsp; \\end{split} \\]</p>\n<p>Now, if we solve the equaction for t we will find the number of days.</p>\n<p>\\[ \\begin{split} \\var{n0}\\times 2^{\\frac{t}{\\var{d}}} &amp;= \\var{n1}&nbsp; \\\\ 2^{\\frac{t}{\\var{d}}} &amp;= \\frac{ \\var{n1}}{\\var{n0} } \\\\ 2^{\\frac{t}{\\var{d}}} &amp;= \\var{n1divn0} \\end{split}\\]</p>\n<p>Taking $log_2()$ of both sides:</p>\n<p>\\[ \\begin{split} \\\\ {\\frac{t}{\\var{d}}} &amp;= \\log_2(\\var{n1divn0}) \\\\ t &amp;= \\var{d}\\times\\log_2(\\var{n1divn0}) \\\\ t &amp;= \\var{adviceB} \\end{split}\\].</p>\n<p>Therefore, it takes approximately $\\var{ansb}$ days for the population to reach $\\var{n1}$ flies.</p>\n<p></p>\n<p></p>\n<p><em><strong>Note:&nbsp;</strong>Alternatively, you might already know the formula the formula</em></p>\n<p><em>\\[n=n_0 \\times 2^\\frac{t}{T} \\] </em></p>\n<p><em>Where</em><em>, $n$ represents the size of the population (e.g., number of flies) at time $t$, $n_0$ the initial population and $T$ is the length of each time step. </em></p>\n<p><em>If so, you can use it to find the same results.</em><em></em></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"d": {"name": "d", "group": "Ungrouped variables", "definition": "random(10..25 except [11,13,17,19,23])", "description": "", "templateType": "anything", "can_override": false}, "d1": {"name": "d1", "group": "Ungrouped variables", "definition": "n0*2^int", "description": "", "templateType": "anything", "can_override": false}, "days": {"name": "days", "group": "Ungrouped variables", "definition": "[\"Thirty\", \"Thirty two\", \"Thirty five\", \"Fourty\", \"Fourty two\", \"Fourty five\"]", "description": "", 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"adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p>Calculate the number of flies in the population at the start of the experiment.</p>\n<p>There were [[0]]flies.</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": "{n0}", "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}, {"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>How many days after the start of the experiment did the population reach $\\var{n1}$ flies?</p>\n<p>The population reached $\\var{n1}$ flies in [[0]] days.</p>\n<p><em>If needed round your answer to the nearest integer.</em></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": 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