// Numbas version: exam_results_page_options
{"name": "13: Vectors - Various Geological Applications", "extensions": [], "custom_part_types": [], "resources": [["question-resources/2016-08-19_5.png", "/srv/numbas/media/question-resources/2016-08-19_5.png"], ["question-resources/2016-08-19_6.png", "/srv/numbas/media/question-resources/2016-08-19_6.png"], ["question-resources/2016-08-19_6_BSFfPeY.png", "/srv/numbas/media/question-resources/2016-08-19_6_BSFfPeY.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "statement": "<p>Solve the following vector&nbsp;application&nbsp;questions.</p>\n<p>Note: Bearing angles are always measured from the north by convention.</p>", "metadata": {"description": "<p>These questions deal with vector operations and its application in geology.</p>\n<p></p>", "licence": "All rights reserved"}, "parts": [{"scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<p>A branch&nbsp;fell off a&nbsp;tree and landed in a river at point A, which took it $\\var{a} \\text{ km}$ with a bearing of $\\var{b}^\\circ$ to point B. From there, the river took it $\\var{c} \\text{ km}$ with a bearing of $\\var{d}^\\circ$ taking the branch to point C.</p>\n<p>Calculate the magnitude of the vector $\\overrightarrow{AC}$.&nbsp;</p>\n<p>[[0]] km</p>\n<p>Now calculate the bearing of&nbsp;the vector $\\overrightarrow{AC}$</p>\n<p>[[1]]</p>\n<p>Give your answers to&nbsp;one decimal place.</p>", "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"precisionType": "dp", "strictPrecision": false, "maxValue": "{s}+0.1", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "variableReplacements": [], "scripts": {}, "notationStyles": ["en", "si-en", "plain-eu"], "minValue": "{s}-0.1", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "<p><img src=\"resources/question-resources/2016-08-19_5.png\" />You have not given your answer to the correct precision.</p>", "precision": "1", "showFeedbackIcon": false, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "en"}, {"precisionType": "dp", "strictPrecision": true, "maxValue": "{rba}+0.1", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": false, "marks": 1, "variableReplacements": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "minValue": "{rba}-0.1", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain"}], "marks": 0, "steps": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "<p><img src=\"resources/question-resources/2016-08-19_6_BSFfPeY.png\" style=\"float: left;\"/></p>\n<p style=\"text-align: left;\"><strong>Step 1:</strong> Make a tail to head vector of distance the&nbsp;branch&nbsp;travels&nbsp;$\\overrightarrow{AB}$ with a bearing of $\\var{b}^\\circ$ (b1) from the north direction.</p>\n<p style=\"text-align: left;\"><strong>Step 2:</strong> Make the second vector $\\overrightarrow{BC}$ from the head of the first vector with a bearing of $\\var{d}^\\circ$ (b2)</p>\n<p style=\"text-align: left;\"><strong>Step 3:</strong> Join the start point to the end point by making a vector called the 'resultant vector' ($\\overrightarrow{AC}$).</p>\n<p style=\"text-align: left;\"><strong>Step 4:</strong> Calculate the angle ($X^\\circ$) between the two vectors $\\overrightarrow{AB}$ and $\\overrightarrow{BC}$&nbsp;using&nbsp;parralelogram law of vectors, using the bearings calibrated from the north.</p>\n<p style=\"text-align: left;\">&nbsp; &nbsp; &nbsp; &nbsp; <em>Parralelogram laws of vectors</em> = angles within the parralelogram add up to 180, giving b1 opposite to (180-b1) as this is the angle parralel to it. This is shown in the diagram to the left. Once you have worked out the value of (180-b1), you can work out angle x: 360-(180-b1)-b2 =x. This is because there are 360 degrees in a circle.</p>\n<p style=\"text-align: left;\"><strong>Step 5:</strong> Find the magnitude of the resultant vector by using the cosine rule: $x^2 =a^2+ b^2 - 2ab\\space cos(X)$ where a and b are the magnitude of the two other vectors ($\\overrightarrow{AB}$ and $\\overrightarrow{BC}$) and $X^\\circ$ is the angle between them.</p>\n<p style=\"text-align: left;\"><strong>Step 6:</strong> To find the bearing of the resultant vector, use the sine rule to obtain the angle between&nbsp;$\\overrightarrow{AB}$ and&nbsp;$\\overrightarrow{AC}$. Then add the answer&nbsp;to b1 to give the overall bearing calibrated from north.</p>", "type": "information", "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}], "stepsPenalty": 0}, {"scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<div class=\"_d97\"><span class=\"_5yl5\">Pahoehoe lava following a volcanic eruption flows slowly from the volcano downhill $\\var{h} \\text{km}$ on a bearing of $\\var{n}^\\circ$. The lava reaches the river where it then flows parallel to the river on a bearing of $\\var{k}^\\circ$, $\\var{j} \\text{ km}$ to the west, where it stops flowing and over time&nbsp;solidifies.&nbsp;</span></div>\n<div class=\"_d97\"><span class=\"_5yl5\">What is the resultant vector the lava has travelled between the volcano and where it solidifies?</span></div>\n<div class=\"_d97\"><span class=\"_5yl5\"></span></div>\n<div class=\"_d97\"><span class=\"_5yl5\">[[0]] km</span></div>\n<div class=\"_d97\"><span class=\"_5yl5\">Give your answer to&nbsp;one decimal place.</span></div>", "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"precisionType": "dp", "strictPrecision": true, "maxValue": "({h}^2 + {j}^2 - (2*{h}*{j}*{l}))^{1/2}+0.2", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "variableReplacements": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "minValue": "({h}^2 + {j}^2 - (2*{h}*{j}*{l}))^{1/2}-0.2", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain"}], "marks": 0, "steps": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "<p><strong>Step 1:</strong> Make the first tail to head vector of distance the&nbsp;lava has travelled&nbsp;between the volcano and where it meets the river (with a bearing of $\\var{n}^\\circ$ from the north direction).</p>\n<p><strong>Step 2:</strong> Make the second vector from the head of the first vector with a bearing of $\\var{k}^\\circ$&nbsp;</p>\n<p><strong>Step 3:</strong> Calculate the angle ($X^\\circ$) between the two vectors using geometry or parallelogram law of vectors.</p>\n<p><strong>Step 4:</strong> Join the start point to the end point by making a vector called the 'resultant vector'.</p>\n<p><strong>Step 5:</strong> Find the magnitude of the resultant vector by using the cosine rule: $x^2 = a^2+ b^2 - 2ab\\space cos(X)$ where a and b are the magnitude of the two other vectors and $X^\\circ$ is the angle between them.</p>", "type": "information", "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}], "stepsPenalty": 0}, {"scripts": {}, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "prompt": "<div class=\"_d97\"><span class=\"_5yl5\"><span class=\"_5yl5\">$\\var{c} km$ east from a siltstone outcrop (on a bearing of $\\var{r}^\\circ$) is an interbedded siltstone and dolomite outcrop. From this interbedded outcrop there is a nonconformity that lies to the south as there is a change in lithology to a volcanic tuff. The nearest volcanic tuff outcrop to the interbedded outcrop is $\\var{a} km$&nbsp;away on a bearing of $\\var{o}^\\circ$.</span></span></div>\n<div class=\"_d97\"><span class=\"_5yl5\"><span class=\"_5yl5\"></span></span></div>\n<div class=\"_d97\"><span class=\"_5yl5\"><span class=\"_5yl5\"> Calculate the direct distance (resultant vector) you would have to walk to get from the siltstone outcrop to the volcanic tuff outcrop.&nbsp;</span></span></div>\n<div class=\"_d97\"></div>\n<div class=\"_d97\">[[0]] km</div>\n<div class=\"_d97\"></div>\n<div class=\"_d97\">Now, deduce the bearing you would have to walk on to go directly from the siltstone outcrop to the volcanic tuff outcrop.&nbsp;&nbsp;</div>\n<div class=\"_d97\"></div>\n<div class=\"_d97\">[[1]] $^\\circ$</div>\n<div class=\"_d97\"></div>\n<div class=\"_d97\">Give your answer to&nbsp;two decimal places</div>", "type": "gapfill", "showCorrectAnswer": true, "variableReplacements": [], "gaps": [{"precisionType": "dp", "strictPrecision": true, "maxValue": "({c}^2 + {a}^2 - (2*{c}*{a}*cos({angle2}/180*pi)))^{1/2}+0.3", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "variableReplacements": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "minValue": "({c}^2 + {a}^2 - (2*{c}*{a}*cos({angle2}/180*pi)))^{1/2}-0.3", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain"}, {"precisionType": "dp", "strictPrecision": false, "maxValue": "{rbc}+0.2", "precisionPartialCredit": 0, "correctAnswerFraction": false, "type": "numberentry", "showCorrectAnswer": true, "marks": 1, "variableReplacements": [], "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "minValue": "{rbc}-0.2", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.", "precision": "1", "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain"}], "marks": 0, "steps": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "<p>+</p>\n<p><strong>Step 1:</strong> Make a tail to head vector of the distance between the siltstone and interbedded&nbsp;outcrop with a bearing of $\\var{r}^\\circ$ from the north direction.</p>\n<p><strong>Step 2:</strong> Make the second vector from the head of the first vector with a bearing of $\\var{o}^\\circ$&nbsp;</p>\n<p><strong>Step 3:</strong> Calculate the angle ($X^\\circ$) between the two vectors using geometry or parallelogram law of vectors.</p>\n<p><strong>Step 4:</strong> Join the start point to the end point by making a vector called the 'resultant vector'.</p>\n<p><strong>Step 5:</strong> Find the magnitude of the resultant vector by using the cosine rule: $x^2=a^2+ b^2 - 2ab\\space cos(X)$ where a and b are the magnitude of the two other vectors and $X^\\circ$ is the angle between them.</p>\n<p><strong>Step 6:</strong> Find the bearing of the resultant vector by using the sine rule to firstly find the angle within the triangle constructed (between the interbedded outcrop and the volcanic tuff outcrop) and add this to the bearing $\\var{r}$ to give the resultant bearing.</p>", "type": "information", "variableReplacements": [], "marks": 0, "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}], "stepsPenalty": 0}], "variables": {"m": {"description": "", "name": "m", "templateType": "anything", "definition": "(180-{k}+{n})", "group": "Ungrouped variables"}, "BC": {"description": "", "name": "BC", "templateType": "anything", "definition": "precround(arcsin((sin(angle2/180*pi)/ansc)*a)/pi*180,5)", "group": "Unnamed group"}, "j": {"description": "", "name": "j", "templateType": "anything", "definition": "random(10..19)", "group": "Ungrouped variables"}, "t": {"description": "", "name": "t", "templateType": "anything", "definition": "(({h}^2+{j}^2)- (2*{h}*{j}*{l}))^(1/2)", "group": "Ungrouped variables"}, "BA": {"description": "", "name": "BA", "templateType": "anything", "definition": "precround(arcsin((sin(g/180*pi)/s)*c)/pi*180,5)", "group": "Unnamed group"}, "f": {"description": "", "name": "f", "templateType": "anything", "definition": "cos((radians(g)))", "group": "Ungrouped variables"}, "RBA": {"description": "", "name": "RBA", "templateType": "anything", "definition": "precround(b+BA,2)", "group": "Unnamed group"}, "k": {"description": "", "name": "k", "templateType": "anything", "definition": "random(250..280)", "group": "Ungrouped variables"}, "angle2": {"description": "", "name": "angle2", "templateType": "anything", "definition": "360-((180-r)+o)", "group": "Unnamed group"}, "n": {"description": "", "name": "n", "templateType": "anything", "definition": "random(120..160)", "group": "Ungrouped variables"}, "l": {"description": "", "name": "l", "templateType": "anything", "definition": "cos((radians(m)))", "group": "Ungrouped variables"}, "d": {"description": "", "name": "d", "templateType": "anything", "definition": "random(110..140)", "group": "Ungrouped variables"}, "ACMIN": {"description": "", "name": "ACMIN", "templateType": "anything", "definition": "sqrt(({a}*{a}+{c}*{c}-2*{a}*{c}*{f}))-0.3", "group": "Ungrouped variables"}, "o": {"description": "", "name": "o", "templateType": "anything", "definition": "random(160..200)", "group": "Ungrouped variables"}, "u": {"description": "", "name": "u", "templateType": "anything", "definition": "(({c}^2+{a}^2)- (2*{c}*{a}*{p}))^(1/2)", "group": "Ungrouped variables"}, "p": {"description": "", "name": "p", "templateType": "anything", "definition": "cos((radians(q)))", "group": "Ungrouped variables"}, "c": {"description": "", "name": "c", "templateType": "anything", "definition": "random(6..15 except[a])", "group": "Ungrouped variables"}, "s": {"description": "", "name": "s", "templateType": "anything", "definition": "precround((({a}^2+{c}^2)- (2*{a}*{c}*{f}))^(1/2),2)", "group": "Ungrouped variables"}, "RBC": {"description": "", "name": "RBC", "templateType": "anything", "definition": "precround(r+bc,1)", "group": "Unnamed group"}, "h": {"description": "", "name": "h", "templateType": "anything", "definition": "random(20..40)", "group": "Ungrouped variables"}, "ACMAX": {"description": "", "name": "ACMAX", "templateType": "anything", "definition": "sqrt(({a}*{a}+{c}*{c}-2*{a}*{c}*{f}))+0.3", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "templateType": "anything", "definition": "random(030..070)", "group": "Ungrouped variables"}, "q": {"description": "", "name": "q", "templateType": "anything", "definition": "(180-{o}+{r})", "group": "Ungrouped variables"}, "g": {"description": "", "name": "g", "templateType": "anything", "definition": "(180-{d}+{b})", "group": "Ungrouped variables"}, "r": {"description": "", "name": "r", "templateType": "anything", "definition": "random(070..110)", "group": "Ungrouped variables"}, "a": {"description": "", "name": "a", "templateType": "anything", "definition": "random(3..9)", "group": "Ungrouped variables"}, "ansc": {"description": "", "name": "ansc", "templateType": "anything", "definition": "precround(sqrt({c}^2+{a}^2-(2*{c}*{a}*{p})),2)", "group": "Ungrouped variables"}}, "extensions": [], "ungrouped_variables": ["a", "b", "c", "d", "h", "n", "j", "k", "f", "g", "l", "m", "r", "o", "p", "q", "s", "t", "u", "ansc", "ACMAX", "ACMIN"], "variablesTest": {"maxRuns": 100, "condition": ""}, "rulesets": {}, "functions": {}, "variable_groups": [{"variables": ["BA", "RBA", "BC", "angle2", "RBC"], "name": "Unnamed group"}], "name": "13: Vectors - Various Geological Applications", "tags": [], "advice": "<p style=\"text-align: left;\"><span style=\"text-decoration: underline;\"><strong>PART A</strong></span></p>\n<p style=\"text-align: left;\"></p>\n<p><em><img src=\"resources/question-resources/2016-08-19_5.png\"/></em></p>\n<p>Angle X =<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $X^\\circ = 360-(180-\\var{b})-\\var{d}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $X^\\circ = \\var{g}^\\circ$<br/>Resultant distance&nbsp;&nbsp;= $\\sqrt{\\var{a}^2 + \\var{c}^2 - (2\\times\\var{a}\\times\\var{c}\\times cos(\\var{g}))}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= $\\sqrt{\\var{a}^2 + \\var{c}^2 - (2\\times\\var{a}\\times\\var{c}\\times\\var{f}}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;=&nbsp; $\\var{s} km $<br/>Resultant bearing = $sin^{-1}\\left[\\frac{sin(\\var{f}}{\\var{s}}\\times\\var{c}\\right]$ = $\\var{ba}^\\circ$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $\\var{ba}^\\circ + \\var{b} = \\var{rba}$ to two decimal places.<br/><br/><br/></p>\n<p><span style=\"text-decoration: underline;\"><strong>PART B</strong></span></p>\n<p><em>Angle X</em> =<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $X^\\circ = 180-\\var{k}^\\circ+\\var{n}^\\circ$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $X^\\circ = \\var{m}^\\circ$<br/><em>Resultant distance</em>&nbsp;= $\\sqrt{\\var{h}^2 + \\var{j}^2 - (2\\times\\var{h}\\times\\var{j}\\times cos(\\var{m}))}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= $\\sqrt{\\var{h}^2 + \\var{j}^2 - (2\\times\\var{h}\\times\\var{j}\\times\\var{l}}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= $\\var{t} = \\var{precround({t},2)} km $<br/><br/><strong>PART C</strong></p>\n<p><em>Angle X</em> =&nbsp;<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $X^\\circ = 180-\\var{o}^\\circ+\\var{r}^\\circ$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; $X^\\circ = \\var{q}^\\circ$<br/><em>Resultant distance</em>&nbsp;= $\\sqrt{\\var{c}^2 + \\var{a}^2 - (2\\times\\var{c}\\times\\var{a}\\times cos(\\var{q}))}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= $\\sqrt{\\var{c}^2 + \\var{a}^2 - (2\\times\\var{c}\\times\\var{a}\\times\\var{p})}$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= $\\var{u} = \\var{precround({u},2)} km $<br/><em>Resultant bearing</em> = $sin^{-1}\\left[\\frac{sin(\\var{angle2}}{\\var{ansc}}\\times\\var{a}\\right]$ = $\\var{bc}^\\circ$<br/>&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;= $\\var{bc}^\\circ + \\var{r} = \\var{rbc}$ to two decimal places.</p>", "type": "question", "contributors": [{"name": "Jean jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/353/"}]}]}], "contributors": [{"name": "Jean jinhua Mathias", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/353/"}]}