[npl] / trunk / NationalProblemLibrary / WHFreeman / Rogawski_Calculus_Early_Transcendentals_Second_Edition / 12_Vector_Geometry / 12.6_A_Survey_of_Quadric_Surfaces / 12.6.15.pg Repository: Repository Listing bbplugincoursesdistsnplrochestersystemwww

 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();