View of /trunk/NationalProblemLibrary/WHFreeman/Rogawski_Calculus_Early_Transcendentals_Second_Edition/11_Parametric_Equations_Polar_Coordinates_and_Conic_Sections/11.2_Arc_Length_and_Speed/11.2.1.pg

    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('1')
   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 $coeff = Formula("($a**2 + (-1 * $b)**2) ** .5");
   25 
   26 #$ans = Formula("$coeff * 2");
   27 $ans = Compute("2*sqrt($a**2 + $b**2)");
   28 
   29 
   30 Context()->texStrings;
   31 BEGIN_TEXT
   32 \{ beginproblem() \}
   33 \{ textbook_ref_exact("Rogawski ET 2e", "11.2","1") \}
   34 $PAR
   35 Use equation 4 to calculate the length of the path over the given interval.
   36 \[ c(t) = ($a t + 1, 9 - $b t), \, 0 \le t \le 2 \]
   37 $PAR
   38 \{ ans_rule() \}
   39 $PAR
   40 END_TEXT
   41 Context()->normalStrings;
   42 
   43 ANS($ans->cmp);
   44 
   45 Context()->texStrings;
   46 SOLUTION(EV3(<<'END_SOLUTION'));
   47 $PAR
   48 $SOL
   49 Since \( x = $a t + 1 \) and \( y = 9 - $b t \) we have \( x' = $a \) and \( y' = - $b \).  Hence, the length of the path is
   50 \[ S = \int ^2 _0 \sqrt{$a^2 + (- $b)^2} \, dt = $coeff \int ^2 _0 \, dt = 2\sqrt{\{$a**2 + $b**2\}} = $ans \]
   51 END_SOLUTION
   52 
   53 ENDDOCUMENT();