## DBsubject('Calculus') ## DBchapter('Vector Geometry') ## DBsection('A Survey of Quadratic Surfaces') ## KEYWORDS('calculus') ## TitleText1('Calculus: Early Transcendentals') ## EditionText1('2') ## AuthorText1('Rogawski') ## Section1('12.6') ## Problem1('15') ## Author('JustAsk - Vladimir Finkelshtein') ## Institution('W.H.Freeman') DOCUMENT(); loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl"); loadMacros("Parser.pl"); loadMacros("freemanMacros.pl"); loadMacros("PGauxiliaryFunctions.pl"); loadMacros("PGgraphmacros.pl"); loadMacros("PGchoicemacros.pl"); TEXT(beginproblem()); $a=random(2,8,1);$b=random(2,8,1); $z=random(2,4,2);$ans=list_random(1,4); if ($ans==1) {$z=1/$z};$d=1-$z**2; if ($d<0) {$ans2='empty set'}; if ($d>0) {$ans2='ellipse'}; Context()->texStrings; BEGIN_TEXT \{ textbook_ref_exact("Rogawski ET 2e", "12.6","15") \}$PAR State the type of the quadratic surface: $BR $$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + z^2=1$$$PAR 1. Hyperboloid of two sheets $BR 2. Hyperboloid of one sheet$BR 3. Ellipsoid $BR 4. None of these$BR \{ans_rule(1)\} $PAR Describe the trace obtained by intersecting with the plane $$z=z$$:$PAR 1. Ellipse $BR 2. Hyperbola$BR 3. Circle $BR 4. Empty set$BR \{ans_rule(1)\} $BR END_TEXT Context()->normalStrings; ANS(Real(3)->cmp); ANS(Real($ans)->cmp); Context()->texStrings; SOLUTION(EV3(<<'END_SOLUTION')); $PAR$SOL $BR 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 $$a=a, b=b, c=1$$.$PAR 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 $$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2 + z^2=1$$$BR $$\left( \frac{x}{a} \right)^2 + \left( \frac{y}{b} \right)^2=d$$ $BR We conclude that the trace is an$ans2. \$BR END_SOLUTION ENDDOCUMENT();