// Numbas version: exam_results_page_options {"name": "13: Vectors and Trigonometry: Subduction Zone", "extensions": [], "custom_part_types": [], "resources": [["question-resources/93e3bc2ef7cb323f434ca53208c4a9ad.jpg", "/srv/numbas/media/question-resources/93e3bc2ef7cb323f434ca53208c4a9ad.jpg"], ["question-resources/2016-08-18_3.png", "/srv/numbas/media/question-resources/2016-08-18_3.png"], ["question-resources/2016-08-18_6.png", "/srv/numbas/media/question-resources/2016-08-18_6.png"], ["question-resources/2016-08-19_3.png", "/srv/numbas/media/question-resources/2016-08-19_3.png"], ["question-resources/2016-08-19_3_QIXNOoD.png", "/srv/numbas/media/question-resources/2016-08-19_3_QIXNOoD.png"], ["question-resources/2016-08-19_4.png", "/srv/numbas/media/question-resources/2016-08-19_4.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"preamble": {"css": "", "js": ""}, "statement": "

Below is an image illustrating Plate A being subducted beneath Plate B.

\n

\n

S$^\\circ%$ = The subduction angle

\n

$\\overrightarrow{XY}$ = The vector from the GPS tracker at location (X) to the location of the end point (Y).

", "variables": {"yminusz": {"description": "", "name": "yminusz", "templateType": "anything", "definition": "coordy1-coordz1", "group": "Ungrouped variables"}, "j": {"description": "", "name": "j", "templateType": "anything", "definition": "(({a}^2+{b}^2)- (2*{a}*{b}*{c}))^(1/2)", "group": "Ungrouped variables"}, "coordz1": {"description": "", "name": "coordz1", "templateType": "anything", "definition": "random(100..200)", "group": "Ungrouped variables"}, "coordx2": {"description": "", "name": "coordx2", "templateType": "anything", "definition": "0", "group": "Ungrouped variables"}, "f": {"description": "", "name": "f", "templateType": "anything", "definition": "(({h}^2+{b}^2-{d}^2)/(2*{h}*{b}))", "group": "Ungrouped variables"}, "x": {"description": "", "name": "x", "templateType": "anything", "definition": "random(110..150)", "group": "Ungrouped variables"}, "coordy1": {"description": "", "name": "coordy1", "templateType": "anything", "definition": "coordz1+random(50..100)", "group": "Ungrouped variables"}, "subangle": {"description": "", "name": "subangle", "templateType": "anything", "definition": "precround(180-(arctan(compsubangle)/pi*180),1)", "group": "Ungrouped variables"}, "d": {"description": "", "name": "d", "templateType": "anything", "definition": "random(200..300)", "group": "Ungrouped variables"}, "coordx1": {"description": "", "name": "coordx1", "templateType": "anything", "definition": "0", "group": "Ungrouped variables"}, "direction": {"description": "", "name": "direction", "templateType": "anything", "definition": "random(\"west\",\"east\",\"north\",\"south\")", "group": "Ungrouped variables"}, "compsubangle": {"description": "", "name": "compsubangle", "templateType": "anything", "definition": "coordy2/veczy", "group": "Ungrouped variables"}, "c": {"description": "", "name": "c", "templateType": "anything", "definition": "cos((radians(x)))", "group": "Ungrouped variables"}, "a": {"description": "", "name": "a", "templateType": "anything", "definition": "random(100..300 except[b])", "group": "Ungrouped variables"}, "coordy2": {"description": "", "name": "coordy2", "templateType": "anything", "definition": "random(100..200)", "group": "Ungrouped variables"}, "h": {"description": "", "name": "h", "templateType": "anything", "definition": "random(100..150 except[b])", "group": "Ungrouped variables"}, "coordz2": {"description": "", "name": "coordz2", "templateType": "anything", "definition": "0", "group": "Ungrouped variables"}, "veczy": {"description": "", "name": "veczy", "templateType": "anything", "definition": "precround(sqrt(coordy2^2+(coordy1-coordz1)^2),1)", "group": "Ungrouped variables"}, "b": {"description": "", "name": "b", "templateType": "anything", "definition": "random(101..150)", "group": "Ungrouped variables"}, "vecxy": {"description": "", "name": "vecxy", "templateType": "anything", "definition": "precround(sqrt({coordy1}^2+{coordy2}^2),1)", "group": "Ungrouped variables"}, "g": {"description": "", "name": "g", "templateType": "anything", "definition": "precround(arccos({f})/pi*180,0)", "group": "Ungrouped variables"}, "ans1": {"description": "", "name": "ans1", "templateType": "anything", "definition": "dpformat(sqrt({a}^2+{b}^2-(2*{a}*{b}*{c})),2)", "group": "Unnamed group"}}, "parts": [{"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

Plate A subducts beneath Plate B. From the starting point X, the plate moves laterally $\\var{a} km$ east and then subducts $\\var{b} km$ downwards until point Y. The angle of subduction, $S^\\circ$, is $\\var{x}^\\circ$.

\n

Deduce the magnitude of $\\overrightarrow{XY}$

\n

[[0]] km

\n

", "steps": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "

To find the magnitute of the vector $\\overrightarrow{XY}$, we use the cosine rule:

\n

$c^2=a^2+b^2-2\\space{a}\\space{b}\\space{cos(C)}$

\n

If we consider $\\overrightarrow{XY}$ to be side $c$, and $a$ is $\\var{a}$ and $b$ is $\\var{b}$, you can substitute in the known values to obtain a value for the magnitude.

\n
\n
• Remember to square root the whole term at the end to give the value of $c$ rather than $c^2$.
• \n
", "type": "information", "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}], "type": "gapfill", "marks": 0, "gaps": [{"precisionPartialCredit": 0, "precisionType": "dp", "strictPrecision": false, "maxValue": "({a}^2 + {b}^2 - (2*{a}*{b}*{c}))^{1/2}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "type": "numberentry", "marks": 1, "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "minValue": "({a}^2 + {b}^2 - (2*{a}*{b}*{c}))^{1/2}", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "

", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "precision": "2"}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0}, {"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

From location X, plate A moves $\\var{h} km$ east and then subducts $\\var{b} km$ downwards beneath plate B. If the overall magnitude of $\\overrightarrow{XY}$ is $\\var{d} km$, what is the angle of subduction $S^\\circ$?

\n

\n

[[0]]$^\\circ$

", "steps": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "

To calculate the subduction angle, $S$, we use the cosine rule.

\n

If we take side $c$ to be the vector $\\overrightarrow{XY}$, then starting from the original equation with side $c$ being the subject; $c^2=a^2+b^2-2\\space{a}\\space{b}\\space{cos(C)}$, we need to make $cos(C)$ the subject.

\n

$cos(C^\\circ)= \\frac{a^2+b^2-c^2}{2ab}$

\n

$C^\\circ = cos^{-1}[\\frac{a^2+b^2-c^2}{2ab}]$

\n

where $a$ is $\\var{h} km$, and $b$ is $\\var{b} km$ and $c$ is $\\var{d} km$.

\n

where $C^\\circ$ = subduction angle $S$.

\n

", "type": "information", "marks": 0, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showFeedbackIcon": true}], "type": "gapfill", "marks": 0, "gaps": [{"precisionPartialCredit": 0, "precisionType": "dp", "strictPrecision": false, "maxValue": "{g}", "correctAnswerStyle": "plain", "showCorrectAnswer": true, "correctAnswerFraction": false, "type": "numberentry", "marks": 1, "showFeedbackIcon": true, "scripts": {}, "notationStyles": ["plain", "en", "si-en"], "minValue": "{g}", "allowFractions": false, "showPrecisionHint": false, "precisionMessage": "You have not given your answer to the correct precision.", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "precision": "0"}], "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "stepsPenalty": 0}, {"scripts": {}, "showFeedbackIcon": true, "showCorrectAnswer": true, "prompt": "

You can alternatively view the subduction zone in graphical format as illustrated below.

\n

You are given the coordinates for X, Y and Z.
X = ($\\var{coordx1},\\var{coordx2}$)
Y = ($\\var{coordy1},\\var{coordy2}$)
Z = ($\\var{coordz1},\\var{coordz2}$)

\n

Find the magnitude of $\\overrightarrow{ZY}$ =[[0]]
Find the overal magnitude of $\\overrightarrow{XY}$ =[[1]]

\n

", "steps": [{"scripts": {}, "showCorrectAnswer": true, "prompt": "

Part one

\n

To find the magnitude of vector $\\overrightarrow{ZY}$, it will help to construct a right angled triangle as shown in the illustration. You know the depth of Y as it is the $y$ variable in the coordinates ($x,y$), and you can work out the distance horizontally between Z and Y as this is the $x$ coordinate of Y minus the $x$ coordinate of Z, as the $x$ coordinate of each point is the distance from X, hence giving the distance between Z and Y.
You can now use Pythagoras' theorem $a^2 + b^2 = c^2$ to determine the magnitude of $\\overrightarrow{ZY}$

\n

Part two

\n

Now construct a large right angled triangle to deduce the magnitude of vector $\\overrightarrow{XY}$ like the one illustrated.The overall length is distance Y (from the x-coordinate of the point Y), and the depth is the depth of Y (from the y-coordinate of the point Y) Use Pythagoras' theorem to find the magnitude of $\\overrightarrow{XY}$.

This exercise will help you solve questions based on subduction zones using  the Cosine rule.

\n

Part a)

\n

To find the magnitute of the vector $\\overrightarrow{XY}$, we use the cosine rule:

\n

$c^2=a^2+b^2-2\\space{a}\\space{b}\\space{cos(C)}$

\n

If we consider $\\overrightarrow{XY}$ to be side $c$, you can substitute in the known values to obtain a value for the magnitude.

\n

$c^2=\\sqrt{\\var{a}^2+\\var{b}^2-2\\times\\var{a}\\times\\var{b}\\times{cos\\var{x}}}$

\n

$c = \\var{ans1}$km.

\n

\n

Part b)

\n

To calculate the subduction angle, $S$, we use the cosine rule.

\n

If we take side $c$ to be the vector $\\overrightarrow{XY}$, then starting from the original equation with side $c$ being the subject; $c^2=a^2+b^2-2\\space{a}\\space{b}\\space{cos(C)}$, we need to make $cos(C)$ the subject.

\n

$cos(C^\\circ)= \\frac{a^2+b^2-c^2}{2ab}$

\n

$C^\\circ = cos^{-1}\\left[\\frac{a^2+b^2-c^2}{2ab}\\right]$, where $C^\\circ = S^\\circ$

\n

$S^\\circ = cos^{-1}\\left[\\frac{\\var{h}^2+\\var{b}^2-\\var{d}^2}{2\\times\\var{h}\\times\\var{b}}\\right]$

\n

= $\\var{g}^\\circ$ to the nearest whole number.

\n

\n

\n

Part c)

\n

Using the diagrams and the explanations illustrated in the steps to part c we obtain the following solutions:

\n
\n
• Magnitude of $\\overrightarrow{ZY}$ = $\\sqrt{\\var{coordy2}^2 + \\var{yminusz}^2}= \\var{veczy}$ km.
• \n
• Magnitude of $\\overrightarrow{XY}$ = $\\sqrt{\\var{coordy1}^2+\\var{coordy2}^2}$ = $\\var{vecxy}$ km.
• \n
", "rulesets": {}, "functions": {}, "variable_groups": [{"variables": ["ans1"], "name": "Unnamed group"}], "name": "13: Vectors and Trigonometry: Subduction Zone", "tags": [], "variablesTest": {"maxRuns": 100, "condition": "h + b > d and h + d > b and b + d > h and b^2 + h^2 < d^2"}, "type": "question", "contributors": [{"name": "Nasir Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}]}]}], "contributors": [{"name": "Nasir Khan", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/909/"}]}