// Numbas version: exam_results_page_options
{"name": "Trigonometry:  Introduction to Wave Functions - Drug Concentration", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trigonometry:  Introduction to Wave Functions - Drug Concentration", "tags": [], "metadata": {"description": "<p>Using given information to complete the equation $c= A \\cos{ \\left( \\frac{2 \\pi}{P} \\left( t-H \\right) \\right) }+V $ that describes the concentration, $c$, of perscribed drug in a patient's drug over time, $t$. Calculating the maximum concentration and the concentration at a specific time.&nbsp;</p>", "licence": "None specified"}, "statement": "<p>A patient takes a drug every $\\var{P}$ hours each day. The concentration, $c$, of the drug in the patient&rsquo;s blood $t$ hours after the start of the treatment is modelled by the equation:<br/>$c=\\var{A}\\cos{\\left(k\\left(t-\\var{H}\\right)\\right)}+\\var{V}$</p>", "advice": "<p>a) The equation</p>\n<p>\\[ \\begin{split} c=\\var{A}\\cos{\\left(k\\left(t-\\var{H}\\right)\\right)}+\\var{V} \\end{split} \\]</p>\n<p>is a wave function.</p>\n<p>The general form of a wave function can be written as:</p>\n<p>\\[ \\begin{split} c=A\\cos{\\left(\\frac{2 \\pi}{P}\\left(t-H\\right)\\right)}+V \\end{split} \\]</p>\n<p>Where $V$ is the average value, $A$ is the amplitute, $H$ the phase and $P$ the period.&nbsp;</p>\n<p>By comparying the equation with the general form we can notice that&nbsp;</p>\n<p>\\[ k= \\frac{2 \\pi}{P} \\]</p>\n<p>We know that the patient takes the drug every $\\var{P}$ hours each day. Therefore, the period is $P=\\var{P}$. Therefore,&nbsp;</p>\n<p>\\[ \\begin{split} k&amp;= \\frac{2 \\pi}{\\var{P}} \\\\ &amp;=\\simplify{&nbsp; {2*pi} / {P}} \\end{split} \\]</p>\n<p>So, $k=\\simplify{&nbsp; {2*pi} / {P}}$ and we can now rewrite the equation as:</p>\n<p>\\[ \\begin{split} c=\\var{A}\\cos{\\left(\\simplify{&nbsp; {2*pi} / {P}}\\left(t-\\var{H}\\right)\\right)}+\\var{V} \\end{split} \\]</p>\n<p></p>\n<p>b) We know that the wave function is a transformation of the trigonometric function cos(x).&nbsp;</p>\n<p>Clink on the link for a visual representation of the wave function.</p>\n<p>https://www.desmos.com/calculator/ssqdx7ys7k</p>\n<p>We also know that the cosine function has a maximum value of 1. So, the maximum value of the wave function will occure when $ \\cos{\\left(\\simplify{&nbsp; {2*pi} / {P}}\\left(t-\\var{H}\\right)\\right)}=1$. Therefore,&nbsp;</p>\n<p>\\[ \\begin{split} c_{max}&amp;=\\var{A}\\times 1 +\\var{V} \\\\ &amp;= \\var{A} +\\var{V} \\\\ &amp;=\\var{A+V} \\end{split} \\]</p>\n<p>So, $c_{max}=\\var{max}$ mg/L.</p>\n<p></p>\n<p>c) To calculate the concentration of the drug $\\var{t1}$ hours after taking it for the first time, we need to subtitute $t=\\var{t1}$ in the equation</p>\n<p>\\[ \\begin{split} c=\\var{A}\\cos{\\left(\\simplify{&nbsp; {2*pi} / {P}}\\left(t-\\var{H}\\right)\\right)}+\\var{V} \\end{split} \\]</p>\n<p>Therefore, when $t=\\var{t1}$ the equation become</p>\n<p>\\[ \\begin{split} c_{\\var{t1}}&amp;=\\var{A}\\cos{\\left(\\simplify{&nbsp; {2*pi} / {P}}\\left(\\var{t1}-\\var{H}\\right)\\right)}+\\var{V} \\\\ &amp;=\\var{A}\\cos{\\left(\\simplify{&nbsp; {2*pi} / {P}} \\times \\var{t1-H}\\right)}+\\var{V} \\\\ &amp;=\\var{A}\\cos{\\left(\\simplify{&nbsp; {2*pi*(t1-H)}/{P}}\\right)}+\\var{V} \\\\ &amp;=\\var{ansC} &nbsp;\\end{split} \\]</p>\n<p>So, $ c_{\\var{t1}}=\\var{ansc} $ mg/L.</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"p": {"name": "p", "group": "Ungrouped variables", "definition": "Random([6,8,12])", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "h": {"name": "h", "group": "Ungrouped variables", "definition": "random(10..20)", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "random(5..10)", "description": "", "templateType": "anything", "can_override": false}, "t1": {"name": "t1", "group": "Ungrouped variables", "definition": "random (0..24)", "description": "", "templateType": "anything", "can_override": false}, "max": {"name": "max", "group": "Ungrouped variables", "definition": "a+v", "description": "", "templateType": "anything", "can_override": false}, "ansC": {"name": "ansC", "group": "Ungrouped variables", "definition": "precround( a*cos(2*pi/p*(t1-h))+V,2)", "description": "", "templateType": "anything", "can_override": true}}, "variablesTest": {"condition": "(ansc>0)", "maxRuns": 100}, "ungrouped_variables": ["p", "a", "h", "v", "t1", "max", "ansC"], "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 value of the constant $k$.</p>\n<p>$k=$[[0]]</p>\n<p><em>Give your answer as a fraction in terms of $\\pi$ (you can type: pi)</em><em>.</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": null, "answer": "{2/{p}}*{pi}", "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": [], 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"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>What is the concentration of the drug $\\var{t1}$ hours after taking it for the first time?</p>\n<p>$c_{\\var{t1}}=$[[0]]</p>\n<p><em>Give your answer rounded to 2 decimal places, if needed.</em></p>", "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, 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