// Numbas version: exam_results_page_options
{"name": "Trig application question (tides)", "extensions": ["geogebra"], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Trig application question (tides)", "tags": [], "metadata": {"description": "<p>given trig function applied to tides. students need to find max depth, time of low tide and times between which a boat of given depth can use the port.</p>", "licence": "Creative Commons Attribution-NonCommercial 4.0 International"}, "statement": "", "advice": "<p><span style=\"text-decoration: underline;\"><strong>Part a</strong></span></p>\n<p>Low tide will be at the minimum value on the graph. This will be the central line $\\var{a}$ minus the amplitude of $\\var{b}$ which gives $\\simplify{{a}-{b}}$</p>\n<p></p>\n<p><span style=\"text-decoration: underline;\"><strong>Part b</strong></span></p>\n<p>The graph is shown below.</p>\n<p>It can be found by applying the following transformations to $y=cosx$</p>\n<ul>\n<li>stretch in the $y$ direction by scale factor of $\\var{b}$</li>\n<li>translate up by $\\var{a}$</li>\n<li>stretch scale factor of $\\frac{6}{\\pi}$ in the $x$ direction</li>\n</ul>\n<p>{geogebra_applet('https://www.geogebra.org/m/fzfgusaj',defs)}</p>\n<p></p>\n<p>From the graph you can see that the first high tide after midnight is at $t=12$ which is 12pm.</p>\n<p></p>\n<p>You could also have solved the equation equal to the maximum value as shown below</p>\n<p>\\[\\simplify{{a}+{b}}=\\var{a}+\\var{b}cos \\left( \\frac{\\pi t}{6} \\right) \\]</p>\n<p></p>\n<p><span style=\"text-decoration: underline;\"><strong>Part c</strong></span></p>\n<p>To do this you need to solve as follows</p>\n<p>\\[\\var{c}=\\var{a}+\\var{b}cos \\left( \\frac{\\pi t}{6} \\right) \\]</p>\n<p><br/>Rearranging this gives</p>\n<p>\\begin{align}<br/>cos \\left( \\frac{\\pi t}{6} \\right) &amp; =\\left( \\frac{\\var{c}-\\var{a}}{\\var{b}} \\right) \\\\\\\\<br/>\\left( \\frac{\\pi t}{6}\\right) &amp; = cos^{-1}\\left( \\frac{\\var{c}-\\var{a}}{\\var{b}} \\right)=\\var{pt_6} \\text{&nbsp; or&nbsp; }2\\pi-\\var{pt_6} \\\\\\\\<br/>t &amp; = \\var{t1} \\text{&nbsp; or } \\var{t2}<br/>\\end{align}</p>\n<ul>\n<li>For $\\var{t1}$, you can multiply the $\\simplify{{t1}-{t1_hour}}$ by $60$ to get $\\var{t1_min}$ minutes</li>\n<li>For $\\var{t2}$, you can multiply the $\\simplify{{t2}-{t2_hour}}$ by $60$ to get $\\var{t2_min}$ minutes</li>\n</ul>\n<p></p>\n<p>So the boat cannot use the port between $\\var{t1_min} \\text{&nbsp; minutes past&nbsp; } \\var{t1_hour}$ and $\\var{t2_min} \\text{&nbsp; minutes past&nbsp; } \\var{t2_hour}$</p>", "rulesets": {}, "extensions": ["geogebra"], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Original Equation", "definition": "random(2 .. 3#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "b": {"name": "b", "group": "Original Equation", "definition": "random(1 .. 2#0.1)", "description": "", "templateType": "randrange", "can_override": false}, "c": {"name": "c", "group": "Original Equation", "definition": "random(2.21 .. 2.79#0.02)", "description": "", "templateType": "randrange", "can_override": false}, "pt_6": {"name": "pt_6", "group": "Solution steps to original equation", "definition": "arccos((c-a)/b)", "description": "", "templateType": "anything", "can_override": false}, "t1": {"name": "t1", "group": "Solution steps to original equation", "definition": "(6*pt_6)/pi", "description": "", "templateType": "anything", "can_override": false}, "t2": {"name": "t2", "group": "Solution steps to original equation", "definition": "(2pi-pt_6)*6/pi", "description": "", "templateType": "anything", "can_override": false}, "defs": {"name": "defs", "group": "Original Equation", "definition": "[\n  ['a',a],\n  ['b',b],\n  ]", "description": "", "templateType": "anything", "can_override": false}, "t1_hour": {"name": "t1_hour", "group": "Times", "definition": "trunc(t1)", "description": "", "templateType": "anything", "can_override": false}, "t2_hour": {"name": "t2_hour", "group": "Times", "definition": "trunc(t2)", "description": "", "templateType": "anything", "can_override": false}, "t1_min": {"name": "t1_min", "group": "Times", "definition": "int((t1-trunc(t1))*60)", "description": "", "templateType": "anything", "can_override": false}, "t2_min": {"name": "t2_min", "group": "Times", "definition": "int((t2-trunc(t2))*60)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Original Equation", "variables": ["a", "b", "c", "defs"]}, {"name": "Solution steps to original equation", "variables": ["pt_6", "t1", "t2"]}, {"name": "Times", "variables": ["t1_hour", "t2_hour", "t1_min", "t2_min"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "information", "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>The depth of water ($x$ m) in a port is given by the equation below where $t$ represents the time in hours after midnight</p>\n<p>\\[x=\\var{a}+\\var{b}cos \\left( \\frac{\\pi t}{6} \\right) \\]</p>"}, {"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 depth of water at the low tide? (you do not need to include the units in your answer)</p>\n<p>[[0]]</p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "low", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{a}-{b}", "maxValue": "{a}-{b}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "1_n_2", "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 time of the first high tide after $t=0$?</p>", "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "dropdownlist", "displayColumns": 0, "showCellAnswerState": true, "choices": ["1am", "2am", "3am", "4am", "5am", "6am", "7am", "8am", "9am", "10am", "11am", "12pm"], "matrix": [0, 0, 0, 0, 0, "0", "0", "0", 0, 0, 0, "1"], "distractors": ["", "", "", "", "", "", "", "", "", "", "", ""]}, {"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 are the first two times after midnight and between which a boat with a draught of $\\var{c}$m is not able to use the port?</p>\n<ul></ul>\n<p></p>\n<ul>\n<li>Fill in the gaps below giving each number as an integer</li>\n</ul>\n<p></p>\n<p>The boat is not able to use the port between [[2]] minutes past [[0]] and [[3]] minutes past [[1]]</p>\n<p></p>", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "t1h", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{t1_hour}", "maxValue": "{t1_hour}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "showFractionHint": true, "notationStyles": ["plain", 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