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| 1 : | aubreyja | 2584 | ## DBsubject('Calculus') |
| 2 : | ## DBchapter('Vector Geometry') | ||
| 3 : | ## DBsection('A Survey of Quadratic Surfaces') | ||
| 4 : | ## KEYWORDS('calculus') | ||
| 5 : | ## TitleText1('Calculus: Early Transcendentals') | ||
| 6 : | ## EditionText1('2') | ||
| 7 : | ## AuthorText1('Rogawski') | ||
| 8 : | ## Section1('12.6') | ||
| 9 : | ## Problem1('15') | ||
| 10 : | ## Author('JustAsk - Vladimir Finkelshtein') | ||
| 11 : | ## Institution('W.H.Freeman') | ||
| 12 : | |||
| 13 : | DOCUMENT(); | ||
| 14 : | |||
| 15 : | loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); | ||
| 16 : | loadMacros("Parser.pl"); | ||
| 17 : | loadMacros("freemanMacros.pl"); | ||
| 18 : | loadMacros("PGauxiliaryFunctions.pl"); | ||
| 19 : | loadMacros("PGgraphmacros.pl"); | ||
| 20 : | loadMacros("PGchoicemacros.pl"); | ||
| 21 : | |||
| 22 : | TEXT(beginproblem()); | ||
| 23 : | |||
| 24 : | $a=random(2,8,1); | ||
| 25 : | $b=random(2,8,1); | ||
| 26 : | $z=random(2,4,2); | ||
| 27 : | $ans=list_random(1,4); | ||
| 28 : | if ($ans==1) {$z=1/$z}; | ||
| 29 : | $d=1-$z**2; | ||
| 30 : | if ($d<0) {$ans2='empty set'}; | ||
| 31 : | if ($d>0) {$ans2='ellipse'}; | ||
| 32 : | Context()->texStrings; | ||
| 33 : | |||
| 34 : | BEGIN_TEXT | ||
| 35 : | \{ textbook_ref_exact("Rogawski ET 2e", "12.6","15") \} | ||
| 36 : | $PAR | ||
| 37 : | State the type of the quadratic surface: $BR | ||
| 38 : | \( \left( \frac{x}{$a} \right)^2 + \left( \frac{y}{$b} \right)^2 + z^2=1\) | ||
| 39 : | $PAR | ||
| 40 : | 1. Hyperboloid of two sheets | ||
| 41 : | $BR | ||
| 42 : | 2. Hyperboloid of one sheet | ||
| 43 : | $BR | ||
| 44 : | 3. Ellipsoid | ||
| 45 : | $BR | ||
| 46 : | 4. None of these | ||
| 47 : | $BR | ||
| 48 : | \{ans_rule(1)\} | ||
| 49 : | $PAR | ||
| 50 : | Describe the trace obtained by intersecting with the plane \(z=$z\): | ||
| 51 : | $PAR | ||
| 52 : | 1. Ellipse | ||
| 53 : | $BR | ||
| 54 : | 2. Hyperbola | ||
| 55 : | $BR | ||
| 56 : | 3. Circle | ||
| 57 : | $BR | ||
| 58 : | 4. Empty set | ||
| 59 : | $BR | ||
| 60 : | \{ans_rule(1)\} | ||
| 61 : | $BR | ||
| 62 : | |||
| 63 : | END_TEXT | ||
| 64 : | |||
| 65 : | Context()->normalStrings; | ||
| 66 : | |||
| 67 : | ANS(Real(3)->cmp); | ||
| 68 : | ANS(Real($ans)->cmp); | ||
| 69 : | |||
| 70 : | Context()->texStrings; | ||
| 71 : | SOLUTION(EV3(<<'END_SOLUTION')); | ||
| 72 : | $PAR | ||
| 73 : | $SOL | ||
| 74 : | $BR | ||
| 75 : | The quadratic surface is an ellipsoid, since its equation has the form \( \left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + \left( \frac{z}{c} \right)^2=1\), for \(a=$a, b=$b, c=1\). | ||
| 76 : | $PAR | ||
| 77 : | To find the trace obtained by intersecting the ellipsoid with the plane \(z=$z\), we set \(z=$z\) in the equation of the ellipsoid. This gives $PAR | ||
| 78 : | \( \left( \frac{x}{$a} \right)^2 + \left( \frac{y}{$b} \right)^2 + $z^2=1\) | ||
| 79 : | $BR | ||
| 80 : | \( \left( \frac{x}{$a} \right)^2 + \left( \frac{y}{$b} \right)^2=$d\) | ||
| 81 : | $BR | ||
| 82 : | We conclude that the trace is an $ans2. | ||
| 83 : | $BR | ||
| 84 : | END_SOLUTION | ||
| 85 : | |||
| 86 : | ENDDOCUMENT(); |
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