In this last exercise, you are given four domains in the $xy$-plane. Your task is to describe them in polar coordinates, by specifying the range for the radius $r$ and for the phase $\\theta$.
\nAll the radii are integers less than 10. The possibilities for the phases are: $0,\\frac{\\pi}{6},\\frac{\\pi}{4},\\frac{\\pi}{3},\\frac{\\pi}{2}$, and the corresponding shifts into the other quadrants of the $xy$-plane. Enter phases rounded to two decimals in the range $[-\\pi,\\pi]$ (it is not possible to enter $\\frac{5\\pi}{6}$ -- you would have to evaluate that).
\nFor example, the unit disk ($\\rightarrow$ area inside the circle of radius $1$ centered at $(0,0)$) should be entered as
\n$\\begin{pmatrix} r_{\\text{lower}} & r_{\\text{upper}} \\\\ \\theta_{\\text{lower}} & \\theta_{\\text{upper}} \\end{pmatrix} = \\begin{pmatrix} 0 & 1 \\\\ -3.14 & 3.14 \\end{pmatrix}$.
", "variable_groups": [], "ungrouped_variables": ["q1_r2", "q2_r1", "q2_r2", "q3_r2", "q3_t1", "q3_t2", "q4_r1", "q4_r2", "q4_t1", "q4_t2"], "rulesets": {}, "name": "Polar domains", "tags": [], "parts": [{"showCorrectAnswer": true, "gaps": [{"correctAnswer": "matrix([0,q1_r2],[-pi,pi])", "variableReplacements": [], "marks": 1, "tolerance": "0.01", "type": "matrix", "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "numColumns": "2", "allowResize": false, "showFeedbackIcon": true, "numRows": "2", "markPerCell": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "Describe the area inside the circle ($\\rightarrow$ disk) in polar coordinates.
\n{polar_domains(0,q1_r2,0,0,1)}
\n$\\begin{pmatrix} r_{\\text{lower}} & r_{\\text{upper}} \\\\ \\theta_{\\text{lower}} & \\theta_{\\text{upper}} \\end{pmatrix} = $ [[0]]
", "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"correctAnswer": "matrix([min(q2_r1,q2_r2),max(q2_r1,q2_r2)],[-pi,pi])", "variableReplacements": [], "marks": 1, "tolerance": "0.01", "type": "matrix", "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "numColumns": "2", "allowResize": false, "showFeedbackIcon": true, "numRows": "2", "markPerCell": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "Describe the area between the two circles ($\\rightarrow$ ring) in polar coordinates.
\n{polar_domains(q2_r1,q2_r2,0,0,1)}
\n$\\begin{pmatrix} r_{\\text{lower}} & r_{\\text{upper}} \\\\ \\theta_{\\text{lower}} & \\theta_{\\text{upper}} \\end{pmatrix} = $ [[0]]
", "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"correctAnswer": "matrix([0,q3_r2],[min(q3_t1,q3_t2),max(q3_t1,q3_t2)])", "variableReplacements": [], "marks": 1, "tolerance": "0.01", "type": "matrix", "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "numColumns": "2", "allowResize": false, "showFeedbackIcon": true, "numRows": "2", "markPerCell": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "Describe the disk segment in polar coordinates.
\n{polar_domains(0,q3_r2,q3_t1,q3_t2,0)}
\n$\\begin{pmatrix} r_{\\text{lower}} & r_{\\text{upper}} \\\\ \\theta_{\\text{lower}} & \\theta_{\\text{upper}} \\end{pmatrix} = $ [[0]]
", "type": "gapfill", "variableReplacements": [], "variableReplacementStrategy": "originalfirst"}, {"showCorrectAnswer": true, "gaps": [{"correctAnswer": "matrix([min(q4_r1,q4_r2),max(q4_r1,q4_r2)],[min(q4_t1,q4_t2),max(q4_t1,q4_t2)])", "variableReplacements": [], "marks": 1, "tolerance": "0.01", "type": "matrix", "scripts": {}, "showCorrectAnswer": true, "allowFractions": true, "numColumns": "2", "allowResize": false, "showFeedbackIcon": true, "numRows": "2", "markPerCell": false, "correctAnswerFractions": false, "variableReplacementStrategy": "originalfirst"}], "showFeedbackIcon": true, "marks": 0, "scripts": {}, "prompt": "Describe the ring segment in polar coordinates.
\n{polar_domains(q4_r1,q4_r2,q4_t1,q4_t2,0)}
\n$\\begin{pmatrix} r_{\\text{lower}} & r_{\\text{upper}} \\\\ \\theta_{\\text{lower}} & \\theta_{\\text{upper}} \\end{pmatrix} = $ [[0]]
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