// Numbas version: exam_results_page_options
{"name": "Logarithms: Solving equations 5", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "name": "Logarithms: Solving equations 5", "tags": ["exponential function", "logarithmic equation", "logarithms", "logs", "natural logarithms", "solving equations", "solving logarithmic equations"], "advice": "\n    \n    \n    <p>We have on solving the equation $\\rho(t)=\\rho_0e^{kt}$ by taking natural logs of both sides that:<br/>\\[k = \\frac{1}{t}\\ln\\left(\\frac{\\rho}{\\rho_0}\\right)\\]</p>\n    \n    \n    \n    <p>a) <br/>\\[\\begin{eqnarray*}\n    \n    k&#38;=&#38; \\frac{1}{\\var{c}}\\ln\\left(\\frac{\\var{a*r} \\times 10^{\\var{b}}}{\\var{a} \\times 10^{\\var{b}}}\\right)\\\\\n    \n    &#38;=&#38; \\frac{1}{\\var{c}}\\ln(\\var{r})\\\\\n    \n    &#38;=&#38;\\var{kvalue}\n    \n    \\end{eqnarray*}\n    \n    \\] to 3 decimal places.</p>\n    \n    \n    \n    <p>b)<br/>\\[\\begin{eqnarray*}\n    \n    k&#38;=&#38; \\frac{1}{\\var{c1}}\\ln\\left(\\frac{\\var{a1*r1} \\times 10^{\\var{b1}}}{\\var{a1} \\times 10^{\\var{b1}}}\\right)\\\\\n    \n    &#38;=&#38; \\frac{1}{\\var{c1}}\\ln(\\var{r1})\\\\\n    \n    &#38;=&#38;\\var{kvalue1}\n    \n    \\end{eqnarray*}\n    \n    \\] to 3 decimal places.</p>\n    \n    \n    ", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "parts": [{"prompt": "\n        \n        \n        <p>$\\rho_0=\\var{a} \\times 10^{\\var{b}},\\;\\;\\;\\;\\;\\rho(\\var{c})=\\var{a*r} \\times 10^{\\var{b}}$</p>\n        \n        \n        \n        <p>$k=\\;\\;$[[0]]</p>\n        \n        \n        ", "gaps": [{"minvalue": "{kvalue}", "type": "numberentry", "maxvalue": "{kvalue}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}, {"prompt": "\n        \n        \n        <p>$\\rho_0=\\var{a1} \\times 10^{\\var{b1}},\\;\\;\\;\\;\\;\\rho(\\var{c1})=\\var{a1*r1} \\times 10^{\\var{b1}}$</p>\n        \n        \n        \n        <p>$k=\\;\\;$[[0]]</p>\n        \n        \n        ", "gaps": [{"minvalue": "{kvalue1}", "type": "numberentry", "maxvalue": "{kvalue1}", "marks": 1.0, "showPrecisionHint": false}], "type": "gapfill", "marks": 0.0}], "extensions": [], "statement": "\n    <p>The size $\\rho(t)$ of a population satisfies:<br/>\\[\\rho(t)=\\rho_0e^{kt}\\]<br/>where $\\rho$ and $k$ are constants.</p>\n  <p>Given the following data, find $k$ in each case. Input your answers to 3 decimal places.</p>\n    ", "variable_groups": [], "progress": "ready", "type": "question", "variables": {"a": {"definition": "random(1..6)", "name": "a"}, "c": {"definition": "random(3..15)", "name": "c"}, "b": {"definition": "random(5..7)", "name": "b"}, "r1": {"definition": "if(r2=r,r+1,r)", "name": "r1"}, "r2": {"definition": "max(round(e^(k1*c1)),2)", "name": "r2"}, "k": {"definition": "random(0.005..0.15#0.005)", "name": "k"}, "a1": {"definition": "a+random(1..3)", "name": "a1"}, "k1": {"definition": "random(0.005..0.15#0.005)", "name": "k1"}, "r": {"definition": "max(round(e^(k*c)),2)", "name": "r"}, "b1": {"definition": "b+random(1,2)", "name": "b1"}, "kvalue1": {"definition": "precround(ln(r1)/c1,3)", "name": "kvalue1"}, "c1": {"definition": "c+random(1..3)", "name": "c1"}, "kvalue": {"definition": "precround(ln(r)/c,3)", "name": "kvalue"}}, "metadata": {"notes": "\n                \t\t<p><strong>2/07/2012:</strong></p>\n                \t\t    <p>Added tags</p>\n                \t\t    <p>Forced answers to both parts to be exact to 3 decimal places.</p>\n                \t\t    <p><strong>19/07/2012:</strong></p>\n                \t\t    <p>Added description.</p>\n                \t\t    <p>Checked calculation.</p>\n                \t\t    <p><strong>25/07/2012:</strong></p>\n                \t\t    <p>Added tags.</p>\n                \t\t    <p>Question appears to be working correctly.</p>\n                \t\t", "description": "<p>Given $\\rho(t)=\\rho_0e^{kt}$, and values for $\\rho(t)$ for $t=t_1$ and a value for $\\rho_0$, find $k$. (Two examples).<br /><br /></p>", "licence": "Creative Commons Attribution 4.0 International"}, "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}]}], "contributors": [{"name": "Bill Foster", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/6/"}]}