// Numbas version: finer_feedback_settings
{"name": "Optimisation: Find quantity to maximise profits", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Optimisation: Find quantity to maximise profits", "tags": [], "metadata": {"description": "<p>Given price, marginal cost and fixed cost, find the quantity that maximises profits.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>A demand function for a generic good, where $P$ is the price and $q$ is the quantity is: $P = \\var{p} - q$. The Marginal Cost is given by, $MC = \\simplify{{x}q+{y}}$, and fixed cost is, $FC = \\var{z}$. Find the quantity $q$ to maximise profits.</p>", "advice": "<p>The total profit can be found by calculating the total revenue (price times quantity) subtract the total cost (marginal cost times quantity add fixed costs).</p>\n<p>As an equation this looks like:</p>\n<p>\\[ \\begin{split} \\pi &amp;= TR - TC \\\\ &amp;= Pq - (MC \\times q + FC) \\end{split} \\]</p>\n<p>Using the equations in this question we have:</p>\n<p>\\[ \\pi = \\simplify{({p} -q)q - ({x}q+{y})q - {z}} \\]</p>\n<p>Collecting like terms:</p>\n<p>\\[ \\pi = \\simplify[!noLeadingMinus]{-{x+1}q^2 + {p-y}q +{z}} \\]</p>\n<p>Differentiating with respect to $q$ gives:</p>\n<p>\\[ \\frac{d\\pi}{dq} = \\var{-2*(x+1)}q + \\var{p-y} \\]</p>\n<p>Using the first order condition $ \\frac{d\\pi}{dq} = 0 $ we have:</p>\n<p>\\[ \\var{-2*(x+1)}q + \\var{p-y}= 0 \\]</p>\n<p>Solving to find $q$:</p>\n<p>\\[ q = \\var{-(p-y)/(-2*(x+1))} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"y": {"name": "y", "group": "Ungrouped variables", "definition": "random(20..60)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(100..400)", "description": "", "templateType": "anything", "can_override": false}, "x": {"name": "x", "group": "Ungrouped variables", "definition": "random(1..y-10)", "description": "", "templateType": "anything", "can_override": false}, "z": {"name": "z", "group": "Ungrouped variables", "definition": "random(10..50)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "-(p-y)/(-2*(x+1))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "2(x+1)|(p-y) ", "maxRuns": "500"}, "ungrouped_variables": ["p", "y", "x", "z", "ans"], "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>$q$ = [[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ans", "maxValue": "ans", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "type": "question", "contributors": [{"name": "Ed Southwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2415/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}], "resources": []}]}], "contributors": [{"name": "Ed Southwood", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/2415/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}