// Numbas version: exam_results_page_options {"name": "Find Cartesian form of a surface, and a normal vector", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"variable_groups": [], "variables": {"b": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "b"}, "t": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "", "name": "t"}, "d": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "name": "d"}, "a": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "a"}, "c": {"templateType": "anything", "group": "Ungrouped variables", "definition": "random(1..9)*sign(random(-1,1))", "description": "", "name": "c"}}, "ungrouped_variables": ["a", "c", "b", "d", "t"], "question_groups": [{"pickingStrategy": "all-ordered", "questions": [], "name": "", "pickQuestions": 0}], "name": "Find Cartesian form of a surface, and a normal vector", "functions": {}, "showQuestionGroupNames": false, "parts": [{"scripts": {}, "gaps": [{"answer": "{d}*sqrt(x^2/{a^2}+y^2/{c^2})", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not enter decimals in your answer.

Express the surface in the form $z=g(x,y)$.

$z=g(x,y)=$ [[0]]. (Do not enter decimals in your answer.)

\n ", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "-({d^2}/{a^2})*x", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not enter decimals in your answer.

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Do not enter decimals in your answer.

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Do not enter decimals in your answer.

Using the above Cartesian form $z=g(x,y)$ of the surface you have just found, a normal vector $\\boldsymbol{n}$ with $z$-component equal to $1$, can be written in the form $\\boldsymbol{n}=\\pmatrix{\\frac{p(x)}{z},\\frac{q(y)}{z},k}$. Fill in the values for $p(x)$, $q(y)$, and $k$ below.

$\\boldsymbol{n}=($[[0]]$/z,$[[1]]$/z,$[[2]]$)$. (Do not enter decimals in your answer.)

", "showCorrectAnswer": true, "marks": 0}, {"scripts": {}, "gaps": [{"answer": "sqrt(({d^4}/{a^4})*(x^2/z^2)+({d^4}/{c^4})*(y^2/z^2)+1)", "vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "expectedvariablenames": [], "notallowed": {"message": "

Do not enter decimals in your answer.

Calculate the magnitude $|\\boldsymbol{n}|$ of the normal vector $\\boldsymbol{n}$, using the expression for $z$ to simplify your answer.

$|\\boldsymbol{n}|=$ [[0]]. (Do not enter decimals in your answer.)

", "showCorrectAnswer": true, "marks": 0}], "statement": "

You are given the following surface, defined in parametric form

\\[\\pmatrix{u,v}\\rightarrow\\pmatrix{\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\simplify{{d}*v}}, \\quad 0\\leqslant u\\leqslant 2\\pi, 0\\leqslant v\\leqslant 1\\]

", "tags": ["checked2015", "MAS2104"], "rulesets": {}, "preamble": {"css": "", "js": ""}, "type": "question", "metadata": {"notes": "", "licence": "Creative Commons Attribution 4.0 International", "description": "

Cartesian form of the parametric representation of a surface, normal vector, and magnitude.

"}, "variablesTest": {"condition": "", "maxRuns": 100}, "advice": "

The given surface has components

\\[\\begin{align}x&=\\simplify{{a}*v*({1-t}*cos({b}*u)+{t}*sin({b}*u))},\\\\y&=\\simplify{{c}*v*({1-t}*sin({b}*u)+{t}*cos({b}*u))},\\\\z&=\\simplify{{d}*v}.\\end{align}\\]

Then

\\[\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=(\\simplify{v*({1-t}*cos({b}*u)+{t}*sin({b}*u))})^2+(\\simplify{v*({1-t}*sin({b}*u)+{t}*cos({b}*u))})^2=v^2,\\]

but we know that $z=\\simplify{{d}*v}$, so

\\[\\simplify{(x/{a})^2}+\\simplify{(y/{c})^2}=\\simplify{(z/{d})^2},\\]

and hence

\\[z=\\var{d}\\sqrt{\\simplify{x^2/{a^2}}+\\simplify{y^2/{c^2}}},\\]

where we have taken the positive square root because $v\\geqslant 0\\implies z\\geqslant 0$.

Given the above form for $z=g(x,y)$, a normal vector $\\boldsymbol{n}$, with positive $z$-component is

\\[\\boldsymbol{n}=\\pmatrix{-\\frac{\\partial g}{\\partial x},-\\frac{\\partial g}{\\partial y},1}.\\]

In this case

\\[\\boldsymbol{n}=\\pmatrix{\\simplify{{-d^2}/{a^2}}\\frac{x}{z},\\simplify{{-d^2}/{c^2}}\\frac{y}{z},1}\\]

by straightforward partial differentiation.

The magnitude $\\lvert\\boldsymbol{n}\\rvert$ of the normal vector $\\boldsymbol{n}$ is given by $\\lvert\\boldsymbol{n}\\rvert=\\sqrt{n_1^2+n_2^2+n_3^2}$, and hence

\\[\\lvert\\boldsymbol{n}\\rvert=\\sqrt{\\simplify{{d^4}/{a^4}}\\frac{x^2}{z^2}+\\simplify{{d^4}/{c^4}}\\frac{y^2}{z^2}+1}.\\]

", "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/"}]}