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    1 ## DBsubject('Calculus')
    2 ## DBchapter('Parametric Equations, Polar Coordinates, and Conic Sections')
    3 ## DBsection('Arc Length and Speed')
    4 ## KEYWORDS('calculus', 'parametric', 'polar', 'conic')
    5 ## TitleText1('Calculus: Early Transcendentals')
    6 ## EditionText1('2')
    7 ## AuthorText1('Rogawski')
    8 ## Section1('11.2')
    9 ## Problem1('3')
   10 ## Author('Christopher Sira')
   11 ## Institution('W.H.Freeman')
   12 
   13 DOCUMENT();
   14 loadMacros("PG.pl","PGbasicmacros.pl","PGanswermacros.pl");
   15 loadMacros("PGchoicemacros.pl");
   16 loadMacros("Parser.pl");
   17 loadMacros("freemanMacros.pl");
   18 loadMacros("PGchoicemacros.pl");
   19 $context = Context();
   20 
   21 
   22 $a = random(2, 9);
   23 $b = random(2, 9);
   24 $a2 = 2 * $a;
   25 $b2 = 2 * $b;
   26 $a2sq = $a2 ** 2;
   27 $b2sq = $b2 ** 2;
   28 $coeff = Formula("$a2sq + $b2sq");
   29 
   30 #$ans = Formula("$coeff**.5 * 8");
   31 $ans = Compute("8*sqrt($a2 ** 2 + $b2 ** 2)");
   32 
   33 Context()->texStrings;
   34 BEGIN_TEXT
   35 \{ beginproblem() \}
   36 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","3") \}
   37 Use equation 4 to calculate the length of the path over the given interval.
   38 \[ ($a t^2, $b t^2 - 1), \, 0 \le t \le 4 \]
   39 $PAR
   40 \{ ans_rule() \}
   41 $PAR
   42 END_TEXT
   43 Context()->normalStrings;
   44 
   45 ANS($ans->cmp);
   46 
   47 Context()->texStrings;
   48 SOLUTION(EV3(<<'END_SOLUTION'));
   49 $PAR
   50 $SOL
   51 Since \( x = $a t^2 \) and \( y = $b t^2 - 1 \) we have \( x' = $a2 t \) and \( y' = $b2 t \).  By the formula for the arc length we get
   52 \[ S = \int ^4 _0 \sqrt{x'(t)^2 + y'(t)^2} \, dt = \int ^4 _0 \sqrt{$a2sq t^2 + $b2sq t^2} \, dt = \sqrt{$coeff} \int ^4 _0 t \, dt = 8\sqrt{$coeff} = $ans \]
   53 END_SOLUTION
   54 
   55 ENDDOCUMENT();