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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.
", "licence": "None specified"}, "statement": "A patient takes a drug every $\\var{P}$ hours each day. The concentration, $c$, of the drug in the patient’s blood $t$ hours after the start of the treatment is modelled by the equation:
$c=\\var{A}\\cos{\\left(k\\left(t-\\var{H}\\right)\\right)}+\\var{V}$
a) The equation
\n\\[ \\begin{split} c=\\var{A}\\cos{\\left(k\\left(t-\\var{H}\\right)\\right)}+\\var{V} \\end{split} \\]
\nis a wave function.
\nThe general form of a wave function can be written as:
\n\\[ \\begin{split} c=A\\cos{\\left(\\frac{2 \\pi}{P}\\left(t-H\\right)\\right)}+V \\end{split} \\]
\nWhere $V$ is the average value, $A$ is the amplitute, $H$ the phase and $P$ the period.
\nBy comparying the equation with the general form we can notice that
\n\\[ k= \\frac{2 \\pi}{P} \\]
\nWe know that the patient takes the drug every $\\var{P}$ hours each day. Therefore, the period is $P=\\var{P}$. Therefore,
\n\\[ \\begin{split} k&= \\frac{2 \\pi}{\\var{P}} \\\\ &=\\simplify{ {2*pi} / {P}} \\end{split} \\]
\nSo, $k=\\simplify{ {2*pi} / {P}}$ and we can now rewrite the equation as:
\n\\[ \\begin{split} c=\\var{A}\\cos{\\left(\\simplify{ {2*pi} / {P}}\\left(t-\\var{H}\\right)\\right)}+\\var{V} \\end{split} \\]
\n\nb) We know that the wave function is a transformation of the trigonometric function cos(x).
\nClink on the link for a visual representation of the wave function.
\nhttps://www.desmos.com/calculator/ssqdx7ys7k
\nWe 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{ {2*pi} / {P}}\\left(t-\\var{H}\\right)\\right)}=1$. Therefore,
\n\\[ \\begin{split} c_{max}&=\\var{A}\\times 1 +\\var{V} \\\\ &= \\var{A} +\\var{V} \\\\ &=\\var{A+V} \\end{split} \\]
\nSo, $c_{max}=\\var{max}$ mg/L.
\n\nc) 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
\n\\[ \\begin{split} c=\\var{A}\\cos{\\left(\\simplify{ {2*pi} / {P}}\\left(t-\\var{H}\\right)\\right)}+\\var{V} \\end{split} \\]
\nTherefore, when $t=\\var{t1}$ the equation become
\n\\[ \\begin{split} c_{\\var{t1}}&=\\var{A}\\cos{\\left(\\simplify{ {2*pi} / {P}}\\left(\\var{t1}-\\var{H}\\right)\\right)}+\\var{V} \\\\ &=\\var{A}\\cos{\\left(\\simplify{ {2*pi} / {P}} \\times \\var{t1-H}\\right)}+\\var{V} \\\\ &=\\var{A}\\cos{\\left(\\simplify{ {2*pi*(t1-H)}/{P}}\\right)}+\\var{V} \\\\ &=\\var{ansC} \\end{split} \\]
\nSo, $ c_{\\var{t1}}=\\var{ansc} $ mg/L.
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\n$k=$[[0]]
\nGive your answer as a fraction in terms of $\\pi$ (you can type: pi).
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\n$c_{max}=$[[0]] mg/L
", "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": "{max}", "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": "What is the concentration of the drug $\\var{t1}$ hours after taking it for the first time?
\n$c_{\\var{t1}}=$[[0]]
\nGive your answer rounded to 2 decimal places, if needed.
", "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": "{ansc}", "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": "always", "penaltyVisibility": "always"}, {"name": "Trigonometry: Introduction to Wave Functions - Electromagnetic Waves", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Evi Papadaki", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/18113/"}], "tags": [], "metadata": {"description": "Using the given information to complete the equation $y= A \\cos{ \\left( \\frac{2 \\pi}{P} x \\right) }+V $ that describes an electromagnetic wave and calculating the smallest angle, $x$, for which $y=y_0$.
", "licence": "None specified"}, "statement": "The shape of an electromagnetic wave is modelled by the equation:
$y=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}+\\var{v}$
where $x$ is measured in radians.
", "advice": "The equation
\n\\[ \\begin{split} y=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}+\\var{v} \\end{split} \\]
\nis a wave function.
\nThe general form of a wave function can be written as:
\n\\[ \\begin{split} y=A\\cos{\\left(\\frac{2 \\pi}{P}\\left(x-H\\right)\\right)}+V \\end{split} \\]
\nWhere $V$ is the average value, $A$ is the amplitute, $H$ the phase and $P$ the wavelenght (also called period).
\nBy comparying the equation with the general form we can notice that:
\na) The amplitude of the wave is $A=\\var{a}$
\nb)The period can be found by solving the equation
\n\\[ \\begin{split} \\frac{\\pi}{\\simplify{{p}/2}} &= \\frac{2 \\pi}{P} \\\\ \\pi P&= \\simplify{{p}/2}\\times 2\\pi \\\\ \\pi P&= {p}\\pi \\\\ P&=\\frac{{p} \\pi}{\\pi} \\\\ P &=\\var{p} \\end{split} \\]
\nSo, the wavelenght is $P=\\var{p}$.
\nTo find the smallest value of $x$ for which $y=\\var{y1}$. We need to substitute $y=\\var{y1}$ in the equation and solve for $x$.
\n\\[ \\begin{split} \\var{y1}&=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}+\\var{v} \\\\ \\var{y1}-\\var{v} &=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)} \\\\ \\var{y1-v} &=\\var{a}\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)} \\\\ \\frac{\\var{y1-v}}{\\var{a}} &=\\cos{\\left(\\frac{\\pi x}{\\simplify{{p}/2}}\\right)}\\end{split} \\]
\nWe take $\\cos^{-1}$ from both sides:
\n\\[ \\begin{split} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)&=\\frac{\\pi x}{\\simplify{{p}/2}} \\\\ \\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)&=\\pi x \\\\ \\frac{\\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)}{\\pi}&=x \\end{split} \\]
\nWe can use the calculator to find that $x= \\frac{\\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)}{\\pi}=\\var{ansC}$ radians (rounded to 2 decimal places).
\nWe can use the calculator to find that $x= \\frac{\\simplify{{p}/2} \\cos^{-1} \\left(\\frac{\\var{y1-v}}{\\var{a}} \\right)}{\\pi}=\\var{ansC}$ radians.
\nWrite down the amplitude of this wave.
\n$A=$[[0]]
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\nThe wavelength is [[0]].
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\n$x=$[[0]] radians.
\nGive your answer rounded to 2 decimal places, if needed.
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