// Numbas version: exam_results_page_options
{"name": "Find normals to surface enclosing a region, and volume", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"g1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "g1"}, "c1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "a1*d1+b1*g1+random(1..9)", "description": "", "name": "c1"}, "a1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "a1"}, "gradfhat": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(precround(gradf[0]/lengradf,3),0,precround(gradf[2]/lengradf,3))", "description": "", "name": "gradfhat"}, "f1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "e1+random(2..9)", "description": "", "name": "f1"}, "e1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "e1"}, "b1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b1"}, "d1": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "d1"}, "gradf": {"templateType": "anything", "group": "Ungrouped variables", "definition": "vector(a1,0,b1)", "description": "", "name": "gradf"}, "lengradf": {"templateType": "anything", "group": "Ungrouped variables", "definition": "abs(gradf)", "description": "", "name": "lengradf"}, "vol": {"templateType": "anything", "group": "Ungrouped variables", "definition": "precround(((f1-e1)/(2*a1*b1))*(c1-b1*g1-a1*d1)^2,3)", "description": "", "name": "vol"}}, "ungrouped_variables": ["f1", "g1", "vol", "gradfhat", "gradf", "lengradf", "a1", "b1", "c1", "e1", "d1"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find normals to surface enclosing a region, and volume", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "-1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "-1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "0", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"answer": "-1", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "checkvariablenames": false, "expectedvariablenames": [], "showpreview": true, "checkingtype": "absdiff", "scripts": {}, "type": "jme", "showCorrectAnswer": true, "marks": 1, "vsetrangepoints": 5}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "gradfhat[0]+0.001", "minValue": "gradfhat[0]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "0", "minValue": "0", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}, {"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "gradfhat[2]+0.001", "minValue": "gradfhat[2]-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>Find the unit outward normal to each component of the region's boundary.</p>\n      <ul>\n      <li>Unit outward normal at $x=\\var{d1}$: $($[[0]]$,$[[1]]$,$[[2]]$)$</li>\n      <li>Unit outward normal at $y=\\var{e1}$:&nbsp;$($[[3]]$,$[[4]]$,$[[5]]$)$</li>\n      <li>Unit outward normal at $y=\\var{f1}$:&nbsp;$($[[6]]$,$[[7]]$,$[[8]]$)$</li>\n      <li>Unit outward normal at $z=\\var{g1}$:&nbsp;$($[[9]]$,$[[10]]$,$[[11]]$)$</li>\n      <li>Unit outward normal at $\\simplify{{a1}*x+{b1}*z}=\\var{c1}$:&nbsp;$($[[12]]$,$[[13]]$,$[[14]]$)$ (Enter your answer to 3d.p.)</li>\n      </ul>\n      <p>&nbsp;</p>", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"showCorrectAnswer": true, "allowFractions": false, "scripts": {}, "type": "numberentry", "maxValue": "vol+0.001", "minValue": "vol-0.001", "correctAnswerFraction": false, "marks": 1, "showPrecisionHint": false}], "type": "gapfill", "prompt": "<p>By calculating a volume integral, find the volume $V$ of the region enclosed by the above surfaces.</p>\n        <p>$V=$ [[0]]. &nbsp;(Enter your answer to 3d.p.)</p>", "showCorrectAnswer": true, "marks": 0}], "statement": "<p>A region is enclosed by the surfaces $\\simplify{{a1}*x+{b1}*z}=\\var{c1}$, $x=\\var{d1}$, $y=\\var{e1}$, $y=\\var{f1}$, and $z=\\var{g1}$.</p>", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "<p>Outward normals to the surfaces enclosing a region; volume of that enclosed region.</p>"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "<p>a)</p>\n    <p>The unit outward normals can most easily be identified by sketching the region bounded by the given surfaces which, in this case, is a wedge.</p>\n    <p>Then the unit outward normals to the non-slanted surfaces are given by</p>\n    <ul>\n    <li>Unit outward normal at $x=\\var{d1}$: $\\pmatrix{-1,0,0}$</li>\n    <li>Unit outward normal at $y=\\var{e1}$:&nbsp;$\\pmatrix{0,-1,0}$</li>\n    <li>Unit outward normal at $y=\\var{f1}$:&nbsp;$\\pmatrix{0,1,0}$</li>\n    <li>Unit outward normal at $z=\\var{g1}$:&nbsp;$\\pmatrix{0,0,-1}$</li>\n    </ul>\n    <p>&nbsp;</p>\n    <p>The outward normal to the final, slanted surface, is given by</p>\n    <p>\\[\\boldsymbol{\\nabla}(\\simplify{{a1}*x+{b1}*z-{c1}})=\\pmatrix{\\var{a1},0,\\var{b1}},\\]</p>\n    <p>and so the unit outward normal is given by</p>\n    <p>\\[\\frac{1}{\\sqrt{(\\var{a1})^2+(\\var{b1})^2}}\\pmatrix{\\var{a1},0,\\var{b1}}=\\pmatrix{\\var{gradfhat[0]},\\var{gradfhat[1]},\\var{gradfhat[2]}}.\\]</p>\n    <p>&nbsp;</p>\n    <p>b)</p>\n    <p>The volume of a region $V$ bounded by some particular surfaces is given by</p>\n    <p>\\[V=\\int_V\\mathrm{d}x\\mathrm{d}y\\mathrm{d}z.\\]</p>\n    <p>The relevant integral for the wedge in this question is therefore</p>\n    <p>\\[V=\\int_{x=\\var{d1}}^{\\simplify{{c1-b1*g1}/{a1}}}\\mathrm{d}x\\int_{y=\\var{e1}}^{\\var{f1}}\\mathrm{d}y\\int_{z=\\var{g1}}^{\\simplify{{c1}/{b1}-{a1}/{b1}*x}}\\mathrm{d}z.\\]</p>\n    <p>The integrals in $y$ and $z$ are straight forward, and we are left with</p>\n    <p>\\[\\begin{align}V&amp;=\\var{f1-e1}\\int_{x=\\var{d1}}^{\\simplify{{c1-b1*g1}/{a1}}}\\left(\\simplify{{-a1}/{b1}*x+{c1}/{b1}-{g1}}\\right)\\mathrm{d}x\\\\&amp;=\\var{f1-e1}\\left[\\simplify{{-a1}/{2*b1}*x^2+{c1-g1*b1}/{b1}*x}\\right]_{x=\\var{d1}}^{\\simplify{{c1-b1*g1}/{a1}}}\\\\&amp;=\\var{f1-e1}\\left\\{\\left(\\simplify{{-(c1-b1*g1)^2}/{2*a1*b1}+{(c1-g1*b1)^2}/{a1*b1}}\\right)-\\left(\\simplify{{-a1*d1^2}/{2*b1}+{d1*(c1-g1*b1)}/{b1}}\\right)\\right\\}\\\\&amp;=\\var{vol}\\;\\text{to 3d.p.}\\end{align}\\]</p>\n    <p>&nbsp;</p>", "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}]}], "contributors": [{"name": "Newcastle University Mathematics and Statistics", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/697/"}]}